PProper Orbifold Cohomology
Hisham Sati, Urs SchreiberSeptember 29, 2020
Abstract
The concept of orbifolds should unify differential geometry with equivariant homotopy theory, so that orb-ifold cohomology should unify differential cohomology with proper equivariant cohomology theory. Despitethe prominent role that orbifolds have come to play in mathematics and mathematical physics, especially instring theory, the formulation of a general theory of orbifolds reflecting this unification has remained an openproblem. Here we present a natural theory argued to achieve this. We give both a general abstract axiomati-zation in higher topos theory, as well as concrete models for ordinary as well as for super-geometric and forhigher-geometric orbifolds. Our first main result is a fully faithful embedding of the 2-category of orbifoldsinto a singular-cohesive ∞ -topos whose intrinsic cohomology theory is proper globally equivariant differentialgeneralized cohomology, subsuming traditional orbifold cohomology, Chen-Ruan cohomology, and orbifoldK-theory. Our second main result is a general construction of orbifold ´etale cohomology which we show tonaturally unify (i) tangentially twisted cohomology of smooth but curved spaces with (ii) RO-graded properequivariant cohomology of flat but singular spaces. As a fundamental example we present J-twisted orbifoldCohomotopy theories with coefficients in shapes of generalized Tate spheres. According to “Hypothesis H” thisincludes the proper orbifold cohomology theory that controls non-perturbative string theory.
Contents
Appendix A Model category presentations 104Appendix B Equivariant homotopy theory 106 a r X i v : . [ m a t h . A T ] S e p Introduction
The concept of orbifolds [Sa56][Sa57][Th80][Hae84] – manifolds with singularities modeled on fixed points offinite group actions (review in [MM03][Ka08, §6][BG08, §4][IKZ10]) – has become commonplace in mathematics(e.g. [BLP05][Rat06, §13][JY11]), and plays a central role in theoretical physics (see [AMR02]), notably so instring theory ([DHVW85][DHVW86][BL99][SS19]). However, the definition of the homotopy theory, i.e. of the ∞ -category ([Lu09a], see §2.1) of orbifolds, hence in particular of orbifold cohomology , is subtle, as witnessed bythe convoluted history of the concept; see [Le08, Intro.][IKZ10, §1]. In fact, the issue has remained open: Orbifolds as ´etale stacks?
A proposal popular among Lie theorists [MP97] (see [Mo02][Le08][Am12]) is toregard an orbifold with local charts G i (cid:121) U i (347) as• the ´etale groupoid; specifically: Lie groupoid (see [MM03][TX06]) or topological groupoid (see [CPRST14]);• equivalently, the ´etale geometric stack; specifically: differentiable or topological stack ([Ca11][Ca19][Gi13])obtained by gluing the corresponding homotopy quotient stacks U i (cid:12) G i (17).This proposal is directly modeled (explictly so in [Jo12, §8]) on the concept of Deligne-Mumford stacks in al-gebraic geometry ([DM69], review in [Kr09]) and extends to a concept of general ´etale ∞ -stacks [Ca20][Ca16].It relies on the fact that ´etale stacks, in their role as homotopy-theoretic generalizations of sheaves, fully cap-ture geometric aspects (via generalized sheaf cohomology [Br73], see [NSS12a]), while in their role as geometricrefinements of classifying spaces they support Borel equivariant cohomology (see [Tu11]). However, Borel coho-mology is coarser than the proper equivariant cohomology that is generally relevant in theory and in applications:
Proper equivariant cohomology , formulated in equivariant homotopy theory (review in [Blu17][May96]), isobtained by refining the purely homotopy-theoretic nature of Borel cohomology by the geometric (‘cohesive”, see§1.2) nature of fixed loci (see Example 3.71) of topological group actions – hence by the characteristic nature oforbifold geometry – as encoded in the category of orbits of the equivariance group (recalled in §B). The properequivariant version of ordinary cohomology is known as Bredon cohomology [Br67a][Br67b] (review in [Blu17,§1.4][tD79, §7]); beyond that, there is a wealth of proper equivariant generalized cohomology theories (Def.B.6 below) such as equivariant K-theory [Se68][AS69] (which is proper equivariant by [AS04, A3.2][FHT07,A.5][DL98]) and equivariant Cohomotopy theory [Se71][tD79, §8][SS19][BSS19].However, if orbifolds are modeled just by ´etale stacks, then their proper equivariant cohomology remains, byand large, invisible. This is true even for Chen-Ruan orbifold cohomology:
Traditional orbifold cohomology and its shortcomings.
Given an orbifold X , we write (see §1.2) ⊂ X for the´etale stack underlying it, and S ⊂ X for its geometric realization or classifying space (often denoted B X ). In thecase that X is the global quotient orbifold of a G -space X , this is the homotopy type of the Borel construction ; sothat we may generally call S ⊂ X the Borel space of the orbifold. Now, traditional orbifold cohomology is [ALR07,p. 38] just the ordinary cohomology (e.g. singular cohomology) of this Borel space, hence is
Borel cohomology : traditionalorbifold cohomology H • trad (cid:0) X , A (cid:1) : = Borel cohomology H • sing singularcohomology (cid:0) S ⊂ X Borelspace , A (cid:1) . (1)This can be considered with any kind of coefficients A , notably in the generality of local coefficient systems[MP99], but it always remains an invariant of just the Borel space. Moreover, for a coefficient ring that invertsthe order of the isotropy groups of X , hence in particular for rational, real and complex number coefficients A ∈ { Q , R , C } , the purely torsion cohomology of the orbifold’s finite isotropy groups becomes invisible, andtraditional orbifold cohomology reduces further (e.g. [ALR07, Prop. 2.12]) to an invariant of just the shape S < X of the singular quotient space < X (the “coarse moduli space”) underlying the orbifold (often denoted | X | ): We follow [DHLPS19] with this terminology, see Remark 4.60 below. The “esh”-symbol “ S ” stands for shape [Sc13, 3.4.5][Sh15, 9.7], following [Bo75], which for well-behaved topological spaces isanother term for their homotopy type [Lu09a, 7.1.6][Wa17, 4.6]; see Example 3.18. ationalorbifold cohomology H • trad (cid:0) X , Q (cid:1) (cid:39) ordinary cohomology H • sing singularcohomology (cid:0) S < X naive/coarsequotientspace , Q (cid:1) . (2)It is in this form that orbifold cohomology was originally introduced in [Sa56, Thm. 1] (following [Ba54], reviewedin [ALR07, 2.1]).Of course it did not go unnoticed that this coarse notion of orbifold cohomology is insensitive to the actualnature of orbifolds. In reaction to this (and motivated by algebraic constructions [DHVW85][DHVW86] on 2dconformal field theories interpreted as describing strings propagating on orbifold spacetimes), Chen and Ruanfamously introduced a new orbifold cohomology theory in [CR04]. But in fact (see [Cl14, p. 4,7] for review)Chen-Ruan cohomology of an orbifold is just Satake’s coarse cohomology (2) (typically considered with complexcoefficients), but applied to the corresponding “inertia orbifold” Maps (cid:0) S S , X (cid:1) : Chen-Ruanorbifold cohomology H • CR (cid:0) X (cid:1) (cid:39) traditional orbifold cohomology H • trad (cid:0) Maps ( S S , X ) inertia orbifold , C (cid:1) . (3)Still, it turns out that, for global G -quotient orbifolds X = ≺ ( X (cid:12) G ) , Chen-Ruan cohomology is equivalent to aproper equivariant cohomology theory, namely to Bredon cohomology with coefficient system given specificallyby: A CR : G / H ClassFunctions ( H , C ) . (4)This was observed in [Mo02, p. 18], using [Ho90, Thm. 5.5] with [Ho90, Prop. 6.5 b)]: Chen-Ruanorbifold cohomology H • CR (cid:0) ≺ ( X (cid:12) G ) global quotientorbifold , C (cid:1) (cid:39) Bredon cohomology H • G (cid:0) X , A CR specific systemof coefficients (4) (cid:1) . (5)Thus the success of Chen-Ruan cohomology (surveyed in [ALR07, §4,5]) highlights the relevance of proper equiv-ariance in orbifold cohomology. At the same time, this means that to detect the full proper equivariant homotopytype of orbifolds, one needs an orbifold cohomology theory that induces Bredon coefficient systems more generalthan (4); and, in fact, one that subsumes also generalized equivariant cohomology theories such as equivariantK-theory. In [AR01] the authors define orbifold K-theory to be the equivariant K-theory of any global quotientpresentation (see also [ARZ06][BU09][HW11]): traditionalorbifold K-theory K • (cid:0) ≺ ( X (cid:12) G ) global quotientorbifold (cid:1) : = equivariantK-theory K • G ( X ) . (6)This works for the case of K-theory, because it has been proven explicitly [PS10, Prop. 4.1] that the right handside of (6) is independent of the choice of global quotient presentation. However, in general, this approach ofcircumventing an intrinsic definition of orbifold cohomology by just defining it to be equivariant cohomology ofglobal quotient presentations is, besides being somewhat unsatisfactory, in need of justification: Orbifolds in global equivariant homotopy theory?
That orbifold cohomology should also capture proper equiv-ariant cohomology was suggested explicitly in [PS10]. However, the fundamental issue remained that a quotientpresentation X (cid:39) ≺ ( X (cid:12) G ) of an orbifold is not intrinsic to the orbifold, similarly to a choice of coordinate at-las, while in equivariant cohomology theory the equivariance group G is traditionally taken to be fixed. But thissuggests [Schw17, Intro.][Schw18, p. ix-x] (details in [Ju20]) that the right context for orbifold cohomology is“global” equivariant homotopy theory [Schw18] (following [HG07] and originally motivated from patterns seen ingenuine equivariant stable homotopy theory [Se71][LMS86]) where the equivariance group G is allowed to vary ina prescribed class of groups. On the other hand, plain global homotopy theory retains no geometric information! The open problem is thus to set up a mathematical theory of proper orbifold cohomology which unifies: (i) the higher geometric (differential, ´etale) aspects of orbifolds captured by geometric ∞ -stack theory; and (ii) the singular (equivariant) aspects of orbifolds captured by proper and global equivariant homotopy theory.3o achieve this, we look to higher topos theory [TV05][Lu09a][Re10] (more pointers in §2.1 below) as anambient foundational homotopy theory of higher geometric spaces [Sc13][Sc19]: ∞ -Toposes as collections of generalized higher geometric spaces. Viewed from the outside (i.e., “externally”),an ∞ -topos is a collection of geometric spaces of a given flavor, which may be: (a) generalized geometric, as well as (b) higher geometric. (a) Here “ generalized geometry ” refers to what Grothedieckcalled functorial geometry [Gr65] (review in [DG80]), whichhe urged in [Gr73] should supercede any point-set (e.g. lo-cally ringed) definition of geometric spaces (further ampli-fied by Lawvere, e.g. [La86][La91]). In hindsight, the ba-sic idea here is just that of how physicists describe emergentspacetimes X in terms of what (classical) p -brane sigma-models on worldvolumes Σ detect when probing X (see[FSS13a][FSS13b][JSSW19]): Charts = CartesianSpaces (Def. 2.5)
JetsOfCartesianSpaces (Def. 3.22)
SuperCartesianSpaces (Def. 3.41)
Singularities (Def. 3.46)
SingularCharts (Lem. 3.60) . . . (7)Given any category of local model spaces (often: “affine spaces”, here: “charts”; see Def. 3.9 below), such asthose shown in (7), one may encode a would-be generalized (“target”-)space X by assigning to each Σ ∈ Chartsthe collection probe space Σ (cid:31) (cid:47) (cid:47) collection of probes of generalized space X by Σ X ( Σ ) : = (cid:8) “ Σ ! X ” (cid:9) (8)of geometric (e.g. smooth, super-geometric, etc.) maps into X ; where the quotation marks indicate that, at thispoint of bootstrapping X into existence, the category in which these maps are actual morphisms is yet to bespecified. To that end, one observes that a minimal set of consistency conditions on such an abstract assignment(8) to be anything like collections of maps into a space X are: (1) Functoriality of probes. For every morphism φ of Charts there is an opera-tion of “pre-composition of probe maps by φ ”: map ofprobe spaces Σ φ (cid:15) (cid:15) pre-composition operationon collections of probes X ( Σ ) Σ X ( Σ ) X ( φ ) = “ ( − ) ◦ φ ” (cid:79) (cid:79) such that X ( φ ) ◦ X ( φ ) (cid:39) X ( φ ◦ φ ) (9) (2) Gluing of probes. If { U i (cid:47) (cid:47) Σ } i ∈ I is a cover of Σ ∈ Charts, then probes of X by Σ should be equivalent to those tuples of probes bythe U i which are coherently identified on intersections: X ( Σ ) (cid:39) tuples of probes U i −! X identified on intersections U i ∩ U j compatibly on U i ∩ U j ∩ U k etc. (10)In the jargon of topos theory (see [MLM92][Joh02]), condition (9) says that the collection X ( − ) of probes of X is a pre-sheaf on Charts, while condition (10) says that this is in fact a sheaf . Hence the category of generalizedgeometric spaces probeable by Charts is the category of sheaves (the Grothendieck topos ) on Charts: topos of generalizedgeometric spaces
GeneralizedSpaces : = Sheaves (cid:0)
Charts (cid:1) category of sheaveson site of charts (11)Now, every Σ ∈ Charts is itself canonically regarded as a generalized space y ( Σ ) ∈ GeneralizedSpaces, by takingits probes to be those given by morphisms of Charts (this is the
Yoneda embedding , recalled as Prop. 2.37 below): chart regarded asgeneralized space y ( Σ ) : Σ (cid:48) (cid:31) (cid:47) (cid:47) (cid:8) Σ (cid:48) ! Σ (cid:9) = : Charts ( Σ (cid:48) , Σ ) collection of its Σ (cid:48) -shaped probes (12)Hence we have completed the bootstrap construction of generalized spaces X in (8) if we may remove the quota-tion marks there, hence if for X ∈ GeneralizedSpaces there is a natural equivalence collection of Σ -shapedprobes of X X ( Σ ) (cid:39) (cid:8) y ( Σ ) ! X (cid:9) : = GeneralizedSpaces (cid:0) y ( Σ ) , X (cid:1) collection of mapsfrom y ( Σ ) to X . (13)That this is indeed the case is the statement of the Yoneda lemma (recalled as Prop. 2.38 below), which thus impliesconsistency and existence of generalized geometry. Shown here for sub-canonical Grothendieck topologies on Charts, which is the case in all examples of interest here. b) On the other hand, “ higher geometry ” (see [FSS13a][FSS19a][JSSW19] for exposition and applications)refers to the refinement of the above theory of generalized geometric spaces, where the collection of probes (8) of ageneralized space is not necessarily just a set, but may be a set equipped with equivalences (gauge transformations)between its elements, and with higher order equivalences (higher gauge transformations) between these, etc. –called an ∞ -groupoid (e.g., modeled as a Kan simplicial set, see [GJ99, I.3]). For example, for X ∈ Sets and G a discrete group acting on S , the corresponding action groupoid (Example 2.15 below) consists of the elements x ∈ X , but equipped with an equivalence between x and x for every group element whose action takes x to x : homotopyquotient X (cid:12) G (cid:39) y g i (cid:4) (cid:4) g · x g · g (cid:15) (cid:15) g (cid:32) (cid:32) x g (cid:66) (cid:66) g · g · g (cid:36) (cid:36) g · g (cid:47) (cid:47) g · g · x g (cid:120) (cid:120) g · g · g · x z g (cid:28) (cid:28) (cid:94) (cid:94) g − g · z g (cid:33) (cid:33) (cid:100) (cid:100) g − g · g · z g (cid:37) (cid:37) (cid:103) (cid:103) g − g · g · g · z · · · plainquotient X / G (cid:39) (cid:110) [ y ] [ x ] [ z ] · · · (cid:111) τ (14)This is a model for the homotopy quotient of X by G , which resolves the plain quotient X / G (the set of equivalenceclasses) by remembering not only that but how two elements are equivalent. More precisely, the action groupoidremembers the graph and syzygies of the G -action, encoded in its Kan simplicial nerve (Example 2.69 below): X (cid:12) G (cid:39) X × G × G ( x , g , g ) ( g · x , g ) (cid:47) (cid:47) (cid:111) (cid:111) ( x , g , g ) ( x , g · g ) (cid:47) (cid:47) (cid:111) (cid:111) ( x , g , g ) ( x , g ) (cid:47) (cid:47) set ofmorphisms X × G ( x , g ) g · x (cid:47) (cid:47) (cid:111) (cid:111) ( x , e ) (cid:91) x ( x , g ) x (cid:47) (cid:47) set ofobjects X (15)In particular, if an element y ∈ X is fixed by the group action, thenin the homotopy quotient it appears as the one-object groupoid alsoknown as K ( G , ) or (since G is assumed to be discrete here) as BG . K ( G , ) (cid:39) BG (cid:39) (cid:110) ∗ g (cid:6) (cid:6) | g ∈ G (cid:111) (16)More generally, if X ∈ Charts in the list (7) is equipped with the action of a discrete group G , then we obtaina higher generalized space X : = X (cid:12) G whose ∞ -groupoid of Σ -shaped probes (8) is the action groupoid of theinduced action on the set of Σ -shaped probes of X (the following formula is for contractible charts, Lemma 3.12): global quotientorbifold X (cid:12) G : Σ (cid:0) X (cid:12) G (cid:1) ( Σ ) : = X ( Σ ) (cid:12) G = Charts ( Σ , X ) (cid:12) G . groupoid of its Σ -shaped probes (17)Such a higher generalized space with collections of probes (8) being groupoids, and satisfying the appropriategluing condition (10), may be called a 2 -sheaf or sheaf of groupoids [Br93] on Charts, in generalization of (11),but is commonly known as a stack [DM69][Gi72][Ja01][Ho08], following champ [Gi66]. Generally, a highergeneralized space with ∞ -groupoids of probes is thus an ∞ -sheaf or ∞ -stack on Charts, in generalization of (11): ∞ -topos H : = context for higher generalized geometry HigherGeneralizedSpaces : = ∞ -category of ∞ -stacks Sheaves ∞ (cid:0) Charts ∞ -site of probe spaces (cid:1) . (18)The theory of ∞ -stacks originates with [Br73], developed in [Ja87][Ja96] (survey in [Ja15]) and brought into themore abstract form of ∞ -topos theory in [TV05][Lu09a][Re10]. In fact, finitary constructions internal to ∞ -toposesbehave so well that they may naturally be formulated [Sh19] in a kind of programming language now known as homotopy type theory [UFP13]. While we will not dwell on this here, we do focus on elegant internal constructions.For some of these, a homotopy type-theoretic formulation has already been explored in the literature: Theory internalto an ∞ -topos Internal formulation intraditional mathematics Partial formulation inhomotopy type theory Galois theory §2.2 [NSS12a] [BvDR18]modalities & cohesion §3.1 [SSS12][Sc13] [RSS17][Sh15]´etale ∞ -stacks §4.2 [KS17] [We18][CRi20]cohomology §5 [SSS12][NSS12a][Sc13] [Cav15][BH18] ifferential topology in an ∞ -topos. As a consequence of the above, every ∞ -topos H behaves like a homotopytheory of generalized geometric spaces. In order to narrow back in, among these generalized spaces, on thosewith some minimum properties, we may, following [La91][La94][La07], axiomatize qualities of geometric objects(such as being discrete , smooth , ´etale , reduced , bosonic , singular , etc.) via the systems of (co-)reflective sub- ∞ -categories H (cid:35) , H (cid:50) , · · · ⊂ H , that the objects with these properties (should) form inside H [SSS12][Sc13]: ambient ∞ -topos ofgeneralized geometric spaces H i ! ⊥ (cid:47) (cid:47) (cid:111) (cid:111) i ∗ (cid:63) (cid:95) i ∗ ⊥ (cid:47) (cid:47) H (cid:50) or H (cid:111) (cid:111) i ∗ ⊥ (cid:63) (cid:95) i ∗ (cid:47) (cid:47) (cid:111) (cid:111) i ! ⊥ (cid:63) (cid:95) H (cid:35) sub- ∞ -category ofobjects of pure (cid:35) -nature (19)This induces systems of adjoint (co-)projection operators (cid:35) (cid:97) (cid:50) : H ! H , the associated idempotent (co-)monads : (cid:35) : = i ∗ ◦ i ! , (cid:50) : = i ∗ ◦ i ∗ or (cid:35) : = i ∗ ◦ i ∗ , (cid:50) : = i ! ◦ i ∗ , (20)to which we refer as modal operators or just modalities [Sc13][RSS17][Co20]. These are idempotent (Prop. 2.29) (cid:35)(cid:35) X (cid:39) (cid:35) X , (cid:50)(cid:50) X (cid:39) (cid:50) X , (21)which means that they act like projecting out certain qualitative aspects of generalized spaces, while them beingadjoint means that they project out an opposite pair of such qualities. Therefore, their (co-)unit transformations η (cid:50) (48) and ε (cid:35) (49) exhibit every X ∈ H as carrying a quality intermediate to these two opposite extreme aspects[LR03, p. 245]: (cid:35) X pure (cid:35) -aspect ε (cid:35) X (cid:47) (cid:47) X generalizedgeometric space η (cid:50) X (cid:47) (cid:47) (cid:50) X pure (cid:50) -aspect . For example, any adjoint modality (cid:91) (cid:97) (cid:93) (see Def. 3.1 below) that contains the initial modality ∅ (cid:97) ∗ (whichglobally projects to the initial and the terminal object, respectively) acts like projecting out discrete and purelycontinuous (co-discrete, chaotic) aspects of a space. Consequently, the existence of such a modality on H exhibitseach space X ∈ H as carrying quality intermediate to these extremes, hence, in this example, as equipped with akind of topology (see [Sh15, §3], following [La94]).We observe here that extending this basic example to a larger system of adjoint modalities allows to abstractlyencode the presence of differential geometric structure (Def. 3.21 below) and of super-geometric structure (Def.3.40 below) in ∞ -toposes, and hence on higher generalized spaces. Generalized cohomology in an ∞ -topos. Following [SSS12][NSS12a][Sc13], we may regard the concept of ∞ -toposes H as the ultimate notion of generalized cohomology theory , subsuming and combining all of: Sheaf hypercohomology in non-discrete ∞ -toposes [Br73] Non-abelian cohomology in general ∞ -toposes [SSS12][NSS12a, 3] Twisted non-abelian cohomology in slice ∞ -toposes Prp. 2.46, Rem. 2.94 Twisted abelian cohomology in tangent ∞ -toposes Exl. 2.51, Rem. 2.96 Differential cohomology in cohesive ∞ -toposes Def. 3.1, Rem. 3.20 ´Etale cohomology in elastic ∞ -toposes Def. 3.21, Def. 5.11 Superspace cohomology in solid ∞ -toposes Def. 3.40, Rem. 3.44 Proper equivariant cohomology in singular ∞ -toposes Def. 3.48, Rem. 5.4, Thm. 5.9In all these cases, for X , A ∈ H any two objects, with X regarded as a domain “space” and A as the “coefficients”,the A-cohomology of X is embodied by the homomorphisms from X to A : (i) a morphism X c (cid:47) (cid:47) A is a cocycle ; (ii) a homotopy X c (cid:36) (cid:36) c (cid:58) (cid:58) A (cid:11) (cid:19) is a coboundary ; (iii) the homotopy groups of the cocycle space H − n ( X , A ) : = π n H ( X , A ) (cid:39) π H ( X , Ω n A ) (22)are the cohomology sets of X with coefficients in A . (Here Ω n ( − ) is the n -fold based loop space.)This is the intrinsic cohomology theory of the ∞ -topos H – we discuss various examples below in §5.6 .2 Results We survey the results presented below:
Axiomatic orbifold geometry in modalhomotopy theory.
Building on the above, therefore, to formu-late proper orbifold cohomology we ask for ∞ -toposes (18) equipped with a system of ad-joint modalities (20) that capture both aspectsof proper orbifold cohomology: (i) the geometric (differential, ´etale) aspectand (ii) the singular (proper equivariant) aspect. Modalities for Singular Super-Geometry (§3) τ n n -groupoidal S shaped (cid:91) discrete (cid:93) continuous ℜ reduced ℑ ´etale L locally constant ⇒ even (cid:32) bosonic Rh rheonomic < singular ⊂ smooth ≺ orbi-singular
1. The geometric aspect of orbifoldtheory.
In order to formulate, internal tosuitable ∞ -toposes, the (a) differential topol-ogy, (b) differential geometry, and (c) super-geometry of orbifolds (hence of manifolds,super-manifolds, super-orbifolds, etc.) intheir smooth guise as ´etale ∞ -stacks (18), weconsider a corresponding progression of ad-joint modalities (20), which starts out in theform of the “axiomatic cohesion” of [La07],on to a second layer that contains a “de Rhamshape” operation ℑ as considered in [Si96][ST97], and then to a third layer which cap-tures super-geometry in a new axiomatic way. id (cid:97)∨ id ∨ ⇒ (cid:97) (cid:32) (cid:97)∨ Rh ∨ for super-geometry insolid ∞ -toposes (Def. 3.40) ℜ (cid:97) ℑ (cid:97)∨ L ∨ for differential geometry inelastic ∞ -toposes (Def. 3.21) S (cid:97) (cid:91) (cid:97)∨ (cid:93) ∨ for differential topology incohesive ∞ -toposes (Def. 3.1) ∅ (cid:97) ∗
2. The singular aspect of orbifold theory.
Envision the picture of an orbifold singularity ≺ and a mathematical magnifying glass heldover the singular point. Under this magnifi-cation, one sees resolved the singular point asa fuzzy fattened point , to be denoted ≺ G . Re-moving the magnifying glass, what one seeswith the bare eye depends on how one squints: (i) The physicists (see, e.g., [BL99, §1.3])and the classical geometers (see, e.g.,[IKZ10][Wat15]) say that they see anactual singular point, such as the tip of acone < . This is the plain quotient < G : = ∗ / G = ∗ , a point. (ii) The higher geometers (see, e.g., [MP97][CPRST14]) say that they see thesmooth G -action around that point,hence a smooth stacky geometry ⊂ .This is the homotopy quotient ⊂ G : = ∗ (cid:12) G = BG = K ( G , ) (16). Singular quotient Smooth homotopy quotientorbi-singularity ≺ G (cid:10) p r o j ec t o n t o pu re l y s m oo t h a s p ec t ⊂ (cid:37) (cid:37) (cid:52) p r o j e c t o n t o p u r e l y s i n g u l a r a s p e c t < (cid:121) (cid:121) opposite extremeaspects of orbifold singularities < G = ∗ / G = ∗ singularquotient ⊂ G = ∗ (cid:12) G = BG smoothhomotopy quotient We observe in §3.2 that just this is captured by the cohesive structure on global equivariant homotopy theory thathad been observed in [Re14], but whose conceptual interpretation had remained open [Re14, Footnote 8].7 ifferential geometry of ´etale ∞ -stacks. We present, in §4.2, a general theory of higher differential geometryformulated internally to these elastic ∞ -toposes (§3.1). This deals with ´etale ∞ -stacks locally modeled on any group ∞ -stack V (“ V -folds”, Def. 4.14). For the special case V = ( R n , +) , this subsumes ordinary manifolds (Example4.17) and ordinary ´etale Lie groupoids (Example 4.18). For V a super-symmetry group (197), this produces atheory of super-orbifolds (Example 4.20), capturing, for instance, those that appear as target spaces in superstringtheory (e.g. [PR04][GIR08]) and M-theory [HSS18], or those that appear as moduli spaces of super-Riemannsurfaces [Ra87][LBR88][Wi12][CV17]. (cid:111) (cid:111) coarser finer (cid:47) (cid:47) ( X / G ) top − (cid:91) X / G − (cid:91) X (cid:12) G − (cid:91) ≺ (cid:0) X (cid:12) G (cid:1)(cid:8) underlyingtopological spaces (cid:9) (cid:111) (cid:111) Dtplg
Prop. 2.7 (cid:8) orbifolds asdiffeological spaces (cid:9) (cid:115) (cid:115)
Snglr
Prop. 3.50, Prop. 4.9 (cid:111) (cid:111) τ (cid:127) (cid:95) (cid:15) (cid:15) (cid:8) orbifolds as´etale stacks (cid:9) (cid:111) (cid:111) Smth
Prop. 3.50, Prop. 4.7 (cid:47) (cid:47) (cid:127) (cid:95)
Def. 4.14 (cid:15) (cid:15) (cid:8) orbifolds ascohesive orbi-spaces (cid:9) (cid:127) (cid:95)
Def. 4.58 (cid:15) (cid:15) (cid:8) sheaves on
CartesianSpaces (cid:9) (cid:31) (cid:127) (cid:47) (cid:47) (cid:8) ∞ -stacks on CartesianSpaces (cid:9) (cid:31) (cid:127)
OrbSnglr
Prop. 3.50, Def. 4.58 (cid:47) (cid:47) (cid:8) ∞ -stacks on SingularCartesianSpaces (cid:9)
SmoothGroupoids ∞ Example 3.18 (cid:31) (cid:127) (cid:47) (cid:47)
SingularSmoothGroupoids ∞ Example 3.56
Gerbes on ´etale ∞ -stacks. With orbifolds, in their incarnation as ´etale stacks, thus embedded into a fully-fledged ∞ -topos, the general theory of ∞ -bundles [NSS12a][NSS12b] (see §2.2 below) applies to provide the theory of fiberbundles on orbifolds (e.g. [LGTX04][Se06][BG08]) and of gerbes on orbifolds (e.g. [LU04][Ca10][BX][TT14])naturally generalized to higher, to non-abelian and to twisted gerbes on orbifolds. Differential cohomology of ´etale ∞ -stacks. Moreover, since the intrinsic cohomology theory of cohesive ∞ -toposes is differential cohomology (Remark 3.20), this realization of ´etale ∞ -stacks within cohesive ∞ -toposesimmediately provides their differential cohomology theory (see [SSS12][FSS13a][FSS15]). This includes, in par-ticular (as made explicit in [PR19]), the Borel-equivariant/orbifold Deligne cohomology considered in [KT18](which, for finite groups, coincides with [LU03][Gom05]), given, in low degrees, by: (i) gerbes with connection, hence including what in string theory is known as the discrete torsion classificationof the B-field on orbifolds [Va86][VW95][Sha00a][Sha00c][Sha02][Ru03]; or (ii) discrete torsion classificationof the C-field on orbifolds [Sha00b][Se01][dBDHKMMS02, §4.6.2]. Geometric enhancement of global equivariant homotopy theory.
We enhance all of the above to a theory ofproperly orbi-singular spaces, formulated internally to “singular-elastic” ∞ -toposes (§3.2), where a natural notionof orbi-singularization ≺ (Prop. 3.50) promotes (Def. 4.58) any such ∞ -category of ´etale ∞ -groupoids faithfullyto its proper orbifold version (Remark 4.60). This detects geometric fixed point spaces (Def. 3.69) in the sense ofproper equivariant homotopy theory. We show (Prop. 4.2, Lemma 4.7) that the cohesive shape (Def. 3.1) of theorbi-singularization of an ´etale groupoid is its incarnation as an orbispace in global equivariant homotopy theory,in the sense of [HG07][Re14][K¨o16][Schw17] (Remark 4.1). The proper 2-category of orbifolds.
One modelfor the axioms of singular-cohesive homotopy the-ory is the ∞ -topos of singular-smooth ∞ -groupoids (Examples 3.18, 3.56 below). In this model, theproper 2-category (Rem. 4.60) of orbifolds X , ei-ther Lie theoretically (Example 4.10) or topologi-cally (Example 4.11), is equivalent, via passage to (i) their purely smooth aspect ⊂ X , to the tradi-tional 2-category of ´etale stacks (Prop. 4.9), (ii) while their purely singular aspect < X givesthe underlying singular coarse quotient space(Prop. 4.6). properorbifold §4 ≺ ( X (cid:12) G ) (cid:12) p r o j ec t o n t o pu re l y s m oo t h a s p ec t ⊂ (cid:38) (cid:38) (cid:50) p r o j e c t o n t o p u r e l y s i n g u l a r a s p e c t < (cid:121) (cid:121) opposite extremeaspects of orbifolds X / G singularspace [IKZ10] X (cid:12) G smooth´etale stack [MP97][CPRST14] artan geometry of ´etale ∞ -stacks. In this internal formulation we find all fundamental phenomena of differentialgeometry naturally generalized to ´etale ∞ -stacks, hence in particular to orbifolds: §4.2 Cartan geometryfor ´etale ∞ -stacks Discussion for ordinary orbifolds, e.g. in (i)
Def. 4.26
Frame bundles [MM03, p. 42] (ii)
Def. 4.36
G-structures [Wo16][Zh06][BZ03]Def. 4.42,Def. 4.43 - locally - globally integrable (ii.a) Ex. 4.44
Geometric structures [Ap00, §1.8][Wo16]-
Riemannian structure [Bo92][HM04][Rat06][BZ07][He09a][He09b][Ak12][Kan13][BDP17][Lan18]-
Flat structure [BDP17][Ref06][IU12, §8][SS19]-
Complex structure [SW99][FS07]-
Symplectic structure [Ve00][Go01][DE05][HM12][CP14][Ch17][RC19]-
Lorentzian structure [HS91][Ne02][LMS02a][LMS02b][BR07][ZR12]-
Pseudo-Riemannian structure [Me09][Zh18][BZ19]-
Conformal structure [Ap98][Ap00]-
CR-structure [DM02]-
Hypercomplex structure [BGM98]- . . . (ii.b)
Ex. 4.44
Special holonomy [Jo00][CT05]-
K¨ahler structure [Fu83][Je97][Ab01][BBFMT16]-
Calabi-Yau structure [Ro91][Jo98][Jo99a][Jo99b][Jo00, §6.5.1][St10][RZ11][CDR16]-
Quaternionic K¨ahler [GL88][Jo00, §7.5.2]-
Hyper-K¨ahler struc. [BD00]- G -structure [Jo00, §11][Rei15]- Spin ( ) -structure [Jo00, §13][Ba07] (iii) Def. 4.41
Local isometries [BZ07] (iv)
Def. 4.45
Haefliger stacks [Hae71][Hae84][Ca19, §2.2, §3][Ca16]). (v)
Def. 4.48
Tangential structures [Wee18][Pa20] (v.a)
Ex. 4.52
Higher Spin-structures - Orientation [Dr94]-
Spin structure [Ve96][Ac01][BGR07][DLM02]-
Spin c structure [Du96, §14]- String structure [PW88][LU04][LU06]-
Fivebrane structure [BL12] (cf. [SSS09][SSS12]). . .
Orbifold ´etale cohomology.
Based on this, we give a natural general definition of ´etale cohomology of V -´etale ∞ -stacks (Def. 5.11) hence in particular of orbifold ´etale cohomology , which is sensitive to the above (integrable) G -structures, and hence to geometry/special holonomy on orbifolds. For example, in the case of complex structure,this orbifold ´etale cohomology subsumes traditional notions of complex-geometric orbifold cohomology such asorbifold Dolbeault cohomology [Ba54][Ba56][CR04][Fe03] and orbifold Bott-Chern cohomology [An12][Ma05].Abstractly, orbifold ´etale cohomology is the intrinsic cohomology (22) of integrably G -structured ´etale ∞ -stackswhen regarded in the slice ∞ -topos (Prop. 2.46) over the G -Haefliger stack (Def. 4.45) via the classifying mapof their G -structure (Prop. 4.47). As such, orbifold ´etale cohomology is the progenitor of tangentially twisted (proper) orbifold cohomology (Def. 5.13, Def. 5.15), to which we turn next.9 roper equivariant cohomology. While the proper 2-category of orbifolds is equivalent to the traditional oneof orbifolds as ´etale stacks, its full embedding into an ambient singular-cohesive ∞ -topos (§3.2) provides formore general coefficient objects, each of which is guaranteed to produce a proper orbifold Morita-class invariant(Remark 4.60). Our first main Theorem G -equivariant cohomology theories: Bredoncohomology with any coefficient system, as well as proper equivariant generalized cohomology theories. Traditional orbifold cohomology.
In particular, Prop. 4.2 and Theorem 5.9 imply, via [Ju20] (Remark 4.1),that proper orbifold cohomology in singular-cohesive homotopy theory subsumes Chen-Ruan orbifold cohomol-ogy, via (5), and Adem-Ruan orbifold K-theory, via (6). Hence it also subsumes Freed-Hopkins-Teleman orb-ifold K-theory [FHT07] (reviewed in [Nu13, §3.2.2]) and Jarvis-Kaufmann-Kimura’s “full orbifold K-theory”[JKK05][GHHK08] for orbifolds with global quotient presentations (by [FHT07, Prop. 3.5] and [JKK05, 3], re-spectively). Moreover, singular-cohesion provides a natural transformation ⊂ X ε ⊂ X (cid:47) (cid:47) ≺ X which restricts thisproper orbifold cohomology to the underlying ´etale stack, where it reduces to traditional Borel orbifold cohomol-ogy (1) and, in particular, to Satake cohomology (2) (see also, e.g., [ADG11][BNSS18]). Twisted orbifold cohomology.
All these cohomology theories generalize to their twisted cohomology versions(e.g., local coefficients for ordinary cohomology, as in [MP99], or twisted K-theory, as in [AR01]), by passage toslice ∞ -toposes of the ambient singular-cohesive ∞ -topos (Remark 2.94). In particular, slicing of orbifolds over ≺ Z via their orientation bundle promotes them (Example 5.10) to orientifolds [DFM11][FSS15, 4.4][SS19]. Proper orbifold ´etale cohomology.
Finally, wepromote (Def. 5.15) orbifold ´etale cohomology,in its guise as tangentially twisted cohomology,to a proper orbifold cohomology theory in theabove sense (Remark 4.60). Our second maintheorem proper orbifold´etale cohomology unifies: (i) ( ⊂ ) ´etale cohomology (Def. 5.11) ofsmooth V -folds (Def. 4.14). (ii) ( (cid:91) ) proper equivariant cohomology (Def.5.2) of flat orbifolds (Def. 4.53), i.e., oftheir flat frame bundles (Prop. 4.54). properorbifold ´etale cohomology Def. 5.15 H S ≺ τ (cid:0) X , A (cid:1) (cid:69) ( i ⊂ ) ∗ s m oo t h o r b i f o l d s (cid:2) (cid:2) (cid:6) ( i (cid:91) ) ∗ fl a t o r b i f o l d s (cid:35) (cid:35) H S τ (cid:0) X , A (cid:1) smooth´etale cohomology Def. 5.13 H (cid:91) G (cid:0) ( (cid:91) G ) Frames ( X ) , A (cid:1) properequivariant cohomology Def. 5.2
J-twisted orbifold Cohomotopy.
We constructa fundamental class of examples of such properorbifold ´etale cohomology theories, which wecall
J-twisted orbifold Cohomotopy theories (Def. 5.28). Their coefficients are
Tate spheres (Def. 5.19), in the sense of (unstable) motivichomotopy theory (Example 5.20), with twist-ing via an intrinsic
Tate J-homomorphism (Def.5.24). Specified to ordinary orbifolds (Example5.29), Theorem 5.16 shows that J-twisted orb-ifold Cohomotopy subsumes: (i) ( ⊂ ) J-twisted Cohomotopy theory ofsmooth but curved spaces, as introduced in[FSS19b][FSS19c]. (ii) ( (cid:91) ) RO-graded equivariant Cohomotopytheory of flat orbifolds, as discussed in[SS19][BSS19]. J-twistedorbifold Cohomotopy
Def. 5.28 π S ≺ τ (cid:0) X (cid:1) (cid:74) ( i ⊂ ) ∗ s m oo t h o r b i f o l d s (cid:4) (cid:4) (cid:0) ( i (cid:91) ) ∗ fl a t o r b i f o l d s (cid:32) (cid:32) π S τ (cid:0) X (cid:1) J-twistedCohomotopy theory [FSS19b][FSS19c] π (cid:91) G (cid:0) ( (cid:91) G ) Frames ( X ) (cid:1) RO-gradedequivariant Cohomotopy [SS19][BSS19]
We conclude with a Remark 5.30 on the impact of this unification.10 ther approaches and outlook.
We briefly comment on relations of our constructions to other approaches in theliterature, further discussion of which is beyond the scope of this article.
Proper ∞ -categories of general ´etale ∞ -stacks. Another general theory of ´etale ∞ -stacks has been presentedin [Ca20], generalizing an elegant characterization of ´etale 1-stacks due to [Ca19] by following the discussion ofderived Deligne-Mumford stacks conceived as structured ∞ -toposes in [Lu09b]. This approach proceeds externallyvia characterizing the sites (recalled below as Prop. 2.41) which present ∞ -toposes of ´etale ∞ -stacks; and is thuscomplementary to the internal perspective proceeding from inside an ambient ∞ -topos which we are presentinghere. We briefly indicate the relation between the two: ◦ The approach in [Ca20] is to pick an ∞ -site of ModelSpaces (denoted “ L ” there) which is equipped with asuitable notion of which of its 1-morphisms qualify as being ´etale maps (the external version of our notionDef. 3.26). The inclusion i of the wide subcategory on these ´etale morphisms induces, by left Kan extension,a pair of adjoint ∞ -functors ( i ! (cid:97) i ∗ ) between the corresponding ∞ -stack ∞ -toposes, and the ´etale ∞ -stacksare then characterized as those in the essential image of the left adjoint i ! . This is shown on the right of thefollowing diagram: Sheaves ∞ (cid:0) ModelSpaces × Singularities (cid:1)
Smth (cid:47) (cid:47) (cid:111) (cid:111)
OrbSinglr ⊥ (cid:63) (cid:95) Prop. 3.50
Sheaves ∞ (cid:0) ModelSpaces (cid:1) (cid:111) (cid:111) i ! i ∗ (cid:62) (cid:47) (cid:47) Sheaves ∞ (cid:0) LocalModelSpaces ´et (cid:1) (cid:116) (cid:116) (cid:116) (cid:116)
OrbSinglr (cid:0) ´EtaleStacks ∞ (cid:1) proper ∞ -categoryof higher orbifolds(Remark 4.60) (cid:31) (cid:63) (cid:79) (cid:79) ´EtaleStacks ∞ ∞ -categoryof ´etale ∞ -stacks[Ca20] (cid:39) (cid:111) (cid:111) (cid:31) (cid:63) (cid:79) (cid:79) (23) ◦ Following Remark 4.60, we may and should enhance this construction to the proper ∞ -category of higherorbifolds Def. 3.48, Def. 4.58, as shown on the left in (23). ◦ In fact, the archetypical example of ModelSpaces considered in [Ca20] is SmoothManifolds (Def. 2.9), inwhich case the left hand side of (23) is the singular-cohesive ∞ -topos of our Examples 3.18, 3.56, containingthe proper (Def. 4.58) ∞ -category of orbi- R n -folds in our Example 4.18. ◦ On the other hand, a general ∞ -topos Sheaves ∞ ( ModelSpaces ) is not going to be cohesive (Def. 3.1) or evenelastic (Def. 3.21). This means that various nice geometric properties, which we derive here, of objects in theproper ∞ -category of higher orbifolds, are not guaranteed to exist in the general setup of [Ca20]. Notablythe theory of frame bundles on orbifolds, according to Prop. 4.26, and the main theorem on the induced´etale cohomology of orbifolds (Theorem 5.16) crucially uses the internal modal logic of singular-cohesiveand singular-elastic ∞ -toposes as in §3, which may not exist, or not exist completely, for any given site ofModelSpaces as in [Ca20]. Proper orbifold differential cohomology.
While (i) generalized differential cohomology on smooth manifolds [HS05] is well-understood (see [Bu12]) and (ii) plain global equivariant cohomology has been established [Schw18] and understood to provide proper orb-ifold cohomology ([Ju20], see Remark 4.1 below),their combination to (generalized, global) proper equivariant differential cohomology has remained elusive. Ex-plicit constructions have been explored for the case of equivariant/orbifold differential K-theory [SV07][BS09][Or09], but even these theories do not seem to be well-understood yet [BS12, p. 47]. What has been missing is acoherent theoretical framework for proper equivariant differential cohomology: Since (a) differential cohomology is the intrinsic cohomology (22) of cohesive ∞ -toposes (by Remark 3.20) and (b) proper equivariant cohomology is the intrinsic cohomology of ∞ -toposes over a (global) orbit category (byRemark 5.4),proper equivariant differential cohomology should be the intrinsic cohomology of ∞ -toposes that combine thesetwo properties. This is exactly what our notion of singular-cohesive ∞ -toposes expresses (Def. 3.48), as confirmedby Theorem 5.9. For example, in singular-cohesive ∞ -toposes there exists the (global) proper equivariant versionof twisted differential non-abelian cohomology [FSS20b], now given by homotopy fiber products parametrizedover Singularities (Def. 3.46). Hence singular-cohesive ∞ -toposes constitute a coherent framework in which todiscuss proper equivariant/orbifold differential cohomology in general. We will develop this elsewhere.11 Preliminaries
We recall basics of higher topos theory in §2.1 and lay out in §2.2 the internal formulation , in ∞ -toposes, of groupactions and the classification of fiber bundles. We briefly record basics of ∞ -topos theory [TV05][Lu09a][Re10] (review is in [Re19], exposition with an eyetowards differential geometric applications is in [FSS13a]). This is to set up our notation and to highlight someless widely used aspects that we need further below. Categories.
We make free use of the language and the basic facts of category theory and homotopy theory (see[GJ99][Rie14][Ri20]) as well as of ∞ -category theory (see [Joy08a][Joy08a][Lu09a][Rie14][Ci19]). (i) We write Categories ∞ for the (“very large”) ∞ -category of (large) ∞ -categories [Re98][Be05][Lu09a, Ch. 3],though we only use this for declaring ∞ -categories. Inside Categories ∞ , there is the sequence of full sub- ∞ -categories (Def. 2.1) of n -categories (i.e.: ( n , ) -categories) as well as of n -groupoids (see Def. 2.12) for all n ∈ N ,denoted thus: Sets (cid:31) (cid:127) (cid:47) (cid:47) Categories (cid:31) (cid:127) (cid:47) (cid:47) Categories (cid:31) (cid:127) (cid:47) (cid:47) · · · (cid:31) (cid:127) (cid:47) (cid:47) Categories ∞ Sets (cid:31) (cid:127) (cid:47) (cid:47)
Groupoids (cid:31) (cid:127) (cid:47) (cid:47) (cid:63)(cid:31) (cid:79) (cid:79) Groupoids (cid:31) (cid:127) (cid:47) (cid:47) (cid:63)(cid:31) (cid:79) (cid:79) · · · (cid:31) (cid:127) (cid:47) (cid:47) Groupoids ∞ (cid:63)(cid:31) (cid:97) (cid:79) (cid:79) (cid:15) (cid:15) Core (24) (ii)
Here Core ( C ) denotes the maximal ∞ -groupoid inside an ∞ -category C . (iii) For C ∈ Categories ∞ and for X , Y ∈ C a pair of objects, we write C ( X , Y ) : = Hom C ( X , Y ) ∈ Groupoids ∞ (25)for the hom- ∞ -groupoid , i.e. the ∞ -groupoid of morphisms between them, and higher homotopies between these(see [Lu09a, 1.2.2][DS09]). This is well-defined, up to equivalence of ∞ -groupoids, independently of which modelfor ∞ -categories is used, since these are all equivalent to each other [Be06][Be14]. We have no need to specify anyparticular model for ∞ -categories (except for the construction of examples, in §A). Definition 2.1 (Fully faithful functor [Lu09a, 1.2.10]) . For C , D ∈ Categories ∞ (24), a functor F : C (cid:47) (cid:47) D iscalled fully faithful , to be denoted C (cid:31) (cid:127) F (cid:47) (cid:47) D , (26)if it is an equivalence on all hom- ∞ -groupoids (25): ∀ X , Y ∈ C C ( X , Y ) F X , Y (cid:39) (cid:47) (cid:47) D (cid:0) F ( X ) , F ( Y ) (cid:1) . In this case we also say that (26) exhibits a full sub- ∞ -category inclusion . Topology.
The category of ∆ -generated or D-topological spaces (Remark 2.3) is both: a convenient foundation forhomotopy theory (Prop. 2.4) as well as pivotal for our key example context (Example 3.18):
Definition 2.2 (Topological spaces) . We writeCWComplexes (cid:31) (cid:127) (cid:47) (cid:47)
DTopologcalSpaces (cid:31) (cid:127) (cid:47) (cid:47)
TopologicalSpaces ∈ Categories (27)for (from right to left): (i) the category of all topological spaces with continuous functions between them; (ii) the full subcategory on those spaces whose topology coincides with the final topology on the set of continuousfunctions out of a Euclidean space R n , hence whose open subsets coincide with those subsets whose pre-imagesunder all continuous functions R n ! X are open in R n , for all n ∈ N ; (iii) the further full subcategory on those that admit the structure of a CW-cell complex, hence that are homeomor-phic to topological spaces which are obtained, starting with the empty space, by gluing on standard n -disks alongtheir ( n − ) -sphere boundaries, iteratively for n ∈ N . 12 emark 2.3 (D-topological is ∆ -generated) .(i) Since the topological n -simplex ∆ n top is a retract of the Euclidean space R n , the condition on X ∈ TopologicalSpacesof being D-topological (Def. 2.2) is equivalent to being ∆ -generated , in that the open subsets of X are preciselythose whose pre-images under all continuous functions of the form ∆ n top ! X are open. (ii) The concept of ∆ -generated spaces is due to [Sm][Dug03]; and independently due to [SYH10], where they arecalled numerically generated . (iii) We say
D-topological to better bring out their conceptual role, in view of Prop. 2.7 below.
Proposition 2.4 (D-topological spaces are convenient) . The category of
DTopologicalSpaces (Def. 2.2) is a con-venient category of topological spaces in the sense of [St67] in that it: (i) contains all CW-complexes (27) [SYH10, Cor. 4.4]; (ii) has all small limits and colimits [SYH10, Prop. 3.4]; (iii) is locally presentable [FR07, Cor. 3.7]; (iv) is Cartesian closed [SYH10, Cor. 4.6]: the mapping space between X , Y ∈ DTopologicalSpaces is the reflec-tion (32) of the internal mapping space
Maps (56) of DiffeologicalSpaces [SYH10, Prop. 4.7]:
Maps ( X , Y ) = Dtplg (cid:16)
Maps (cid:0)
Cdfflg ( X ) , Cdfflg ( Y ) (cid:1)(cid:17) . (28) Differential topology.
D-topological spaces lend themselves to differential topology via their joint (co-)reflection(Prop. 2.7) both into all topological spaces and into diffeological spaces (Def. 2.6):
Definition 2.5 (Cartesian spaces) . We writeCartesianSpaces (cid:31) (cid:127) (cid:47) (cid:47)
SmoothManifolds ∈ Categories for the category whose objects are the natural numbers n ∈ N , thought of as representing the Cartesian spaces R n , and whose morphisms are the smooth functions between these. We regard this category as equipped withthe coverage (Grothendieck pre-topology) whose covers are the differentially good open covers (i.e., such that allnon-empty finite intersections of patches are diffeomorphic to a Cartesian space [FSS12, 6.3.9]). Definition 2.6 (Diffeological spaces) . (i)
The category of diffeological spaces ([So80][So84][IZ85], see [BH08][IZ13]) is the full subcategory of sheaves on CartesianSpaces (Def. 2.5)DiffeologicalSpaces (cid:31) (cid:127) (cid:47) (cid:47)
Sheaves ( CartesianSpaces ) (29)on those X ∈ Sheaves ( CartesianSpaces ) which are concrete sheaves [Du79b] supported on their underlying setX s : = Sheaves ( SmthMfd )( ∗ , X ) in that the canonical function X ( U ) (cid:31) (cid:127) (cid:47) (cid:47) Set ( U s , X s ) is an injection, for all U ∈ CartesianSpaces, with U s denoting their underlying set of U . (ii) We call X ( U ) (cid:39) Prop. 2.38
DiffeologicalSpaces ( U , X ) ∈ Set (30)the set of
U -plots of the diffeological space X . Proposition 2.7 (Topological/diffeological adjunction) . (i)
There is an adjunction [SYH10, Prop. 3.1]
TopologicalSpaces (cid:111) (cid:111)
DtplgCdfflg ⊥ (cid:47) (cid:47) DiffeologicalSpaces (31) between the categories of topological spaces (Def. 2.2) and of diffeological spaces (Def. 2.6), where• the right adjoint
Cdfflg sends a topological space to the same underlying set equipped with the topologicaldiffeology whose plots (30) are precisely the continuous functions; the left adjoint Dtplg sends a diffeological space to the same underlying set equipped with the diffeologicaltopology (“D-topology” [IZ13, 2.38][CSW13]), which is the final topology with respect to all plots (30) ,hence such that a subset is open precisely if its pre-image under all plots is open. (ii)
The fixed points X ∈ TopologicalSpaces of this adjunction are the D-topological spaces (Remark 2.3)X is D-topological ⇔ Dtplg (cid:0)
Cdfflg ( X ) (cid:1) ε X (cid:39) (cid:47) (cid:47) X . (iii) The adjunction is idempotent [SYH10, Lemma 3.3], hence factors through the category of D-topologicalspaces, exhibiting them as a co-reflective subcategory of
TopologicalSpaces and a reflective subcategory of
DiffeologicalSpaces : TopologicalSpaces (cid:111) (cid:111) (cid:63) (cid:95)
Cdfflg ⊥ (cid:47) (cid:47) DTopologicalSpaces (cid:111) (cid:111)
Dtplg (cid:31) (cid:127) ⊥ (cid:47) (cid:47) DiffeologicalSpaces . (32)The following Prop. 2.8 is due to [Har13, Thm. 3.3]. Proposition 2.8 (Model structure on D-topological spaces) .(i)
The standard cell inclusions define a cofibrantly generated model category structure on
DTopologicalSpaces (Def. 2.2). (ii)
With respect to this model structure and the standard model structure on
TopologicalSpaces , the co-reflection (32) becomes a Quillen equivalence:
TopologicalSpaces (cid:111) (cid:111) (cid:63) (cid:95)
Cdfflg (cid:39)
Quillen (cid:47) (cid:47)
DTopologicalSpaces . Differential geometry.Definition 2.9 (Smooth Manifolds) . We writeSmoothManifolds ∈ Categories (33)for the category of finite-dimensional paracompact smooth manifolds with smooth functions between them. Weregard this as a site with the Grothendieck topology of open covers.
Proposition 2.10 (Cartesian spaces are dense in the site of manifolds) . With respect to the coverages in Def.2.9 and Def. 2.5, the inclusion
CartesianSpaces i (cid:44) ! SmoothManifolds is a dense sub-site, in that it induces anequivalence of categories of sheaves
Sheaves ( CartesianSpaces ) (cid:111) (cid:111) i ∗ i ∗ (cid:39) (cid:47) (cid:47) Sheaves ( SmoothManifolds ) . Proposition 2.11 (Smooth manifolds inside diffeological spaces) . Every X ∈ SmoothManifolds (33) becomes adiffeological space (Def. 2.6) on its underlying set by taking its plots (30) of shape U ∈ CartesianSpaces to be theordinary smooth functions: X ( U ) : = SmoothManifolds ( U , X ) . More generally, every possibly infinite-dimensional Fr´echet manifold (e.g. [KS17, 2.2]) becomes a diffeologicalspace this way. Moreover, this constitutes fully faithfull embeddings (Def. 2.1) into the category of Diffeologicalspaces [Lo94, Thm. 3.1.1]:
SmoothManifolds finite-dimensional (cid:31) (cid:127) (cid:47) (cid:47)
Fr´echetSmoothManifolds possibly infinite-dimensional (cid:31) (cid:127) (cid:47) (cid:47)
DiffeologicalSpaces . (34) Homotopy theory.Definition 2.12 ( ∞ -Groupoids) . (i) We writeGroupoids ∞ ∈ Categories ∞ (35)for the ∞ -category which is presented by the topologically enriched category whose objects are the CW-complexes(27) and whose hom-spaces are the mapping spaces (28). (ii) The full sub- ∞ -category (Def. 2.1) on the homotopy n -types is that of n-groupoids Groupoids n (cid:31) (cid:127) (cid:47) (cid:47) Groupoids ∞ . efinition 2.13 (Topological shape) . (i) We writeShp
Top : CWComplexes (cid:47) (cid:47)
Groupoids ∞ for the ∞ -functor from the 1-category of CW-complexes (27) to the ∞ -category of ∞ -groupoids (Def. 2.12) which,as a topologically enriched functor, is the identity on objects, and is on hom-spaces the continuous map given bythe identity function from the discrete set of continuous maps to the mapping space (28). (ii) For any choice of CW-approximation functorTopologicalSpaces ( − ) cof (cid:47) (cid:47) CWComplexeswe get the corresponding functor on all topological spaces (Def. 2.2), hence on D-toplogical spaces (Def. 2.2)which we denote by the same symbol:Shp
Top : TopologicalSpaces ( − ) cof (cid:47) (cid:47) CWComplexes
Shp
Top (cid:47) (cid:47)
Groupoids ∞ . (36) Example 2.14 (Delooping groupoids) . For G ∈ Groups fin , consider the groupoid with a single object ∗ , and with G as its set of morphisms, whose composition is given by the product in the group: ∗ g (cid:41) (cid:41) ∗ g (cid:53) (cid:53) g · g (cid:47) (cid:47) ∗ (37)This groupoid is the topological shape (36) of the Eilenberg-MacLane space K ( G , ) as well as (since G is assumedto be finite) the classifying space BG . More intrinsically, this groupoid is, equivalently, the homotopy quotient ofthe point by the trivial G -action: ∗ (cid:12) G ∈ Groupoids (cid:31) (cid:127) (cid:47) (cid:47) Groupoids ∞ . More generally:
Example 2.15 (Action groupoids) . For G ∈ Groups fin a finite group and for X ∈ Set a set equipped with a G -action G × X ρ (cid:47) (cid:47) X ( g , x ) (cid:31) (cid:47) (cid:47) g · x (38)the corresponding action groupoid has as objects the elements of X and its morphisms and their composition aregiven as follows: g · x g (cid:43) (cid:43) x g (cid:53) (cid:53) g · g (cid:47) (cid:47) g · g · x (39)This action groupoid is a model for the homotopy quotient of X by its G -action X (cid:12) G ∈ Groupoids (cid:31) (cid:127) (cid:47) (cid:47) Groupoids ∞ . The following elementary example plays a pivotal role in later constructions (Lemma 4.7):
Example 2.16 (Hom-groupoid into action groupoid) . Let G ∈ Groups fin , X ∈ Set equipped with a G -action (38),hence with action groupoid/homotopy quotient X (cid:12) G ∈ Groupoids (Example 2.15). Let K ∈ Groups fin be anyfinite group, with ∗ (cid:12) K ∈ Groupoids its delooping groupoid (Example 2.14). Then the hom-groupoid (functorgroupoid) of morphisms (functors) ∗ (cid:12) K −! X (cid:12) G is, equivalently, the action groupoid of G acting on the set ofpairs consisting of a group homomorphism φ : K ! G and a point in X fixed by the image of φ :Groupoids (cid:0) ∗ (cid:12) K , X (cid:12) G (cid:1) (cid:39) (cid:18) (cid:70) φ ∈ Groups ( K , G ) X φ ( K ) (cid:19) (cid:12) G . (40)Here• φ ( K ) ⊂ G denotes the subgroup of G which is image of the group homomorphism φ : K ! G ;• X φ ( K ) = (cid:26) x ∈ X (cid:12)(cid:12)(cid:12)(cid:12) ∀ h ∈ φ ( K ) h · x = x (cid:27) denotes the φ ( K ) -fixed-point set in X ;• the G -action by which the homotopy quotient is taken is the conjugation action on φ , hence g · φ : = Ad g ◦ φ ,and the given G -action on x ∈ X . 15his follows by direct unwinding of the definition of functors and of natural transformations between the groupoids(37) and (39). Definition 2.17 (Simplicial-topological shape) . Let X • : ∆ op (cid:47) (cid:47) TopologicalSpaces (41)be a simplicial topological space, for instance the nerve of a topological groupoid. Then we say that its simplicial-topological shape is the homotopy colimit (Prop. 2.36) of its degreewise topological shape (Def. 36):Shp sTop (cid:0) X • (cid:1) : = lim −! (cid:0) Shp
Top ( X ) (cid:1) • ∈ Groupoids ∞ . (42)The following Prop. 2.18 appears as [Wa18, 4.3,. 4.4]: Proposition 2.18 (Simplicial-topological shape of degreewise cofibrant spaces is fat geometric realization) . IfX • is a simplicial topological space (41) which degreewise admits the structure of a retract of a cell complex(for instance: degreewise a CW-complex (27) ), then its simplicial topological shape (42) is equivalent to its fatgeometric realization (cid:107)−(cid:107) [Se74] (see [HG07, 2.3]):X • degreewise cofibrantsimplicial topological spaces ∈ (cid:0) TopologicalSpaces cof (cid:1) ∆ op ⇒ Shp sTop (cid:0) X • (cid:1) simplicialtopological shape (cid:39) (cid:107) X • (cid:107) fat geometricrealization . Definition 2.19 (Diffeological simplices) . (i)
We write ∆ ∆ • smth (cid:47) (cid:47) DiffeologicalSpaces [ n ] (cid:31) (cid:47) (cid:47) (cid:110) (cid:126) x ∈ R n + | ∑ i x i = (cid:111) for the diffeological extended simplicies , hence for the simplicial object in diffeological spaces (Def. 2.6) (in factin smooth manifolds, under Prop. 2.11) which in degree n is the extended n -simplex in R n + , regarded with itssub-diffeology, and whose face and degeneracy maps are the standard ones (see [CW14, Def. 4.3][BEBP19, p. 1]). (ii) The induced nerve/realization construction is a pair of adjoint functors (Def. 2.24)DiffeologicalSpaces (cid:111) (cid:111) |−| diff
Sing diff ⊥ (cid:47) (cid:47) SimplicialSets (43)between the categories of simplicial sets and of diffeological spaces (Def. 2.6), where the right adjoint Sing diff sends X ∈ DiffeologicalSpaces to its smooth singular simplicial set
Sing diff ( X ) • : = DiffeologicalSpaces (cid:0) ∆ • diff , X (cid:1) . The following Prop. 2.20 is due to [CW14, Prop. 4.14]:
Proposition 2.20 (Diffeological singular simplicial set of continuous Diffeology) . For all X top ∈ TopologicalSpaces there is a weak homotopy equivalence between the diffeological singular simplicial set (Def. 2.19) of its continuousdiffeology (Def. 2.7) and its ordinary singular simplicial set:
Sing (cid:0) X top (cid:1) (cid:39) whe Sing diff (cid:0)
Cdfflg ( X top ) (cid:1) . Equivalently this means, in the terminology to be introduced in a moment, that the topological shape (36) oftopological spaces is equivalent to the cohesive shape (Def. 3.1) of their incarnation as continuous-diffeologicalspaces (see Example 3.18 below):
Shp
Top (cid:0) X top (cid:1) (cid:39) Shp (cid:0)
Cdfflg ( X top ) (cid:1) ∈ Groupoids ∞ . Universal constructions.
All diagrams we consider now are homotopy-coherent, even if we do not notationallyindicate the higher cells, unless some are to be highlighted. Similarly, all universal constructions we consider noware ∞ -categorical, even if this is not further pronounced by the terminology. In particular, we say “colimit” lim −! for “homotopy colimit”, “limit” lim − for “homotopy limit” (see Prop. 2.36), “Cartesian square” for “homotopyCartesian square”, etc.: 16 otation 2.21 (Cartesian squares) . We say a square in an ∞ -category is Cartesian , to be denoted X × B Y f ∗ g (cid:15) (cid:15) (cid:47) (cid:47) (pb) Y g (cid:15) (cid:15) X f (cid:47) (cid:47) B (44)if it is an limit cone over the diagram consisting of f and g . We also say this is the pullback square of g along f . Example 2.22 (Pullback of equivalence is equivalence) . Let C ∈ Categories ∞ . Then a square in C whose rightvertical morphism is an equivalence is Cartesian (Notation 2.21) precisely if the left vertical morphism is also anequivalence: A (pb) (cid:47) (cid:47) (cid:15) (cid:15) B (cid:39) (cid:15) (cid:15) C (cid:47) (cid:47) D ⇔ A (cid:39) (cid:15) (cid:15) C (45)hence precisely if C (cid:47) (cid:47) D is equivalent to A (cid:47) (cid:47) B in C ∆ . Proposition 2.23 (Pasting law [Lu09a, Lemma 4.4.2.1]) . In any ∞ -category, consider a diagram of the formA (cid:15) (cid:15) (cid:47) (cid:47) ⇓ B (cid:15) (cid:15) (cid:47) (cid:47) ⇓ C (cid:15) (cid:15) D (cid:47) (cid:47) E (cid:47) (cid:47) Fsuch that the right square is Cartesian (Notation 2.21). Then the left square is Cartesian if and only if the totalrectangle is Cartesian.
Definition 2.24 (Adjoint ∞ -functors [Lu09a, 5.2.2.7, 5.2.2.8][RV13, 4.4.4]) . Let C , D ∈ Categories ∞ (24) and L : C (cid:111) (cid:111) (cid:47) (cid:47) D : R two functors between them, back and forth. This is an adjoint pair with L left adjoint and
Rright adjoint , to be denoted ( L (cid:97) R ) : D (cid:111) (cid:111) LR ⊥ (cid:47) (cid:47) C (46)if there is a natural equivalence of hom- ∞ -groupoids (25) of the form D (cid:0) L ( − ) , − (cid:1) (cid:39) C (cid:0) − , R ( − ) (cid:1) (47)(This is unique when it exists [Lu09a, Prop. 5.2.1.3, 5.2.6.2]). In this case, one says: (i) The adjunction unit is the natural transformation X η X (cid:47) (cid:47) R ◦ L ( X ) (48)which is the (pre-)image under (47) of the identity on R ( X ) . (ii) The adjunction co-unit is the natural transformation L ◦ R ( X ) ε X (cid:47) (cid:47) X (49)which is the image under (47) of the identity on L ( X ) .As in the classical situation of 1-category theory, it follows that: Proposition 2.25 (Triangle identities) . Let D (cid:111) (cid:111) LR ⊥ (cid:47) (cid:47) C be a pair of adjoint ∞ -functors (Def. 2.24). Then theiradjunction unit η (48) and counit ε (49) satisfy the following natural equivalences: (i) for all c ∈ C , L ◦ R ◦ L ( c ) ε L ( c ) (cid:43) (cid:43) L ( c ) L ( η c ) (cid:51) (cid:51) L ( c ) ; (ii) for all d ∈ D , R ◦ L ◦ R ( d ) R ( ε d ) (cid:43) (cid:43) R ( d ) η R ( d ) (cid:51) (cid:51) R ( d ) . roposition 2.26 (Right/left adjoints preserve limits/colimits [Lu09a, 5.2.3.5]) . Let D (cid:111) (cid:111) LR ⊥ (cid:47) (cid:47) C be a pair of ad-joint ∞ -functors (Def. 2.24) and let I ∈ Categories ∞ . (i) If X • : I (cid:47) (cid:47) D is a diagram whose limit exists, then this limit is preserved by the right adjoint R:R (cid:0) lim − X • (cid:1) (cid:39) lim − RX • (50) (ii) If X • : I (cid:47) (cid:47) C is a diagram whose colimit exists, then this colimit is preserved by the left adjoint L:L (cid:0) lim −! X • (cid:1) (cid:39) lim −! LX • (51)Conversely: Proposition 2.27 (Adjoint ∞ -functor theorem [Lu09a, 5.5.2.9]) . Let C , ∈ Categories ∞ be presentable (e.g. ∞ -toposes, Def. 2.30), then an ∞ -functor C (cid:47) (cid:47) C is a: (i) right adjoint (i.e., has a left adjoint, Def. 2.24) precisely if it preserves limits (50) ; (ii) left adjoint (i.e., has a right adjoint, Def. 2.24) precisely if it preserves colimits (51) . Proposition 2.28 (Fully faithful adjoints [Lu09a, 5.2.7.4]) . For adjoint ∞ -functors (Def. 2.24) D (cid:111) (cid:111) LR ⊥ (cid:47) (cid:47) C , (i) L is fully faithful D (cid:111) (cid:111) L (cid:63) (cid:95) C (Def. 2.1) iff the adjunction unit η (48) is an equivalence: id η (cid:39) (cid:47) (cid:47) R ◦ L ; (ii)
R is fully faithful D (cid:31) (cid:127) R (cid:47) (cid:47) C (Def. 2.1) iff the adjunction counit ε (49) is an equivalence L ◦ R ε (cid:39) (cid:47) (cid:47) id . Proposition 2.29 (Idempotent Monads and Comonads) . For D (cid:111) (cid:111) LR ⊥ (cid:47) (cid:47) C a pair of adjoint ∞ -functors (Def. 2.24): (i) If R is fully faithful (Def. 2.1) then (cid:35) : = R ◦ L is idempotent, exhibited by the (cid:35) -image of the adjunction unit η (48) : (cid:35) ( c ) (cid:35) ( η L ( c ) ) (cid:39) (cid:47) (cid:47) (cid:35) ◦ (cid:35) ( c ) . (52) (ii) If L is fully faithful (Def. 2.1) then (cid:50) : = L ◦ R is idempotent, exhibited by the (cid:50) -image of the adjunction counit η (49) : (cid:50) ◦ (cid:50) ( d ) (cid:50) ( ε R ( d ) ) (cid:39) (cid:47) (cid:47) (cid:50) ( d ) . (53) Proof.
Consider case (i) , the other case is formally dual. Since R is fully faithful, by assumption, the conditionthat (cid:35) ( η L ( c ) ) : = R ◦ L ( η L ( c ) ) is an equivalence is equivalent to L ( η L ( c ) ) being an equivalence. But, by the triangleidentity (Prop. 2.25), we have that the composite ε L ( L ( c )) ◦ L ( η L ( c ) ) is an equivalence, while by Prop. 2.28 thecounit ε is a natural equivalence. By cancellation, this implies that L ( η L ( c ) ) is an equivalence. (cid:3) ∞ -Toposes. For our purposes, we take the following characterization to be the definition of ∞ -toposes. This is dueto Rezk and Lurie [Lu09a, 6.1.6.8]; we follow the presentation in [NSS12a, Prop. 2.2]: Definition 2.30 ( ∞ -topos) . An ∞ -topos H is a presentable ∞ -category with the following properties: (i) Universal colimits. For all morphisms f : X −! B and all small diagrams A : I −! H / B , there is an equiva-lence: lim −! i f ∗ A i (cid:39) f ∗ (cid:0) lim −! i A i (cid:1) (54)between the pullback (44) of the colimit and the colimit over the pullbacks of its components. (ii) Univalent universes. For every sufficiently large regular cardinal κ , there exists a morphism (cid:92) Objects κ −! Objects κ in H , such that for every object X ∈ H , pullback (44) along morphisms X −! Objects κ constitutesan equivalence Core (cid:0) H / κ X (cid:1) (cid:39) H (cid:0) X , Objects κ (cid:1) E (cid:96) E E (pb) (cid:47) (cid:47) (cid:15) (cid:15) (cid:92)
Objects κ (cid:15) (cid:15) X (cid:96) E (cid:47) (cid:47) Objects κ (55)between the ∞ -groupoid core (24) of bundles (Notation 2.45) which are κ -small over X , and the hom- ∞ -groupoid (25) of morphisms from X to the object classifier Objects κ .18 xample 2.31 (Internal mapping space in an ∞ -topos) . Let H be an ∞ -topos (Def. 2.30) and X ∈ H an object.As a special case of universality of colimits (54), we have that the functor X × ( − ) of Cartesian product with X preserves all colimits. Hence, by the adjoint ∞ -functor theorem (Prop. 2.27), this functor has a right adjoint, to bedenoted Maps ( X , − ) , the internal hom - or internal mapping space - or mapping stack -functor: H (cid:111) (cid:111) X × ( − ) Maps ( X , − ) internal mapping space ⊥ (cid:47) (cid:47) H . (56)By adjointness, the probes of the internal mapping space over any U ∈ H are given by H (cid:0) U , Maps ( X , Y ) (cid:1) (cid:39) H (cid:0) U × X , Y (cid:1) . (57) Proposition 2.32 (Colimits and equifibered transformations [Lu09a, 6.1.3.9(4)][Re10, 6.5]) . Let H be an ∞ -topos(Def. 2.30), I a small ∞ -category, X • , Y • : I (cid:47) (cid:47) H two I -shaped diagrams. (i) If X • f • (cid:43) (cid:51) Y • is a natural transformation which is equifibered [Re10, p. 9], in that its value on all mor-phisms i φ (cid:47) (cid:47) i in Y is a Cartesian square (Notation 2.21), then the value of lim −! f • on all colimit componentmorphisms is also Cartesian: ∀ i φ ! i X i f i (cid:47) (cid:47) X φ (cid:15) (cid:15) (pb) Y i Y φ (cid:15) (cid:15) X i f i (cid:47) (cid:47) Y i ⇒ ∀ i X i f i (cid:47) (cid:47) q Xi (cid:15) (cid:15) (pb) Y i q Yi (cid:15) (cid:15) lim −! X • lim −! f • (cid:47) (cid:47) lim −! Y • (58) (ii) Let X (cid:3) • : I (cid:3) (cid:47) (cid:47) H be a cocone under X • , with tip X ∈ H , and let Y (cid:3) • : I (cid:3) (cid:47) (cid:47) H denote the colimitingcocone under Y • with tip lim −! Y • . If X (cid:3) • f (cid:3) • (cid:43) (cid:51) Y (cid:3) • is a natural transformation of cocone diagrams which is equifibered,then X (cid:3) • is a colimiting cocone: ∀ i φ ! i X i f i (cid:47) (cid:47) X φ (cid:15) (cid:15) (pb) Y i Y φ (cid:15) (cid:15) X i f i (cid:47) (cid:47) Y i and ∀ i X i f i (cid:47) (cid:47) q Xi (cid:15) (cid:15) (pb) Y i q Yi (cid:15) (cid:15) X lim −! f • (cid:47) (cid:47) lim −! Y • ⇒ X (cid:39) lim −! X • . (59) Example 2.33 (Initial object in ∞ -topos is empty object [Re19, p. 16]) . Let H be an ∞ -topos (Def. 2.30). Applyingthe implication (59) in Prop. 2.32 to the colimit over the empty diagram, which is the initial object, shows that anyobject with a morphism to the initial object is itself equivalent to the initial object. Hence if we write ∅ ∈ H s.t. ∀ X ∈ H (cid:0) H ( ∅ , X ) (cid:39) ∗ (cid:1) (60)for the initial object, this means that X ∃ (cid:47) (cid:47) ∅ ⇒ X (cid:39) ∅ . (61) Proposition 2.34 (Tensoring of ∞ -toposes over ∞ -groupoids) . Let H be an ∞ -topos (Def. 2.30) with inverse basegeometric morphism (Prop. 2.43) denoted ∆ : Groupoids ∞ −! H . Then, for S ∈ Groupoids ∞ and X , Y ∈ H , thereis a natural equivalence of ∞ -groupoids H (cid:0) ∆ ( S ) × X , Y (cid:1) (cid:39) Groupoids ∞ (cid:0) S , H ( X , Y ) (cid:1) . (62) Proof.
By [Lu09a, Cor. 4.4.4.9] we have, for S ∈ Groupoids ∞ (cid:44) ! Categories ∞ and X , Y ∈ H , natural equivalenceslim −! S const ∗ (cid:39) S and H (cid:16) lim −! S const X , Y (cid:17) (cid:39) Groupoids ∞ (cid:0) S , H ( X , Y ) (cid:1) . (63)This implies the statement in the form (62) by using (a) that ∆ preserves all colimits as well as finite limits (Prop.2.43) and (b) that Cartesian products may be taken inside colimits, as a special case of (54): H (cid:0) ∆ ( S ) × X , Y (cid:1) (cid:39) H (cid:0) ∆ (cid:0) lim −! S ∗ (cid:1) × X , Y (cid:1) (cid:39) H (cid:0)(cid:0) lim −! S ∆ ( ∗ ) (cid:124)(cid:123)(cid:122)(cid:125) (cid:39)∗ (cid:1) × X , Y (cid:1) (cid:39) H (cid:16)(cid:0) lim −! S ( ∗ × X ) (cid:124) (cid:123)(cid:122) (cid:125) (cid:39) X (cid:1) , Y (cid:17) (cid:39) Groupoids ∞ (cid:0) S , H ( X , Y ) (cid:1) . The composite equivalence is (62). (cid:3) heaves.Notation 2.35 ( ∞ -Presheaves) . For C a small ∞ -category, we writePreSheaves ∞ ( C ) : = Functors ∞ (cid:0) C op , Groupoids ∞ (cid:1) (64)for the ∞ -category of ∞ -presheaves on C . More generally, if H is any ∞ -topos (Def. 2.30) we also writePreSheaves ∞ (cid:0) C , H (cid:1) : = Functors ∞ (cid:0) C op , H (cid:1) . (65) Proposition 2.36 (Limits and colimits in an ∞ -topos [Lu09a, Lem. 4.2.4.3]) . Let H be an ∞ -topos (Def. 2.30) and C a small ∞ -category. Then the ∞ -functor which sends an object in H to the H -valued presheaf (65) constant onthis object has a right- and a left-adjoint (Def. 2.24), given by the limit and colimit construction, respectively: Functors ∞ (cid:0) C , H (cid:1) lim −! (cid:47) (cid:47) (cid:111) (cid:111) const ⊥⊥ lim − (cid:47) (cid:47) H (66) Proposition 2.37 ( ∞ -Yoneda embedding [Lu09a, Lemma 5.5.2.1]) . Let C be an ∞ -category. Then the ∞ -functorfrom C to its ∞ -presheaves (64) which assigns representable presheaves C (cid:31) (cid:127) y (cid:47) (cid:47) PreSheaves ∞ ( C ) c (cid:31) (cid:47) (cid:47) C ( − , c ) (67) is fully faithful (Def. 2.1). Proposition 2.38 ( ∞ -Yoneda lemma [Lu09a, Lemma 5.5.2.1]) . Let C be an ∞ -category. Then for X ∈ PreSheaves ∞ ( C ) (64) and c ∈ C , there is a natural equivalence PreSheaves ∞ (cid:0) y ( c ) , X (cid:1) (cid:39) X ( c ) , where y is the Yoneda embedding (67) from Prop. 2.37. Proposition 2.39 ((Co-)Limits of presheaves are computed objectwise [Lu09a, Cor. 5.1.2.3]) . Let H be an ∞ -topos, let C and D be small ∞ -categories, and letI : D (cid:47) (cid:47) PreSheaves ∞ ( C , H ) be a diagram of H -valued ∞ -presheaves over C . Then the limit and colimit over I exist and are given objectwiseover c ∈ C by the limit and colimit of the components in Groupoids ∞ : (cid:0) lim −! I (cid:1) : c (cid:0) lim −! I c (cid:1) , (cid:0) lim − I (cid:1) : c (cid:0) lim − I c (cid:1) . Lemma 2.40 (Colimit of representable functor is contractible) . Let C be a small ∞ -category, and consider an ∞ -functor yC : C op −! Groupoids ∞ to the ∞ -category of ∞ -groupoids (35) , which is representable, hence which isin the essential image of the ∞ -Yoneda embedding (67) . Then the colimit of this functor is contractible: lim −! C ( yC ) (cid:39) ∗ . (68) Proof.
The terminal ∗ ∈
Groupoids ∞ is characterized by the fact that for S ∈ Groupoids ∞ there is a natural equiva-lence S (cid:39) Groupoids ∞ (cid:0) ∗ , S (cid:1) . Hence it is sufficient to see that lim −! ( yC ) satisfies the same property. But we have the following sequence of naturalequivalences: Groupoids ∞ (cid:16) lim −! ( yC ) , S (cid:17) (cid:39) Functors ∞ (cid:0) C op (cid:1)(cid:0) yC , const (cid:1) (cid:39) ( const S )( C ) (cid:39) S . Here the first step is the adjunction (66), while the second step is the ∞ -Yoneda lemma (Prop. 2.38). (cid:3) roposition 2.41 (Topos is accessibly lex reflective in presheaves over site [Lu09a, 6.1.0.6]) . Let H be an ∞ -topos(Def. 2.30). (i) Then there exists an ∞ -site for H , namely a small C ∈ Categories ∞ equipped with a pair of adjoint ∞ -functors(Def. 2.24) between H and PreSheaves ∞ ( C ) (Notation 2.35): H (cid:111) (cid:111) L (cid:31) (cid:127) ⊥ (cid:47) (cid:47) PreSheaves ∞ (cid:0) C (cid:1) (69) such that (a) the right adjoint is accessible and fully faithful (Def. 2.1) and (b) the left adjoint preserves finitelimits (in addition to preserving all colimits, by Prop. 2.26). (ii) Conversely, any such accessibly embedded lex reflective sub- ∞ -category of an ∞ -category of ∞ -presheaves isan ∞ -topos. Definition 2.42 (Sheaf ∞ -topos [Lu09a, 6.2]) . An ∞ -topos H (Def. 2.30) is called an ∞ -category of ∞ -sheaves or of ∞ -stacks , or just a sheaf topos for short, to be denoted H (cid:39) Sheaves ∞ (cid:0) C (cid:1) (70)if there exists a site C , namely a small C ∈ Categories ∞ with a reflection ( L const (cid:97) Γ ) (69) as in Prop. 2.41, suchthat L const exhibits localization at a set (cid:110) U (cid:31) (cid:127) (cid:47) (cid:47) y ( c ) covering sieves (cid:111) ⊂ (cid:116) c ∈ C SubObjects (cid:0) y ( c ) (cid:1) of monomorphisms (Def. 2.59) into representable presheaves (67). Proposition 2.43 (Base geometric morphism [Lu09a, 6.3.4.1]) . Let H be an ∞ -topos (Def. 2.30). There is anessentially unique pair of adjoint ∞ -functors (Def. 2.24) between H and Groupoids ∞ (Def. 2.12) H (cid:111) (cid:111) L const ⊥ Γ (cid:47) (cid:47) Groupoids ∞ (71) such that the left adjoint L const preserves finite limits (in addition to preserving all colimits, by Prop. 2.26). Example 2.44 (Base geometric morphism via site) . Let H be an ∞ -topos (Def. 2.30) and C a site (Prop. 2.41).Then the composite of pairs of adjoint ∞ -functors (Def. 2.24) H (cid:111) (cid:111) L (cid:31) (cid:127) ⊥ (cid:47) (cid:47) PreSheaves ∞ (cid:0) C (cid:1) (cid:111) (cid:111) constlim − ⊥ (cid:47) (cid:47) Groupoids ∞ (72)of (a) the reflection into presheaves over the site (Prop. 2.41) with (b) the limit-construction on presheaves (Prop.2.36) is such that the composite left adjoint L const preserves finite limits (since L does by Prop. 2.41 and constdoes by Prop. 2.26 with Prop. 2.36). Hence, by the essential uniqueness of Prop. 2.43, the composite (72) is afactorization of the base geometric morphism of H . Bundles.Notation 2.45 (Bundles and slicing.) . Let H an ∞ -topos (Def. 2.30) and X ∈ H an object. We write: (i) ( X , p ) ∈ H / X for objects in the slice ∞ -category of H over X , corresponding to morphisms p to X in H ( bundles over X ): E p (cid:15) (cid:15) X (ii) ( f , α ) ∈ H / X (cid:0) ( E , p ) , ( E , p ) (cid:1) for morphisms in the slice ∞ -category, corresponding to diagrams in H of theform E p (cid:39) (cid:39) f (cid:47) (cid:47) E p (cid:119) (cid:119) X α (cid:48) (cid:56) (73)21 roposition 2.46 (Slice ∞ -topos [Lu09a, Prop. 6.3.5.1 (1)]) . Let H be an ∞ -topos (Def. 2.30) and X ∈ H an object.Then the slice ∞ -category H / X (Notation 2.45) is also an ∞ -topos. Example 2.47 (Iterated slice ∞ -topos) . Let H be an ∞ -topos (Def. 2.30), X ∈ H and ( Y , p ) ∈ H / X an object in theslice, hence (Notation 2.45) a morphism Y p (cid:47) (cid:47) X . Then H / X is itself an ∞ -topos, by Prop. 2.46, and we may sliceagain to obtain the iterated slice ∞ -topos (cid:0) H / X (cid:1) / ( Y , p ) ∈ Categories ∞ . (74) (i) an object in (74) is a diagram in H of this form: Z (cid:36) (cid:36) (cid:45) (cid:45) Y p (cid:117) (cid:117) X (cid:42) (cid:50) (ii) a morphism in (74) is a diagram in H of this form:(This is furthermore filled by a 3-morphism,which we notationally suppress, for readability.) Z (cid:32) (cid:32) (cid:45) (cid:45) (cid:47) (cid:47) Z (cid:33) (cid:33) (cid:15) (cid:15) YX (cid:119) (cid:119) p (cid:53) (cid:61) (cid:105) (cid:113) (cid:34) (cid:42) (cid:59) (cid:67) Proposition 2.48 (Hom- ∞ -groupoids in slices [Lu09a, Prop. 5.5.5.12]) . Let H be an ∞ -topos (Def. 2.30) andB ∈ H an object. Then for ( X , p ) , ( X , p ) ∈ H / B two objects in the slice over B (Prop. 2.46) the hom- ∞ -groupoidbetween them is given by the following homotopy fiber-product of hom- ∞ -groupoids of H : H / B (cid:0) ( X , p ) , ( X , p ) (cid:1) (cid:39) { p } × H ( X , B ) H ( X , X ) (75) hence by the ∞ -groupoid given by the following Cartesian square (Notation 2.21): H / B (cid:0) ( X , p ) , ( X , p ) (cid:1) (cid:15) (cid:15) (cid:47) (cid:47) (pb) H ( X , X ) p ◦ ( − ) (cid:15) (cid:15) ∗ (cid:96) p (cid:47) (cid:47) H ( X , B ) Proposition 2.49 (Base change [Lu09a, HTT 6.3.5]) . Let H be an ∞ -topos (Def. 2.30). Then for every morphismX f ! Y in H there is an induced base change adjoint triple (Def. 2.24) between the corresponding slice ∞ -toposes(Prop. 2.46): H / X f ! (cid:47) (cid:47) (cid:111) (cid:111) f ∗ ⊥ f ∗ ⊥ (cid:47) (cid:47) H / Y (76) where, in H , f ! is given by postcomposition with f while f ∗ is given by pullback along f . Example 2.50 (Bundle morphisms covering base morphisms) . For H an ∞ -topos (Def. 2.30), the system of all itsslice ∞ -toposes (Prop. 2.46) H op H / ( − ) (cid:47) (cid:47) Categories ∞ X H / X (77)related via contravariant base change (76) arranges into the “arrow ∞ -topos” [Lu09a, 2.4.7.12]Bundles ( H ) : = (cid:90) X H / X (cid:39) H ∆ [ ] , (78)which, in view of Notation 2.45, may be thought of as the ∞ -category of bundles in H , but now with bundlemorphisms allowed to cover non-trivial base morphisms. Example 2.51 (Spectral bundles and tangent ∞ -topos) . Let H be an ∞ -topos (Def. 2.30). Instead of the system (79)of its plain slices, consider the corresponding system of stabilized slices (stabilized under the suspension/loopingadjunction on pointed objects, e.g. [Lu07, 1.4]): 22 op Stab ( H / ( − ) ) (cid:47) (cid:47) Categories ∞ X Stab (cid:0) H / X (cid:1) (79)The resulting total ∞ -category SpectralBundles ( H ) : = (cid:90) X Stab (cid:0) H / X (cid:1) , (80)is that of bundles of spectra in H (parametrized spectrum objects). Remarkably, this is itself an ∞ -topos [Joy08a,35.5][Lu17, 6.1.1.11], also called the tangent ∞ -topos T H of H (e.g. [Lu07][BM19]). Example 2.52 (Base change along terminal morphism) . Let H be an ∞ -topos (Def. 2.30) and X ∈ H any object.With H (cid:39) H / ∗ regarded as its own slice (Prop. 2.46) over the terminal object, base change (Prop. 2.49) along theterminal morphism X ! ∗ is of the form H / X dom (cid:47) (cid:47) (cid:111) (cid:111) X × ( − ) ⊥ ⊥ (cid:47) (cid:47) H (81)where (a) the top functor sends a morphism Y ! X to its domain object Y , and (b) the middle functor is Cartesianproduct with X . In particular, it follows that: (i) The base geometric morphism (Prop. 2.24) of the slice ∞ -topos H / X (Prop. 2.46) is given by (cid:0) ∆ (cid:97) Γ (cid:1) (cid:39) (cid:0) ( X ! ∗ ) ∗ (cid:97) ( X ! ∗ ) ∗ (cid:1) (82)(since ( X ! ∗ ) ∗ is a left adjoint that also preserves finite limits, as it is also a right adjoint, Prop. 2.26). (ii) The forgetful functor dom : H / X ! H is a left adjoint ( X ! ∗ ) ! and hence preserves all colimits (Prop. 2.26).While dom (81) does not preserve all limits, it does preserve fiber products: Proposition 2.53 (Fiber products in slice ∞ -toposes) . Let H be an ∞ -topos (Def. 2.30), B ∈ H , H / B the slice ∞ -topos (Prop. 2.46) and H / B dom (cid:47) (cid:47) X its forgetful functor (81) from Example 2.52. (i) Given a cospan ( Y , φ Y ) (cid:47) (cid:47) ( X , φ X ) (cid:111) (cid:111) ( Z , φ Z ) in H / B , the underlying object of its fiber product is the fiberproduct of its underlying objects: dom (cid:18) ( Y , φ Y ) × ( X , φ X ) ( Z , φ Z ) (cid:19) (cid:39) Y × X Z . (83) (ii) In particular, since ( X , id X ) is the terminal object in H / X , so that the plain product in the slice is ( Y , φ Y ) × ( Z , φ Z ) = ( Y , φ Y ) × ( X , id X ) ( Z , φ Z ) , we have the that product in H / X is given by the fiber product over X in H : dom (cid:16) ( Y , φ Y ) × ( Z , φ Z ) (cid:17) (cid:39) Y × X Z . Proof.
Generally, limits in H / X are given by limits in H over the underlying co-cone diagram. Specifically: for Y : I (cid:47) (cid:47) H we have dom (cid:0) lim − Y • (cid:1) (cid:39) lim − (cid:0) Y / X (cid:1) • . With this, the claim follows from the fact that the canonicalinclusion of diagram categories (cid:8) y (cid:47) (cid:47) b (cid:111) (cid:111) z (cid:9) (cid:31) (cid:127) (cid:47) (cid:47) (cid:40) t (cid:15) (cid:15) (cid:119) (cid:119) (cid:39) (cid:39) y (cid:47) (cid:47) b (cid:111) (cid:111) z (cid:41) is an initial functor (i.e., under ( − ) op it is a final functor). (cid:3) Proposition 2.54 (Terminal right base change of bare ∞ -groupoids) . In the base ∞ -topos H = Groupoids ∞ (35) ,the right base change along the terminal morphism (Example 2.52) of an object X ∈ Groupoids ∞ is given by thehom- ∞ -groupoid out of X , regarded as the terminal object in the slice: ( X ! ∗ ) ∗ (cid:39) H / X (cid:0) X , − (cid:1) : (cid:0) Groupoids ∞ (cid:1) / X (cid:47) (cid:47) Groupoids ∞ . roof. We have the following chain of natural equivalences:Groupoids ∞ (cid:0) A , ( Groupoids ∞ ) / X ( X , B ) (cid:1) (cid:39) ( Groupoids ∞ ) / X (cid:0) ∆ ( A ) × X X , B (cid:1) (cid:39) ( Groupoids ∞ ) / X (cid:0) ∆ ( A ) , B (cid:1) (cid:39) ( Groupoids ∞ ) / X (cid:0) ( X ! ∗ ) ∗ ( A ) , B (cid:1) . (84)Here the first step observes that the slice ( Groupoids ∞ ) / X is itself an ∞ -topos by Prop. 2.46, so that the tensoringequivalence of Prop. 2.34 applies. The second step uses the fact that X is regarded as the terminal object in itsown slice, so that forming Cartesian product with it is equivalently the identity operation. The last step observesthat for the slice ∞ -topos ∆ (cid:39) ( X ! ∗ ) ∗ (82) by Example 2.52. In summary, the total equivalence of (84) is thehom-equivalence that characterizes H / X ( X , − ) as a right adjoint to ( X ! ∗ ) ∗ . (cid:3) Proposition 2.55 (Base change along effective epi is conservative [NSS12a, 3.15] ) . Let H be an ∞ -topos (Def.2.30). For Y (cid:47) (cid:47) (cid:47) (cid:47) X an effective epimorphism (Def. 2.63) in H , the induced base change (Prop. 2.49) H / X i ∗ (cid:47) (cid:47) H / Y is a conservative ∞ -functor, meaning that a morphism f ∈ H / X is an equivalence if its base change i ∗ ( f ) in H / Y isan equivalence. Proposition 2.56 (Colimits of classifying maps are classifying maps of colimits) . Let H be an ∞ -topos (Def. 2.30), I a small ∞ -category, X • : I ! H a diagram and ( (cid:96) E ) • : X • ! const Objects κ a transformation to the diagramconstant on the object classifier (55) , thus classifying a diagram E • : I ! H of bundles over X • . Then the colimitof ( (cid:96) E ) • formed in the slice H / Objects κ (Prop. 2.46) is the colimit of X • equipped with the classifying map for thecolimit of E • : lim −! ( (cid:96) E ) • (cid:39) (cid:96) (cid:0) lim −! E • (cid:1) . Proof.
Since underlying the colimit lim −! ( (cid:96) E ) • in the slice ∞ -topos H / Objects κ is the colimit lim −! X • in H (by Example 2.52) weare dealing with a situation as shown in the diagram on the right(where a simplicial diagram shape is shown just for definitenessof illustration). We need to demonstrate that the front square inthis diagram is Cartesian. Observe that (a) the vertical squares over each (cid:96) E i are Cartesian by assump-tion, whence (b) also the solid vertical squares over each X i (cid:47) (cid:47) X j areCartesian, by the pasting law (Prop. 2.23).This means that the assumption of Prop. 2.32 is satisfied for theleft part of the diagram (regarded as a transformation of diagramsfrom top to bottom) implying that the dashed square is Cartesian.This implies, together with (a) , that the front square is Cartesian,again the pasting law (Prop. 2.23). (cid:3) (cid:4) (cid:4) (cid:68) (cid:68) (cid:4) (cid:4) (cid:68) (cid:68) (cid:4) (cid:4) E (cid:15) (cid:15) (cid:34) (cid:34) (cid:3) (cid:3) (cid:67) (cid:67) (cid:3) (cid:3) E (cid:15) (cid:15) (cid:43) (cid:43) (cid:4) (cid:4) (cid:92) Objects κ (cid:15) (cid:15) lim −! E • (cid:15) (cid:15) (cid:50) (cid:50) X (cid:96) E (cid:34) (cid:34) (cid:4) (cid:4) (cid:68) (cid:68) (cid:4) (cid:4) X (cid:96) E (cid:43) (cid:43) (cid:3) (cid:3) Objects κ lim −! X • lim −! ( (cid:96) E • ) (cid:50) (cid:50) n -Truncation.Definition 2.57 ( n -truncated objects [Lu09a, Def. 5.5.6.1]) . Let n ∈ {− , − , , , , · · · } . (i) An ∞ -groupoid is called n-truncated for n ≥ > n are trivial. It is called ( − ) -truncated if it is either empty or contractible, and (-2)-truncated if it is (non-empty and) contractible. (ii) Let C be an ∞ -category. Then an object X ∈ C is n-truncated if for all objects U ∈ C the hom- ∞ -groupoid C ( U , X ) is n -truncated, in the above sense. Definition 2.58 ( n -truncated morphisms [Lu09a, Def. 5.5.6.8]) . Let n ∈ {− , − , , , , · · · } . (i) A morphism of ∞ -groupoids is called n-truncated if all its homotopy fibers are n -truncated ∞ -groupoids accord-ing to Def. 2.57. 24 ii) Let C be an ∞ -category. A morphism X f −! Y in C is called n-truncated if for all objects U ∈ C the inducedmorphism of hom- ∞ -groupoids C ( U , X ) C ( U , f ) (cid:47) (cid:47) C ( U , Y ) is n -truncated in the above sense. Definition 2.59 (Monomorphisms) . A (-1)-truncated morphism (Def. 2.58) is also called a monomorphism , to bedenoted X (cid:31) (cid:127) (cid:47) (cid:47) Y . (85) Proposition 2.60 (Monomorphisms are preserved by pushout [Re19, p. 21]) . Let H be an ∞ -topos (Def. 2.30).Then the class of monomorphisms in H (Def. 2.59) is closed under (i) pullback and (ii) composition. Definition 2.61 (Poset of subobjects) . Let H be an ∞ -topos and X ∈ H any object. Then the poset of subobjects of X is the sub- ∞ -category (Def. 2.62) of ( − ) -truncated objects of the slice over X :SubObjects ( X ) (cid:31) (cid:127) (cid:47) (cid:47) H / X (86)whose objects are equivalently the monomorphisms (Def. 2.59) U (cid:44) ! X . Proposition 2.62 ( n -Trucation modality [Lu09a, 5.5.6.18]) . If H is an ∞ -topos (Def. 2.30), for all n ∈ {− , , , , · · · } ,its full sub- ∞ -category (Def. 2.1) of n-truncated objects (Def. 2.57) is reflective, in that the inclusion functor has aleft adjoint (Def. 2.24): H τ n ⊥ (cid:47) (cid:47) (cid:111) (cid:111) i n (cid:63) (cid:95) H n ∞ -topos sub- ∞ -categoryof n -truncated objects (87) We write for the induced n-truncation modality (20) : (cid:0) τττ n : = i n ◦ τ n “ n -truncated” (cid:1) : H −! H . (88) Definition 2.63 (Effective epimorphisms [Lu09a, Cor. 6.2.3.5]) . Let H be an ∞ -topos. A morphism in H is calledan effective epimorphism , to be denoted Y f (cid:47) (cid:47) (cid:47) (cid:47) Z (89)if, when regarded as an object of the slice over X (Prop. 2.46), its ( − ) -truncation (Prop. 2.62) is the terminalobject τ ( − ) ( f ) (cid:39) ∗ ∈ H / X . We write EffectiveEpimorphisms ( H ) ⊂ H ( ! ) ∈ Categories ∞ (90)for the full sub- ∞ -category (Def. 2.1) of the arrow-category of H on those that are effective epimorphisms. Definition 2.64 ( n -Connected morphisms [Lu09a, Prop. 6.5.1.12]) . Let H be an ∞ -topos (Def. 2.30) and n ∈{− , , , · · · } . Then a morphism X f (cid:47) (cid:47) Y in H is called n-connected if, regarded as an object in the slice over X (Prop. 2.46), its n -truncation (Def. 2.62) is the terminal object: Y f (cid:47) (cid:47) X is n -truncated ⇔ τ n ( f ) (cid:39) ∗ ∈ H / X . Hence the ( − ) -connected morphisms are equivalently the effective epimorphisms (Def. 2.63). Lemma 2.65 (Effective epimorphisms are preserved by pullback [Lu09a, 6.2.3.15]) . Let H be an ∞ -topos (Def.2.30). Then the class of effective epimorphisms in H (Def. 2.63) is closed under (i) pullback and (ii) composition. -Image factorization.Proposition 2.66 (Connected/truncated factorization system [Lu09a, Ex. 5.2.8.16][Re10, Prop. 5.8]) . Let H be an ∞ -topos. Then, for all n ∈ {− , , , , · · · } , the pair of classes of n-connected/n-truncated morphisms (Def. 2.64,Def. 2.58) forms an orthogonal factorization system: (i) every morphism f in H factors essentially uniquely asX n -connected (cid:41) (cid:41) f (cid:47) (cid:47) Y im n ( f ) n -truncated (cid:53) (cid:53) (91) (ii) every commuting square as follows has an essentially unique dashed lift:X (cid:47) (cid:47) n -connected (cid:15) (cid:15) A n -truncated (cid:15) (cid:15) Y (cid:54) (cid:54) (cid:47) (cid:47) B (92) Example 2.67 (Epi/mono factorization) . For n = −
1, the connected/truncated factorization system (Prop. 2.66)has as left class the effective epimorphisms (Def. 2.63) and as right class the monomorphisms (Def. 2.59). Hence,with the notation from (89) and (85): (i) the (-1)-image factorization (91) reads: X (cid:42) (cid:42) (cid:42) (cid:42) f (cid:47) (cid:47) Y im − ( f ) (cid:39) (cid:7) (cid:52) (cid:52) (93) (ii) the lifting property (92) for n = − X (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) A (cid:127) (cid:95) (cid:15) (cid:15) Y (cid:55) (cid:55) (cid:47) (cid:47) B (94) Groupoids and Stacks.Definition 2.68 (Groupoids internal to an ∞ -topos [Lu09a, 6.1.2.7]) . Let H be an ∞ -topos (Def. 2.30). (i) A groupoid in H is a simplicial diagram X • : ∆ op (cid:47) (cid:47) H (95)which satisfies the groupoidal Segal condition : For all n ∈ N and for all partitions of the set of n + X • is a Cartesian square (Notation 2.21): { , , · · · , n } S (cid:39) (cid:7) (cid:53) (cid:53) S (cid:55) (cid:87) (cid:106) (cid:106) ∗ (cid:106) (cid:106) (cid:52) (cid:52) (po) X • X n (cid:117) (cid:117) (cid:41) (cid:41) (pb) X | S |− (cid:41) (cid:41) X | S |− (cid:117) (cid:117) X (96) (ii) We write Groupoids ( H ) (cid:31) (cid:127) (cid:47) (cid:47) H ( ∆ op ) ∈ Categories ∞ (97)for the full sub- ∞ -category of that of simplicial diagrams in H on those that are groupoids. Example 2.69 (Nerves) . Let H be an ∞ -topos (Def. 2.30) and X f (cid:47) (cid:47) X a morphism in H . Its nerve is thesimplicial diagram of its iterated homotopy fiber products:Nerve • ( f ) : ∆ op (cid:47) (cid:47) H [ n ] X × X X × X · · · × X X (cid:124) (cid:123)(cid:122) (cid:125) n factors (98)with face maps the projections and degeneracy maps the diagonals. This is evidently a groupoid object accordingto Def. 2.68: Nerve • ( f ) ∈ Groupoids ( H ) . roposition 2.70 (Groupoids equivalent to stacks with atlases [Lu09a, 6.2.3.5]) . Let H be an ∞ -topos (Def. 2.30). Then the ∞ -functor sendingX • ∈ Groupoids ( H ) (Def. 2.68) to the X -component of its col-imiting cocone (i) lands in effective epimorphisms (90) and (ii) constitutes an equivalence of ∞ -categories whose inverseis given by the construction of nerves (Example 2.69): Groupoids ( H ) (cid:39) (cid:47) (cid:47) EffectiveEpimorphisms ( H ) X • (cid:0) X (cid:16) lim −! X • (cid:1) Nerve • ( a ) − (cid:91) (cid:0) X a (cid:16) X (cid:1) (99) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) X × X X (cid:39) pr (cid:15) (cid:15) (cid:79) (cid:79) ∆ pr (cid:15) (cid:15) X s (cid:15) (cid:15) (cid:79) (cid:79) e t (cid:15) (cid:15) “groupoid” X a (cid:15) (cid:15) (cid:15) (cid:15) X (cid:15) (cid:15) (cid:15) (cid:15) “atlas” X (cid:39) lim −! X • “stack” (100) Remark 2.71 (Internal groupoids with prescribed properties) . Often one considers X • ∈ Groupoids ( H ) (Def.2.68) whose simplicial component diagram (95) is inside a chosen sub- ∞ -category of H . Key examples are ´etalegroupoids (Def. 3.35 below) and V -´etale groupoids (Remark 4.15 below). Remark 2.72 (Morita morphisms of groupoids) . A morphism between stacks X : = lim −! X • underlying groupoids X • (according to Prop. 2.70) without (i.e., disregarding) the corresponding atlas is also known as a Morita mor-phism (in particular, a
Morita equivalence if it is an equivalence), or a
Hilsum-Skandalis morphism [HS87][Pr89],or a groupoid bibundle [Bl07][Nu13, Prop. 2.2.34] between the corresponding groupoids:Groupoids ( H ) (cid:39) (cid:47) (cid:47) EffectiveEpimorphisms ( H ) codom (cid:47) (cid:47) H groupoid X • ( X (cid:16) X ) X “stack” f morphism of underlying stacks =“Morita morphism” of groupoids (cid:15) (cid:15) groupoid Y • ( Y (cid:16) Y ) Y “stack” Hence whether or not there is a conceptual distinction between “geometric groupoids” and “stacks” depends onwhether morphisms of groupoids are taken to be their plain morphisms or their Morita morphisms. In practice, oneis typically interested in the latter case. Indeed, the groupoid atlas of a stack, whose preservation restricts Moritamorphisms to plain morphisms of groupoids, by Prop. 2.70, is, in practice, typically required to exist with a certainproperty, but not required to be preserved by morphisms (this is so notably for V -´etale groupoids, Remark 4.15below). In particular, the SmoothGroupoids ∞ of Example 3.18 and the JetsOfSmoothGroupoids ∞ of Example 3.24below are ∞ -groupoids with Morita morphisms understood, hence could also be called ( jets of ) smooth ∞ -stacks . Proposition 2.73 (Equifibered morphisms of groupoids) . Let H be an ∞ -topos (Def. 2.30) and X • , Y • ∈ Groupoids ( H ) (Def. 2.68). Then, under the equivalence (99) between groupoids and their stacks with atlases (Prop. 2.70), wehave that equifibered morphisms of groupoids correspond to Cartesian squares between their atlases:X • f • (cid:43) (cid:51) Y • such that ∀ [ n ] φ ! [ n ] X n f n (cid:47) (cid:47) X φ (cid:15) (cid:15) (pb) Y n Y φ (cid:15) (cid:15) X n f n (cid:47) (cid:47) Y n ⇔ X a X (cid:15) (cid:15) (cid:15) (cid:15) f (cid:47) (cid:47) (pb) Y a Y (cid:15) (cid:15) (cid:15) (cid:15) X lim −! f • (cid:47) (cid:47) Y Proof.
From right to left this follows by the pasting law (Prop. 2.23), while from left to right this is Prop. 2.32. (cid:3)
We discuss here the internal formulation in ∞ -toposes of the theory of groups , group actions , and fiber bundles ,following [NSS12a][SSS12] (see [FSS13a] for exposition). Externally, these concepts are known as grouplikeA ∞ -algebras or equivalently: grouplike E -algebras (here: in ∞ -stacks) and as their A ∞ -modules etc., and are27raditionally presented by simplicial techniques [May72][Lu17]. But internally the theory becomes finitary andelementary, with all concepts emerging naturally from pastings of a few Cartesian squares. Accordingly, much ofthe following constructions may readily be expressed fully formally in homotopy type theory [BvDR18] (see p.5). Thus, the following elegant characterizations of ◦ groups (Prop. 2.74), ◦ group actions (Prop. 2.79), ◦ principal bundles (Prop. 2.88), ◦ fiber bundles (Prop. 2.92),in an ∞ -topos H may be taken to be the definition of these notions for all purposes of internal constructions. Groups.
The following characterization of group ∞ -stacks (Prop. 2.74) is the time-honored May recognitiontheorem [May72] generalized from Groupoids ∞ to general ∞ -toposes [Lu09a, 7.2.2.11][Lu17, 6.2.6.15]: Proposition 2.74 (Groups [NSS12a, Thm. 2.19]) . Let H be an ∞ -topos (Def. 2.30). Then the operation of sendingan ∞ -group G to the homotopy quotient of its action on a point constitutes an equivalence of ∞ -categories: Groups (cid:0) H (cid:1) (cid:111) (cid:111) Ω B (cid:39) (cid:47) (cid:47) H ∗ / ≥ G (cid:31) (cid:47) (cid:47) ∗ (cid:12) G (101) between the ∞ -category of ∞ -group objects and the ∞ -category of pointed and connected objects in H . The inverseequivalence is given by forming the loop space objectG (cid:39) Ω B G (cid:15) (cid:15) (cid:47) (cid:47) (pb) ∗ (cid:15) (cid:15) ∗ (cid:47) (cid:47) B G (102) Example 2.75 (Point in delooping is an effective epi) . For G ∈ Groups ( H ) , the essentially unique morphism thatexhibits its delooping as a pointed object (Prop. 2.74) ∗ (cid:47) (cid:47) (cid:47) (cid:47) B G , (103)is an effective epimorphism (Def. 2.62). Thus, Prop. 2.70 says here that (i) groups in H are, equivalently, the groupoids in H (Def. 2.68) that admit an atlas by the point and, (ii) with (102), we have B G (cid:39) lim −! G × • ∈ H . (104) Example 2.76 (Neutral element) . Let H be an ∞ -topos. Given a group G ∈ Groups ( H ) in the form of a pointed connected object ∗ ! B G , according toProp. 2.74, its neutral element ∗ e −! G is the diagonal morphism into thedefining homotopy fiber product (102), hence the canonical morphism inducedby the universal property of the homotopy fiber product from the equivalencewith itself of the point inclusion into B G (103). ∗ e (cid:15) (cid:15) G (cid:115) (cid:115) (cid:43) (cid:43) (pb) ∗ (cid:43) (cid:43) ∗ (cid:115) (cid:115) B G Example 2.77 (Group division/shear map) . Let H be an ∞ -topos. Given a group G ∈ Groups ( H ) in the form of apointed connected object ∗ −! B G , according to Prop. 2.74, the group division operation G × G ( − ) · ( − ) − (cid:47) (cid:47) G is exhibited by the universal morphism shown dashed in the following diagram: G × G (cid:15) (cid:15) (cid:15) (cid:15) ( − ) · ( − ) − (cid:47) (cid:47) G (cid:15) (cid:15) (cid:15) (cid:15) G (cid:15) (cid:15) (cid:47) (cid:47) ∗ (cid:15) (cid:15) ∗ (cid:47) (cid:47) B G G × G (cid:0) (cid:0) (cid:30) (cid:30) ( − ) · ( − ) − (cid:44) (cid:44) G (cid:2) (cid:2) (cid:28) (cid:28) G (cid:32) (cid:32) (cid:44) (cid:44) G (cid:126) (cid:126) (cid:44) (cid:44) ∗ (cid:29) (cid:29) ∗ (cid:1) (cid:1) ∗ (cid:44) (cid:44) B G (cid:17) (cid:25) (cid:113) (cid:121) (cid:5) (cid:13) (105)28n the left, we are showing this as part of a morphism of ˇCech nerve augmented simplicial diagrams. On the right,the situation is shown in more detail: Here the right and the two bottom squares are all the looping relation (102),while the left square exhibits the plain product of G with itself. With this, the universal property of the right squareimplies the essentially unique dashed morphism making the total diagram homotopy-commute. Notice: (i) The two top squares are also Cartesian: This follows from the pasting law (Prop. 2.23) using, for the top frontsquare, that the left and right and the bottom rear squares are Cartesian; and similarly for the top rear square. (ii)
The total homotopy filling the top and the right faces in (105) is, by commutativity, equivalent to the totalhomotopy filling the left and the bottom faces. But, in performing the composition this way, the direction of one ofthe two bottom homotopies gets reversed. This is why this construction gives the division map ( − ) · ( − ) − (shearmap) instead of the plain group product. Proposition 2.78 (Mayer-Vietoris sequence [Sc13, Prop. 3.6.142]) . Let H be an ∞ -topos (Def. 2.30), G ∈ Groups ( H ) (Prop. 2.74) and ( X , f ) , ( Y , g ) ∈ H / G two objects in the slice (Prop. 2.46) over the underlying ob-ject of G. Then their homotopy fiber product X × G Y pr X (cid:15) (cid:15) pr Y (cid:47) (cid:47) (pb) Y g (cid:15) (cid:15) X f (cid:47) (cid:47) Gis equivalently exhibited by the following Mayer-Vietoris homotopy fiber sequenceX × G Y ( pr X , pr Y ) (cid:15) (cid:15) (pb) (cid:47) (cid:47) ∗ (cid:15) (cid:15) X × Y ( f , g ) (cid:47) (cid:47) f · g − (cid:52) (cid:52) G × G ( − ) · ( − ) − (cid:47) (cid:47) G , (106) where the morphism on the bottom right is the group division map (105) . Group actions.Proposition 2.79 (Group actions [NSS12a, 4.1]) . Let H an ∞ -topos and G ∈ Groups ( H ) (Prop. 2.74). (i) An action ( X , ρ ) of G is an object X ∈ H and homotopy fiber sequence in H of the formX fib ( ρ ) (cid:47) (cid:47) X (cid:12) G ρ (cid:15) (cid:15) B G , (107) where B G is the delooping of G (2.74) . (ii) The object X (cid:12)
G appearing in (107) is, equivalently, the homotopy quotient of the action of G on V :X (cid:12) G (cid:39) lim −! (cid:16) ··· X × G × G (cid:47) (cid:47) (cid:111) (cid:111) (cid:47) (cid:47) (cid:111) (cid:111) (cid:47) (cid:47) X × G (cid:47) (cid:47) (cid:111) (cid:111) (cid:47) (cid:47) X (cid:17) . (108) (iii) Hence the ∞ -category of G-actions is, equivalently, the slice ∞ -topos (Prop. 2.46) of H over B G:G
Actions ( H ) (cid:39) H / B G ∈ Categories ∞ . (109)We record the following immediate but important aspect of this characterization: Lemma 2.80 (Homotopy quotient maps are effective epimorphisms) . Let H be an ∞ -topos, G ∈ Groups ( H ) (Prop.2.74), and ( X , ρ ) ∈ G Actions ( H ) (Prop. 2.79). Then the quotient morphism from X to its homotopy quotient (108) is an effective epimorphism (Def. 2.63): X fib ( ρ ) (cid:47) (cid:47) (cid:47) (cid:47) X (cid:12) G . roof. By (107) in Prop. 2.79, the quotient map sits in a homotopy pullback square of the form X (cid:15) (cid:15) fib ( ρ ) (cid:47) (cid:47) (pb) X (cid:12) G ρ (cid:15) (cid:15) ∗ (cid:47) (cid:47) B G The bottom morphism is an effective epimorphism (Example 2.75). Since these are preserved by pullback (Lemma2.65), the claim follows. (cid:3)
Example 2.81 (Left multiplication action) . Let H be an ∞ -topos (Def. 2.30) and G ∈ Groups ( H ) (Prop. 2.74).The defining looping relation (102) exhibits, by comparison with (107), an action of G on itself: G fib ( ρ (cid:96) ) (cid:47) (cid:47) ∗ ρ (cid:96) (cid:15) (cid:15) B G This is the left multiplication action with G (cid:12) G (cid:39) ∗ . Example 2.82 (Adjoint action) . Let H be an ∞ -topos (Def. 2.30) and G ∈ Groups ( H ) (Prop. 2.74). Then the freeloop space object L B G of the delooping B G (101), defined by the Cartesian square L B G (cid:47) (cid:47) ρ ad (cid:15) (cid:15) (pb) B G ∆ (cid:15) (cid:15) B G ∆ (cid:47) (cid:47) B G × B G sits in a homotopy fiber sequence of the form G fib ( ρ ad ) (cid:47) (cid:47) L B G ρ ad (cid:15) (cid:15) B G By comparison with (107), this exhibits an action of G on itself. This is the adjoint action with G (cid:12) ad G (cid:39) L B G . Definition 2.83 (Equivariant maps) . By the functoriality/universality of the homotopy fiber construction in (107)and using the equivalence (109), we have the ∞ -functor that assigns the underlying objects of the G -actions in Def.2.79: G Actions ( H ) (cid:39) H / B G fib (cid:47) (cid:47) H . (110)With two G -actions ( X i , ρ i ) given, we say that a morphism X ! X ∈ H between their underlying objects is equiv-ariant if it lifts through this functor, hence if it is the image of a morphism ( X , ρ ) ! ( X , ρ ) ∈ GActions ( H ) . Example 2.84 (Group division is equivariant under diagonal left and adjoint action) . Let H be an ∞ -topos (Def.2.30) and G ∈ Groups ( H ) (Prop. 2.74). Then the group division operation (Example 2.77) is equivariant (Def.2.83) with respect to the diagonal left multiplication action ρ (cid:96) (Example 2.81) on its domain and the adjoint action ρ ad (Example 2.82) on its codomain: ( G , ρ (cid:96) ) × ( G , ρ (cid:96) ) ( − ) · ( − ) − (cid:47) (cid:47) ( G , ρ ad ) ∈ G Actions ( H ) . (111) Proof.
Observe the following pasting of Cartesian squares: G × G ( − ) · ( − ) − (cid:47) (cid:47) ( − ) − · ( − ) ◦ σ (cid:15) (cid:15) G (cid:47) (cid:47) (cid:15) (cid:15) ∗ (cid:15) (cid:15) G (cid:15) (cid:15) (cid:47) (cid:47) L B G (cid:47) (cid:47) (cid:15) (cid:15) B G ∆ (cid:15) (cid:15) ∗ (cid:47) (cid:47) B G ∆ (cid:47) (cid:47) B G × B G The middle horizontal composite, regarded as a morphism in the slice over B G and hence as a morphism of G -actions (107), gives (111). (cid:3) roposition 2.85 (Restricted and induced group actions) . Let H be an ∞ -topos. Then, for φ : H ! G a morphism in
Groups ( H ) (Prop. 2.74), there is a triple of adjoint ∞ -functors (Def. 2.24) between the corresponding ∞ -categoriesof group actions (Prop. 2.79) H Actions ( H ) “left-induced” B φ ! (cid:47) (cid:47) (cid:111) (cid:111) B φ ∗ ⊥⊥ B φ ∗ “right-induced” (cid:47) (cid:47) G Actions ( H ) (112) such that B φ ∗ preserves the object being acted on (“restricted action”).Proof. By (109) in Prop. 2.79, an adjoint triple (Def. 2.24) of the form (112) is given by base change (Prop.2.49) of homotopy quotients (108) along the delooped morphism B φ (Prop. 2.74). This means that B φ ∗ is givenby sending the homotopy fiber sequence (107) corresponding to a G -action to the following homotopy pullback(Prop. 2.74): X fib ( φ ∗ ρ ) (cid:47) (cid:47) fib ( φ ) (cid:44) (cid:44) X (cid:12) H (pb) φ ∗ ρ (cid:15) (cid:15) (cid:47) (cid:47) X (cid:12) G ρ (cid:15) (cid:15) B H B φ (cid:47) (cid:47) B G (113)That this preserves the object X being acted on, as indicated, follows by the pasting law (Prop. 2.23). (cid:3) Definition 2.86 (Automorphism group) . Let H be an ∞ -topos and F ∈ H an object. Then the automorphism group Aut ( F ) ∈ Groups ( H ) of F is the looping (Prop. 2.74) of the (-1)-image (91) of the classifying map (55) of F : ∗ (-1)-conn. (cid:47) (cid:47) (cid:47) (cid:47) (cid:96) F (cid:50) (cid:50) B Aut ( F ) (cid:31) (cid:127) (-1)-trunc. (cid:47) (cid:47) Objects κ . The canonical action of this group (Prop. 2.79) on V is exhibited, via (107), by the left square of the followingpasting composite of Cartesian squares: F (cid:15) (cid:15) fib ( ρ Aut ) (cid:47) (cid:47) (pb) F (cid:12) Aut ( F ) (cid:47) (cid:47) ρ Aut (cid:15) (cid:15) (pb) (cid:92)
Objects κ (cid:15) (cid:15) ∗ (cid:96) F (cid:51) (cid:51) (cid:47) (cid:47) (cid:47) (cid:47) B Aut ( F ) (cid:31) (cid:127) (cid:47) Objects κ , (114)where we use the pasting law (Prop. 2.23) to identify F as the homotopy fiber of ρ Aut . Proposition 2.87 (Automorphism group is universal) . Let H be an ∞ -topos, G ∈ Groups ( H ) (Prop. 2.74), and ( X , ρ ) ∈ G Actions ( H ) (Def. 2.79). Then there is a group homomorphism from G to the automorphism group (Def.2.86) G i ρ (cid:47) (cid:47) Aut ( X ) such that the action ρ is the restricted action (Prop. 2.85) along i ρ of the canonical automorphism action (114) ,i.e., such that there is a Cartesian square of this form:X (cid:12) G (cid:47) (cid:47) (pb) ρ (cid:15) (cid:15) X (cid:12) Aut ( X ) ρ Aut (cid:15) (cid:15) B G B i ρ (cid:47) (cid:47) B Aut ( X ) roof. Let κ be a regular cardinal such that X is κ -small, and consider the following solid diagram of classifyingmaps (55) for ρ , ρ Aut and for X : X (cid:47) (cid:47) (cid:36) (cid:36) (cid:15) (cid:15) X (cid:12) Aut ( X ) (cid:40) (cid:40) (cid:15) (cid:15) X (cid:12) G (cid:15) (cid:15) (cid:47) (cid:47) (cid:54) (cid:54) (cid:92) Objects κ (cid:15) (cid:15) ∗ (-1)-connected (cid:37) (cid:37) (cid:47) (cid:47) B Aut ( X ) (cid:22) (cid:118) (-1)-truncated (cid:41) (cid:41) B G B i ρ (cid:54) (cid:54) (cid:96) ρ (cid:47) (cid:47) Objects κ Here the bottom square homotopy-commutes by the essential uniqueness of the classifying map (cid:96) X (55). Hencethe dashed lift exists essentially uniquely (92), by the connected/truncated factorization system (Prop. 2.66). (cid:3) Principal bundles.Proposition 2.88 (Principal bundles [NSS12a, Thm. 3.17]) . Let H be an ∞ -topos, X ∈ H , and G ∈ Groups ( H ) (Prop. 2.74). Then G-principal ∞ -bundles P ! X over X are, equivalently, given by classifying maps (cid:96) P : X ! B G.Forming their homotopy fibers P fib ( (cid:96) P ) (cid:15) (cid:15) X (cid:96) P (cid:47) (cid:47) B Gconstitutes an equivalence of ∞ -groupoids:G Bundles X ( H ) (cid:111) (cid:111) fib (cid:39) H ( X , B G ) . P (cid:96) P (115) Remark 2.89 (Principal base spaces are homotopy quotients) . Comparison of the abstract characterization of (i) group actions (Prop. 2.79) and (ii) principal bundles (Prop. 2.88), reveals that these are about one and the sameabstract concept, just viewed from two different perspectives: In an ∞ -topos, every G -principal bundle is a G -actionwhose homotopy quotient is the given base space; and, conversely, every G -action is that of a principal bundle overits homotopy quotient: principalG-bundle P (cid:121) G (cid:15) (cid:15) G-actionbasespace X (cid:39) P (cid:12) G homotopyquotient Notice (see [NSS12a, 3.1] for exposition) that it is the higher geometry inside an ∞ -topos that makes this work. Definition 2.90 (Atiyah groupoid) . Let H be an ∞ -topos (Def. 2.30), X ∈ H , G ∈ Groups ( H ) (Prop. 2.74), and P ∈ G Bundles X (Prop. 2.88). Then the Atiyah groupoid of P is the groupoid At • ( P ) ∈ Groupoids ( H ) (Def. 2.68)whose corresponding stack with atlas (via Prop. 3.36) is the (-1)-image projection (Example 2.67) of the bundle’sclassifying map (cid:96) P (115): X (cid:47) (cid:47) (cid:47) (cid:47) (cid:96) P (cid:52) (cid:52) A t ( P ) (cid:31) (cid:127) (cid:47) (cid:47) B G . (116) Fiber bundles.Definition 2.91 (Fiber bundle) . Let H be an ∞ -topos (Def. 2.30). (i) morphism Y p −! X in H is a fiber bundle with typical fiber F ∈ H if there exists an effective epimorphism U i (cid:47) (cid:47) (cid:47) (cid:47) X (Def. 2.63) and a Cartesian square (Notation 2.21) of the form U × F (cid:15) (cid:15) (cid:47) (cid:47) (pb) Y p (cid:15) (cid:15) U i (cid:47) (cid:47) (cid:47) (cid:47) X ii) We write F FiberBundles X ( H ) ⊂ Core (cid:0) H / X (cid:1) ∈ Groupoids ∞ for the full ∞ -groupoid of the core (24) of the slice H / X over X (Prop. 2.46) on the F -fiber bundles. Proposition 2.92 (Classification of fiber bundles [NSS12a, Prop. 4.10]) . Let H be an ∞ -topos (Def. 2.30) andX , F ∈ H . Then fiber bundles over X (Def. 2.91) with typical fiber F are equivalent to morphisms X −! B Aut ( F ) from X to the delooping (Prop. 2.74) of the automorphism group (Def. 2.86) of F:F FiberBundles X ( H ) (cid:39) (cid:47) (cid:47) H (cid:0) X , B Aut ( F ) (cid:1) E (cid:96) E (117) Proof.
Let κ be a regular cardinal such that F is κ -small. Then, by assumption, we have the following soliddiagram of classifying maps (55): U × F (cid:15) (cid:15) pr (cid:47) (cid:47) (cid:41) (cid:41) F (cid:12) Aut ( F ) (cid:15) (cid:15) (cid:42) (cid:42) E (cid:15) (cid:15) (cid:47) (cid:47) (cid:92) Objects κ (cid:15) (cid:15) U (cid:47) (cid:47) (-1)-connected (cid:41) (cid:41) (cid:41) (cid:41) B Aut ( F ) (cid:24) (cid:120) (-1)-truncated (cid:43) (cid:43) X (cid:47) (cid:47) (cid:96) E (cid:52) (cid:52) Objects κ Now the (-1)-connected/(-1)-truncated factorization system (Prop. 2.66) implies that the dashed morphism existsessentially uniquely (92).It just remains to see that this assignment is independent of the choice of U : For U (cid:48) (cid:47) (cid:47) (cid:47) (cid:47) X any other effectiveepimorphism with ( (cid:96) E ) (cid:48) the associated classifying map as above, observe that the fiber product U × X U (cid:48) (cid:47) (cid:47) (cid:47) (cid:47) X isagain an effective epimorphism, since the class of effective epimorphisms is closed under pullbacks as well as undercomposition (Lemma 2.65). Therefore (cid:96) E and ( (cid:96) E ) (cid:48) are jointly lifts in a diagram as above but with U × X U (cid:48) inthe top left. Hence, by the essential uniqueness of lifts in the connected/truncated orthogonal factorization system,they are equivalent, ( (cid:96) E ) (cid:39) ( (cid:96) E ) (cid:48) , in an essentially unique way. (cid:3) Notation 2.93 (Associated bundles) . We say that (i) the morphism (cid:96) E in (117) is the classifying map of E and (ii) that E is associated to the Aut ( F ) -principal bundle which is classified by (cid:96) E according to Prop. 2.88. Remark 2.94 (Twisted cohomology in slice ∞ -toposes) . Prop. 2.92 implies (together with the universal propertyof the pullback) that sections σ of A -fiber bundles E over some X are, equivalently, lifts c of the classifying map c : = (cid:96) E (117) through ρ Aut (114): A (cid:12) Aut ( A ) ρ Aut (cid:15) (cid:15) X τ : = (cid:96) E classifying map (cid:47) (cid:47) lift ofclassifying map c (cid:55) (cid:55) BAut ( A ) (cid:39) associated bundle E p (cid:15) (cid:15) (cid:47) (cid:47) (pb) A (cid:12) Aut ( A ) ρ Aut (cid:15) (cid:15) X section σ (cid:63) (cid:63) X τ (cid:47) (cid:47) BAut ( A ) (118) (i) If A is regarded here as a coefficient object for A -cohomology (22), then such a section σ is a locally A -valuedcocycle, which is “twisted” over X according to the classifying map τ . Hence such a σ is a cocycle in (non-abelian) τ -twisted cohomology [NSS12a, 4.2]. But the left hand side of (118) is, equivalently, a morphism (73) inthe slice ∞ -topos (Prop. 2.46) H / BAut ( A ) . It follows that twisted cohomology is the intrinsic cohomology (22) ofslice ∞ -toposes : 33 -twistedcohomology H τ (cid:0) X , A (cid:1) : = π H / BAut ( A ) (cid:16) ( X , τ ) , ( A (cid:12) Aut ( A ) , ρ Aut ) (cid:17) (cid:39) X τ (cid:32) (cid:32) cocycle c (cid:47) (cid:47) A (cid:12) Aut ( A ) ρ Aut (cid:121) (cid:121)
BAut ( A ) (cid:112) (cid:120) (cid:14) ∼ (119) (ii) By the universality of
Aut ( A ) (Prop. 2.87), this holds for slicing over any pointed connected object B G (101). (iii) If the base object is not connected, the intrinsic cohomology of its slice may be thought of as a mixture oftwisted and parametrized cohomology. We encounter an example of this in Def. 5.11 below.
Remark 2.95 (Twisted cohomology as global sections) . The ∞ -groupoid of sections of the associated bundle E : = τ ∗ ( A (cid:12) G ) p (cid:47) (cid:47) X in (118), is equivalently its image Γ X ( E ) under the base geometric morphism (Prop. 2.43) H / X (cid:111) (cid:111) ∆ X Γ X ⊥ (cid:47) (cid:47) Groupoids ∞ of the slice ∞ -topos H X (Prop. 2.46), in that (by Prop. 2.34) Γ X ( E ) (cid:39) H X (cid:0) id X , p (cid:1) . Hence the τ -twisted coho-mology (119) of X is equivalently the set of connected components of the ∞ -groupoid of global sections: H τ (cid:0) X ; A (cid:1) (cid:39) π Γ X (cid:0) τ ∗ ( A (cid:12) G ) (cid:1) . (120) Remark 2.96 (Twisted abelian cohomology in tangent ∞ -toposes) . Let H be an ∞ -topos (Def. 2.30). (i) Notice that the intrinsic cohomology (22) of Bundles ( H ) (Example 2.50) is still twisted cohomology as inRemark 2.94, just up to a change in perspective: now the twisting τ is encoded not in the domain object, but in thecocycles on these (a morphism of the form id X (cid:47) (cid:47) ρ Aut in Bundles ( H ) is still manifestly given by the diagrams in(118)). (ii) Therefore, similarly, the intrinsic cohomology (22) in the tangent ∞ -topos SpectralBundles ( H ) (Example 2.51)is twisted cohomology with local coefficients being spectra [Sc13, 4.1][ABGHR14][GS19a][GS19b], hence is twisted abelian cohomology . (iii) In the case that H = Groupoids ∞ , the base tangent ∞ -topos T Groupoids ∞ = SpectralBundles (cid:0)
Groupoids ∞ (cid:1) (121)is the topic of traditional parametrized stable homotopy theory [Jam95][MSi06][ABGHR14, 2][BM19] and itsintrinsic cohomology theory (22) is traditional twisted generalized cohomology [Do05][ABG10]. Fixed points and fixed loci.Definition 2.97 (Fixed points and fixed loci) . Let H be an ∞ -topos, G ∈ Groups ( H ) (Prop. 2.74) and ( X , ρ ) ∈ G Actions ( H ) (Prop. 2.79). (i) A fixed point of ( X , ρ ) is an element ∗ x (cid:47) (cid:47) X induced from a section x (cid:12) G of ρ in(107), as shown on the right (where we are usingthe pasting law, Prop. 2.23, and Example 2.22 toidentify the top square as Cartesian). ∗ (cid:47) (cid:47) x (cid:15) (cid:15) (pb) B G x (cid:12) G (cid:15) (cid:15) X fib ( ρ ) (cid:47) (cid:47) (cid:15) (cid:15) (pb) X (cid:12) G ρ (cid:15) (cid:15) ∗ (cid:47) (cid:47) B G , (122) (ii) The
G-fixed locus of ( X , ρ ) is the object X G : = B ( G ! ∗ ) ∗ (cid:0) ( X , ρ ) (cid:1) ∈ ( H ) (cid:39) H , (123)that is right induced (Prop. 2.85) along the unique morphism to the trivial group.34 xample 2.98 (Global points of fixed loci are homotopy fixed points) . The global points of a homotopy-fixed locus X G (123) are indeed, equivalently, the fixed points (122). By the adjunction (112), we have the hom-equivalence(47) (cid:0) ∗ (cid:47) (cid:47) X G = B ( G ! ) ∗ ( X , ρ ) (cid:1) ↔ (cid:0) B ( G ! ) ∗ ( ∗ ) (cid:47) (cid:47) ( X , ρ ) (cid:1) and, by Prop. 2.79, the latter morphisms are equivalent to homotopy-commuting diagrams of the form B G B ( G ! ) ∗ ( ∗ ) (cid:39) (cid:38) (cid:38) x (cid:12) G (cid:47) (cid:47) X (cid:12) G ρ (cid:120) (cid:120) B G This is just the type of diagram characterizing homotopy fixed points. as seen vertically on the right in (122).
Example 2.99 (Fixed loci in ∞ -groupoids) . Consider H : = Groupoids ∞ , G ∈ Groups ( Groupoids ∞ ) and ( X , ρ ) ∈ G Actions (cid:0)
Groupoids ∞ (cid:1) . Then the G -fixed locus (Def. 2.97) is given (due to Prop. 2.54) by X G (cid:39) H / ∗ (cid:12) G (cid:0) ∗ (cid:12) G , X (cid:12) G (cid:1) ∈ Groupoids ∞ . Definition 2.100 (Pointed-automorphism group) . Let H be an ∞ -topos and ∗ x (cid:47) (cid:47) X ∈ H ∆ a pointed object in H . Then its pointed-automorphism group Aut ∗ ( X ) ∈ Groups ( H ) is its automorphism group, according to Def.2.86, formed in the arrow ∞ -topos H ∆ . This is characterized by a diagram in H of the form ∗ (cid:33) (cid:33) (cid:47) (cid:47) (cid:15) (cid:15) ∗ (cid:12) Aut ∗ ( X ) (cid:41) (cid:41) X (cid:47) (cid:47) (cid:15) (cid:15) X (cid:12) Aut ∗ ( X ) ρ Aut ∗ (cid:15) (cid:15) ∗ (cid:33) (cid:33) (cid:47) (cid:47) B Aut ∗ ( X ) ∗ (cid:47) (cid:47) B Aut ∗ ( X ) (124)where the front, rear, top and bottom squares are Cartesian: the bottom face trivially, the front face exhibiting theaction on X , the top face exhibiting the given base point as a homotopy fixed point (Def. 2.97) and the rear squareexhibiting the trivial action on that point. Definition 2.101 (Group-automorphism group) . Let H be an ∞ -topos and G ∈ Groups ( H ) (Prop. 2.74). Thenthe group of group-automorphisms of G is the group of pointed-automorphisms (Def. 2.100) of its delooping B G (101): Aut Grp ( G ) : = Aut ∗ ( B G ) ∈ Groups ( H ) . Proposition 2.102 (Canonical action of group-automorphism group) . Let H be an ∞ -topos and G ∈ Groups ( H ) (Prop. 2.74). The group-automorphism group of G (Def. 2.101) has a canonical action (Prop. 2.79) ( G , ρ Aut
Grp ) ∈ Aut
Grp ( G ) Actions ( H ) on the underlying object G ∈ H , which is such that (i) The neutral element ∗ e (cid:47) (cid:47) G (Example 2.76) is a fixed point of the action (Def. 2.97). (ii)
Together with the defining action on the delooping B G of G (101) , the looping equivalence (102) G (cid:39) (cid:47) (cid:47) Ω B Gis
Aut
Grp ( G ) -equivariant (Def. 2.83).Proof. First consider item (ii) : Write G (cid:12) Aut
Grp ( G ) for the homotopy fiber product in the following pullbacksquare G (cid:12) Aut
Grp ( G ) (cid:47) (cid:47) (cid:15) (cid:15) (pb) ∗ (cid:12) Aut
Grp ( G ) (cid:15) (cid:15) ∗ (cid:12) Aut
Grp ( G ) (cid:47) (cid:47) ( B G ) (cid:12) Aut
Grp ( G ) (125)35e need to show that this really is the homotopy quotient of the canonical group-automorphism action with theclaimed property, in that it makes the total solid rear rectangle of the following diagram be Cartesian: ∗ (cid:47) (cid:47) e (cid:15) (cid:15) ∗ (cid:12) Aut
Grp ( G ) ( id , id ) (cid:15) (cid:15) G (cid:33) (cid:33) (cid:47) (cid:47) (cid:15) (cid:15) G (cid:12) Aut
Grp ( G ) (cid:42) (cid:42) (cid:15) (cid:15) ∗ (cid:47) (cid:47) (cid:15) (cid:15) ∗ (cid:12) Aut
Grp ( G ) (cid:15) (cid:15) ∗ (cid:33) (cid:33) (cid:47) (cid:47) (cid:15) (cid:15) ∗ (cid:12) Aut
Grp ( G ) (cid:42) (cid:42) B G (cid:47) (cid:47) (cid:15) (cid:15) ( B G ) (cid:12) Aut
Grp ( B G ) ρ ∗ (cid:15) (cid:15) ∗ (cid:33) (cid:33) (cid:47) (cid:47) B Aut
Grp ( G ) ∗ (cid:47) (cid:47) B Aut
Grp ( G ) (126)Here:• the bottom part is the diagram (124) (for X = B G ) which exhibits the pointed-automorphism action on B G ;• the top front square is Cartesian and exhibits the base point being a homotopy-fixed point; as in (124),• the top left square is Cartesian and exhibits the looping/delooping relation (102);• the top right square is (125) and this Cartesian by definition.Hence the solid top rear square and thus the total solid rear square are Cartesian, by the pasting law (Prop. 2.23).Finally to see item (i) : Observe that there is the dashed morphism shown in the top right of (126), this beingthe diagonal morphism induced from the Cartesian property of the top right square, by the above. This means, byconstruction, that the total vertical morphism on the right is an equivalence. Now define the dashed top square tobe a pullback square. Then, by the pasting law (Prop. 2.23), the pullback object in the top left of the dashed squareis equivalently the pullback of the total rear diagram, hence the pullback of an equivalence to a point, hence is itselfequivalent to the point, as shown. Since the point is terminal, the top left dashed morphism is thus a cone over theCartesian square on the top left. By the universal property of the homotopy fiber product, this means that the topleft dashed morphism must be the neutral element (Example 2.76). The top dashed square hence exhibits this as ahomotopy fixed point. (cid:3) Proposition 2.103 (Group disivion is equivariant under group-automorphisms) . Let H be an ∞ -topos and G ∈ Groups ( H ) (Prop. 2.74). Then the group division morphism G × G ( − ) · ( − ) − (cid:47) (cid:47) G (Example 2.77) is equivariant(Def. 2.83) with respect to the canonical group-automorphism action (Prop. 2.102) of the group-automorphismgroup
Aut
Grp ( G ) (Def. 2.101) acting on all three copies of G: ( G , ρ Aut
Grp ) × ( G , ρ Aut
Grp ) ( − ) · ( − ) − (cid:47) (cid:47) ( G , ρ Aut
Grp ) ∈ Aut
Grp ( G ) Actions ( H ) . Proof.
By (105) the group division morphism is a universal morphism induced from pasting of copies of thelooping square (102). Thus the claim follows by Prop. 2.102. (cid:3) Singular geometry
Here we establish foundations of a geometric homotopy theory of orbifolds which unifies: (i) §3.1 – the cohesive geometric homotopy theory due to [SSS12][Sc19], which reflects the geometric aspects of orbifolds; (ii) §3.2 – the cohesive global-equivariant homotopy theory due to [Re14], understood as reflecting the singularaspects of orbifolds, as in Figure D.
We present axioms internal to ∞ -toposes for- §3.1.1 – Differential topology- §3.1.2 – Differential geometry- §3.1.3 – Super-geometryThis is to provide, in §4 below, a general abstract theory of geometric aspects of orbi-singular spaces and of ´etale ∞ -stacks. We present a formulation of differential topology internal to ∞ -toposes which we call cohesive [Sc13]. In ∞ -categorical generalization of [La94][La07], this involves an abstract shape operation S that relates higher geometricspaces to their bare underlying homotopy type. Definition 3.1 (Cohesive ∞ -topos) . (i) An ∞ -topos H (Def. 2.30) is called cohesive if its base geometric morphism(Prop. 2.43), to be denoted Pnts : H (cid:47) (cid:47) Groupoids ∞ , is part of an adjoint quadruple of ∞ -functors (Def. 2.24) H × “shape” Shp ⊥ (cid:47) (cid:47) (cid:111) (cid:111) “discrete” Disc ⊥ (cid:63) (cid:95) “points” Pnts ⊥ (cid:47) (cid:47) (cid:111) (cid:111) “chaotic” Chtc (cid:63) (cid:95) B cohesive ∞ -topos discretesub-topos (127)such that (a) Disc and Chtc are fully faithful (Def. 2.1), and (b) such that Shp preserves finite products. (ii)
We write (cid:0) S : = Disc ◦ Shp (cid:1) “shape” (cid:97) (cid:0) (cid:91) : = Disc ◦ Pnts (cid:1) “discrete” (cid:97) (cid:0) (cid:93) : = Chtc ◦ Pnts (cid:1) “continuous” : H −! H (128)for the induced adjoint triple (Def. 2.24) of modalities (20) ( cohesive modalities ).The following direct consequence may serve to illustrate how these axioms are put to work: Proposition 3.2 (Composite cohesive modalities) . The cohesive modalities (Def. 3.1) satisfy: S ◦ (cid:91) (cid:39) (cid:91) and (cid:91) ◦ (cid:93) (cid:39) (cid:91) . Proof.
That Disc and Chtc in (127) are fully faithful means, equivalently (Prop. 2.28), that the co-unit morphisms(49) Shp ◦ Disc (cid:39) (cid:47) (cid:47) id , Pnts ◦ Chtc (cid:39) (cid:47) (cid:47) idare natural equivalences. Hence the image under Disc ◦ ( − ) ◦ Pnts of the first of these is a natural equivalence ofthe form S ◦ (cid:91) = Disc ◦ Shp ◦ Disc ◦ Pnts (cid:39) (cid:47) (cid:47)
Disc ◦ Pnts = (cid:91) . while the image of the second is of the form (cid:91) ◦ (cid:93) = Disc ◦ Pnts ◦ Chtc ◦ Pnts (cid:39) (cid:47) (cid:47)
Disc ◦ Pnts = (cid:91) . (cid:3) emma 3.3 (Only the empty object has empty shape) . Let H be a cohesive ∞ -topos (Def. 3.1). Then X ∈ H isempty, i.e., equivalent to the initial object ∅ (60) , precisely if its shape (128) is empty:X (cid:39) ∅ ⇔ S X (cid:39) ∅ . Proof.
In one direction, assume that X (cid:39) ∅ . Noticing that ∅ is the initial colimit and that colimits are preservedby S , this being a left adjoint (Prop. 2.26), it follows that S ( ∅ ) (cid:39) ∅ .In the other direction, assume that the shape of X is empty. Then the shape unit (48) is a morphism of the form X η S X (cid:47) (cid:47) S X (cid:39) ∅ and thus X (cid:39) ∅ follows as in (61), by universality of colimits (Example 2.33). (cid:3) Cohesive ∞ -group actions. The condition that Shp preserves finite products implies the following properties.
Proposition 3.4 (Shape preserves groups, actions and their homotopy quotients) . Let H be a cohesive ∞ -topos(Def. 3.1), G ∈ Groups (cid:0) H (cid:1) (101) and and ( X , ρ ) ∈ G Actions ( H ) (Prop. 2.79). (i) Then the shape S X (128) of X is equipped with an induced S G-action, such that the shape of the homotopyquotient (108) is the homotopy quotient of the shapes. The analogous statement holds for (cid:91) (128) : S (cid:0) X (cid:12) G (cid:1) (cid:39) (cid:0) S X (cid:1) (cid:12) (cid:0) S G (cid:1) and (cid:91) (cid:0) X (cid:12) G (cid:1) (cid:39) (cid:0) (cid:91) X (cid:1) (cid:12) (cid:0) (cid:91) G (cid:1) . (ii) In particular, both S and (cid:91) preserve group objects and their deloopings (Prop. 2.74): S B G (cid:39) B S G and (cid:91) B G (cid:39) B (cid:91) G . Proof.
The homotopy quotient of X by G is, equivalently, a colimit over a simplicial diagram of finite Cartesianproducts of copies of X and G (108). Hence the statement follows for every ∞ -functor that commutes with simpli-cial colimits and with finite products. But, since S is a left adjoint, it commutes with all colimits (Prop. 2.26) andalso with finite products, by assumption on Shp and since Disc is a right adjoint. Similarly, (cid:91) is both left and rightadjoint, and hence preserves all colimits and all limits (again Prop. 2.26). That preservation of homotopy quotientsimplies preservation of ∞ -groups follows by the delooping theorem (Prop. 2.74). (cid:3) Lemma 3.5 (Cohesive shape presverves homotopy fiber products) . In a cohesive ∞ -topos H (Def. 3.1), the shapefunctor Shp (127) preserves homotopy fiber products over cohesively discrete objects. That is, for B ∈ B (cid:31) (cid:127) Disc (cid:47) (cid:47) H and X , Y ∈ H / B , we have a natural equivalence Shp (cid:0) X × B Y (cid:1) (cid:39) Shp ( X ) × B Shp ( Y ) . Proof.
This is proven in [Sc13, Thm. 3.8.19] under the assumption that H admits an ∞ -cohesive site of definition.This assumption was shown to be unnecessary in [BP19, Lemma 3.10]. (cid:3) Lemma 3.6 (Shape of η S -induced action) . Let H be a cohesive ∞ -topos (Def. 3.1), G ∈ Groups ( H ) (Prop. 2.74)and ( X , ρ ) ∈ G Actions ( H ) (Prop. 2.79). (i) The left-induced action (Prop. 2.85) ( (cid:101) X , (cid:101) ρ ) : = B (cid:0) η S G (cid:1) ! ( X , ρ ) ∈ ( S G ) Actions ( H ) along the shape unit morphism (48) G η S G (cid:47) (cid:47) S G acts on an object whose shape (128) is that of X : S (cid:101) X (cid:39) S X , whence (cid:0) S X , S ρ (cid:1) ∈ (cid:0) S G (cid:1) Actions ( H ) . (129) (ii) Similarly, the restricted-induced action (Prop. 2.85) ( (cid:101) X , (cid:101) ρ ) : = B (cid:0) S ε (cid:91) (cid:1) ∗ ◦ B (cid:0) η S G (cid:1) ! ( X , ρ ) ∈ ( (cid:91) G ) Actions ( H ) along the pair of group homomorphisms (using Prop. 3.4) G η S G (cid:47) (cid:47) S G (cid:111) (cid:111) S ε (cid:91) G (cid:91) G acts on an object whoseshape (128) is that of X : S (cid:101) X (cid:39) S X . roof. By Prop. 2.79 and Prop. 2.85, the object (cid:101) X sits in a diagram of Cartesian squares (Notation 2.21) as shownon the left in the following (the full square in case (i) , the pasting decomposition for case (ii) ): (cid:101) X (cid:15) (cid:15) (cid:47) (cid:47) (pb) (cid:101) X (cid:12) (cid:91) G (cid:15) (cid:15) (cid:47) (cid:47) (pb) X (cid:12) G ρ (cid:15) (cid:15) S (cid:101) X (cid:15) (cid:15) (pb) (cid:47) (cid:47) (cid:0) S (cid:101) X (cid:1) (cid:12) (cid:0) (cid:91) G (cid:1) (cid:47) (cid:47) (cid:15) (cid:15) (pb) (cid:0) S X (cid:1) (cid:12) (cid:0) S G (cid:1) S ρ (cid:15) (cid:15) B G B η S G (cid:15) (cid:15) S ∗ (cid:47) (cid:47) B (cid:91) G B S ε (cid:91) G (cid:47) (cid:47) B S G ∗ (cid:47) (cid:47) B (cid:91) G B S ε (cid:91) G (cid:47) (cid:47) B S G (130)But, since the objects in the bottom row B S G (cid:39) S B G and B (cid:91) G (cid:39) (cid:91) B G (equivalences by Prop. 3.4) are bothcohesively discrete, Lemma 3.5 says that the image of these squares under shape are still Cartesian. This is shownon the right in (130), where we have identified the shape of the various objects by using Prop. 3.4 and idempotencyof the modality (Prop. 2.29). With this, the pasting law (Prop. 2.23) implies that the outer right square in (130) isitself Cartesian, hence that S (cid:101) X is the homotopy fiber of S ρ . This implies the claim, by Prop. 2.79. (cid:3) Proposition 3.7 (Automorphisms along shape-unit) . Let H be a cohesive ∞ -topos (Def. 3.1), G ∈ Groups ( H ) (Prop. 2.74) and ( X , ρ ) ∈ G Actions ( H ) (Prop. 2.79). There is a canonical homomorphism Aut ( X ) Aut (cid:0) η S X (cid:1) (cid:47) (cid:47) Aut ( S X ) (131) from the automorphism group (Def. 2.86) of X to that of the shape (128) of X , which is such that the shape unit η S X (48) is equivariant (Def. 2.83) with respect to the canonical automorphism action (114) on X and the restriction(Prop. 2.85) along this morphism (131) of the canonical automorphism action on S X : ( X , ρ Aut ( X ) ) η S X (cid:47) (cid:47) Aut ( η S X ) ∗ (cid:0) S X , ρ Aut ( S X ) (cid:1) ∈ Aut ( X ) Actions ( H ) . Proof.
Take the morphism (131) to be the compositeAut ( X ) Aut (cid:0) η S X (cid:1) (cid:47) (cid:47) η S Aut ( X ) (cid:42) (cid:42) Aut (cid:0) S X (cid:1) S (cid:0) Aut ( X ) (cid:1) Ω (cid:96) S ρ Aut (cid:52) (cid:52) where (a) the left morphism is the shape unit (48), using Prop. 3.4, while (b) the right morphism is that whichexhibits, via Prop. 2.87, the S Aut ( X ) -action S ρ Aut (129) on S X from Lemma 3.6. Then consider the followingdiagram of homotopy fiber sequences: S X (cid:47) (cid:47) ( S X ) (cid:12) Aut (cid:0) S X (cid:1) ρ Aut ( S ( X )) (cid:15) (cid:15) S X (cid:47) (cid:47) ( S X ) (cid:12) (cid:0) S Aut ( X ) (cid:1) S ρ Aut ( X ) (cid:15) (cid:15) (cid:51) (cid:51) (pb) X (cid:47) (cid:47) η S X (cid:53) (cid:53) X (cid:12) Aut ( X ) ρ Aut ( X ) (cid:15) (cid:15) η S X (cid:12) Aut ( X ) (cid:51) (cid:51) B Aut (cid:0) S X (cid:1) B S Aut ( X ) (cid:96) S ρ Aut (cid:51) (cid:51) B Aut ( X ) η S B Aut ( X ) (cid:51) (cid:51) Aut (cid:0) η S X (cid:1) (cid:56) (cid:56) Here (i) the fiber squence in the middle is that from the right of (130), (ii) the right part is the defining pullbackfrom Prop. 2.87, while (iii) the left part exists by the naturality of η S . By the commutativity of the total front squareit factors through the coresponding pullback square, thus implying the claim. (cid:3) oncrete cohesive objects.Definition 3.8 (Concrete objects) . Let H be a cohesive ∞ -topos (Def. 3.1). (i) For X ∈ H (cid:44) ! H X is a concrete object or concrete cohesive space ifthe unit η (cid:93) X (48) of the (cid:93) -modality (128) is (-1)-truncated (Def. 2.58), hence a monomorphism. By the 0-imagefactorization (91), X (-1)-conn. (cid:47) (cid:47) (cid:47) (cid:47) η (cid:93) X unit morphism of (cid:93) -modality (cid:51) (cid:51) image factorization (cid:93) X (cid:31) (cid:127) (-1)-trunc. (cid:47) (cid:47) (cid:93) X this means equivalently that X is equivalent to its 0-image under the (cid:93) -unit (48): X ∈ H : X is concrete ⇔ X (cid:31) (cid:127) η (cid:93) X (cid:47) (cid:47) (cid:93) X ⇔ (cid:93) X (cid:39) X . (132) (ii) We write H ,(cid:93) (cid:44) −! H (cid:44) −! H (133)for the full subcategory of the 0-truncated objects on those which are concrete. (iii) Moreover, for n ∈ N we define, recursively, full sub- ∞ -categories of concrete ( n + ) -truncated objects (Def.2.57) H n + ,(cid:93) (cid:44) −! H n + (cid:44) −! H (134)by declaring that X ∈ H n + is concrete if:• it admits a concrete atlas , namely an effective epimorphism out of a concrete 0-truncated object (132),• such that the homotopy fiber product of the atlas with itself (which is an n -truncated object) is a concrete: X ∈ H n + : X is concrete ⇔ ∃ X ∈ H ,(cid:93) : X ( − ) -trunc. (cid:47) (cid:47) (cid:47) (cid:47) X and X × X X ∈ H n ,(cid:93) . (135) Cohesive charts.Definition 3.9 (Charts) . Let H be a cohesive ∞ -topos (Def. 3.1). We say that an ∞ -category of cohesive charts for H is an ∞ -site Charts for H (Prop. 2.41) H (cid:111) (cid:111) L (cid:31) (cid:127) ⊥ (cid:47) (cid:47) PreSheaves ∞ ( Charts ) all of whose objects (under the ∞ -Yoneda embedding y , Prop. 2.37) have contractible shape (128):Charts (cid:31) (cid:127) y (cid:47) (cid:47) H Shp (cid:47) (cid:47)
Groupoids ∞ U (cid:31) (cid:47) (cid:47) U (cid:31) (cid:47) (cid:47) Shp ( U ) (cid:39) ∗ ⇔ Charts (cid:31) (cid:127) y (cid:47) (cid:47) H S (cid:47) (cid:47) H U (cid:31) (cid:47) (cid:47) U (cid:31) (cid:47) (cid:47) S ( U ) (cid:39) ∗ (136) Lemma 3.10 (Charts are cohesively connected) . Let H be a cohesive ∞ -topos (Def. 3.1) with a site of Charts (Def.3.9). Then, for U ∈ Charts and (cid:8) X i ∈ H (cid:9) i ∈ I an indexed set of objects of H , we have that every morphism from Uinto the coproduct of the X i factors through one of the X i :U f (cid:47) (cid:47) (cid:116) i ∈ I X i ⇔ ∃ i ∈ I U f (cid:52) (cid:52) (cid:47) (cid:47) X i q Xi (cid:47) (cid:47) (cid:116) i ∈ I X i . Proof.
Consider the pullbacks U i q Ui (cid:47) (cid:47) U along f of the canonical inclusions of the X i into their coproduct, givenby these Cartesian squares (Notation 2.21): U iq Ui (cid:15) (cid:15) (cid:47) (cid:47) (pb) X iq Xi (cid:15) (cid:15) U f (cid:47) (cid:47) (cid:70) i ∈ I X i (137)40y Prop. 2.32, this is such that U (cid:39) (cid:71) i ∈ I U i . (138)The image of (138) under shape (128) is ∗ (cid:39) S U (cid:39) (cid:71) i ∈ I S U i ∈ Groupoids ∞ (cid:31) (cid:127) Disc (cid:47) (cid:47) H , where on the left we used the defining property (136) of charts and on the right we used that the shape operation,being a left adjoint, preserves coproducts (Prop. 2.26). But, since ∗ ∈ Groupoids ∞ is connected, this implies thatthere is i ∈ I with S U i (cid:39) (cid:26) ∅ | i (cid:54) = i ∗ | i = i From this, Lemma 3.3 implies that U i (cid:39) ∅ for i (cid:54) = i and, with (138), this implies U i (cid:39) q Ui (cid:47) (cid:47) U . Using this in (137) gives the desired factorization. (cid:3)
Lemma 3.11 (Quotient by cohesively discrete ∞ -group) . Let H be a cohesive ∞ -topos (Def. 3.1) which admits asite of Charts (Def. 3.9). Then, forG ∈ Groups ( Groupoids ∞ ) (cid:31) (cid:127) Disc (cid:47) (cid:47)
Groups ( H ) (139) a cohesively discrete ∞ -group (101) and U ∈ Charts , we have an equivalence H ( U , ∗ (cid:12) G ) (cid:39) ∗ (cid:12) G ∈ Groupoids ∞ . (140) Proof.
Since Disc is both a left and a right adjoint, it preserves (Prop. 2.26) the homotopy quotient that correspondsto the effective epimorphism ∗ (cid:47) (cid:47) (cid:47) (cid:47) ∗ (cid:12) G (Prop. 2.70) so that ∗ (cid:12) G ∈ Groupoids ∞ (cid:31) (cid:127) Disc (cid:47) (cid:47) H is a cohesively discrete object. With this, we have the following sequence of natural equivalences: H (cid:0) U , ∗ (cid:12) G (cid:1) (cid:39) H (cid:0) U , Disc ( ∗ (cid:12) G ) (cid:1) (cid:39) Groupoids ∞ (cid:0) Shp ( U ) , ∗ (cid:12) G (cid:1) (cid:39) Groupoids ∞ (cid:0) ∗ , ∗ (cid:12) G (cid:1) (cid:39) ∗ (cid:12) G where the second step is the hom-equivalence (47) of the Shp (cid:97) Disc-adjunction and the third step is the conditionthat the chart U has contractible shape. (cid:3) Lemma 3.12 (Homming Charts into quotients by discrete groups) . Let H be a cohesive ∞ -topos (Def. 3.1) whichadmits Charts (Def. 3.9). Then, for X ∈ H an object equipped with an ∞ -action (Prop. 107) by a geometricallydiscrete ∞ -group G (139) , the homotopy quotient X (cid:12) G (108) is given as an ∞ -sheaf on Charts , by assigning toU ∈ Charts the homotopy quotient of the ∞ -groupoid of U -shapes plots of X :X (cid:12) G : U H ( U , X ) (cid:12) G . Proof.
Consider the image of the homotopy fiber sequence that characterizes the given ∞ -action (Prop. 2.79) underhomming the chart U into it: X fib ( p ) (cid:47) (cid:47) X (cid:12) G p (cid:15) (cid:15) ∗ (cid:12) G H ( U , − ) H ( U , X ) fib ( H ( U , p )) (cid:47) (cid:47) H ( U , X ) (cid:12) G (cid:39) H (cid:0) U , X (cid:12) G (cid:1) H ( U , p ) (cid:15) (cid:15) ∗ (cid:12) G (cid:39) H ( U , ∗ (cid:12) G ) (141)Since the hom-functor H ( U , − ) preserves limits, the result is again a homotopy fiber sequence, as shown onthe right of (141). Moreover, by the assumption that G is geometrically discrete and that U is geometricallycontractible, we have the equivalence (141) shown on the bottom right. This means that the fiber sequence on theright of (141) exhibits H ( U , X (cid:12) G ) as the homotopy quotient H ( U , X ) (cid:12) G of an ∞ -action by G on H ( U , X ) . (cid:3) emma 3.13 (Fixed locus in 0-truncated objects for discrete groups) . Let H be a cohesive ∞ -topos (Def. 3.1) witha site of Charts (Def. 3.9). Let G ∈ Groups ( H ) (Prop. 2.74) be discrete G (cid:39) (cid:91) G and 0-truncated, G (cid:39) τ G, and let ( X , ρ ) ∈ G Actions ( H ) (Prop. 2.79) with X (cid:39) τ X also 0-truncated. Then the G-fixed locus X G ∈ H (Def. 2.97) isitself 0-truncated and such that, for U ∈ Charts , we have a natural equivalence H (cid:0) U , X G (cid:1) (cid:39) H ( U , X ) G : = (cid:110) φ ∈ H ( U , X ) | ∀ g ∈ G g · φ = φ (cid:111) (142) between (a) the hom-set from U to X G and (b) the naive set of fixed points in the hom-set from U to X , with respectto the restriction (Prop. 2.85) along K (cid:44) ! G of the induced G-action (141) on the latter.Proof.
We claim that we have the following sequence of natural equivalences: H ( U , X G ) = H (cid:0) U , B ( G ! ∗ ) ∗ (cid:0) ( X , ρ ) (cid:1)(cid:1) (cid:39) H / B G (cid:0) B ( G ! ∗ ) ∗ (cid:0) U (cid:1) , X (cid:12) G (cid:1) (cid:39) H / B G (cid:0) ( ∗ (cid:12) G ) × U , X (cid:12) G (cid:1) (cid:39) H (cid:0) ( ∗ (cid:12) G ) × U , X (cid:12) G (cid:1) × H (cid:0) ( ∗ (cid:12) G ) × U , ∗ (cid:12) G (cid:1) (cid:8) pr (cid:9) (cid:39) Groupoids (cid:16) ∗ (cid:12) G , H (cid:0) U , X (cid:12) G (cid:1)(cid:17) × Groupoids (cid:16) ∗ (cid:12) G , H (cid:0) U , ∗ (cid:12) G (cid:1)(cid:17) (cid:8) (cid:102) pr (cid:9) (cid:39) Groupoids (cid:16) ∗ (cid:12) G , H (cid:0) U , X (cid:1) (cid:12) G (cid:17) × Groupoids (cid:0) ∗ (cid:12) G , ∗ (cid:12) G (cid:1) (cid:8) id (cid:9) (cid:39) H ( U , X ) G . (143)Here the first three lines are the definition of fixed loci (123) and the hom-equivalences (47) of the resultingadjunction (81). The fourth line is the characterization (75) of hom- ∞ -groupoids in slices (Prop. 2.48), the fifthline uses the tensoring (62) of H over Groupoids ∞ (Prop. 2.34), and the sixth line follows by Prop. 3.12.To see the last step in (143), use the explicit presentation of the groupoid H ( U , X ) (cid:12) G as an action groupoid,by Example 2.15. This way the projection map in the fiber product in the sixth line in (143) is presented by a Kanfibration, whence this homotopy fiber product may be computed equivalently as a 1-categorical fiber product ofsets of objects and of sets of morphisms, separately. Moreover, since { id } has no non-trivial morphisms and sincethe projection functor itself is faithful, there are in fact no non-trivial morphisms in this fiber product, which ishence just the set whose elements are precisely those functors of action groupoids which are equal to the identityon labels in G :Groupoids (cid:16) ∗ (cid:12) G , H (cid:0) U , X (cid:1) (cid:12) G (cid:17) × Groupoids (cid:0) ∗ (cid:12) G , ∗ (cid:12) G (cid:1) (cid:8) id (cid:9) (cid:39) ∗ (cid:12) G (cid:47) (cid:47) H ( U , X ) (cid:12) G ∗ g ∈ G (cid:15) (cid:15) φ g (cid:15) (cid:15) ∗ g · φ (cid:39) H ( U , X ) G . (cid:3) Lemma 3.14 ( n -Truncated morphisms via n -truncated homotopy fibers) . Let H be an ∞ -topos which is cohesive(Def. 3.1). Let G be a finite group in H (215) . Then, for every n ∈ {− , − , , , · · · } and for any morphism in H to its delooping groupoid (Example 2.14) X p −! ∗ (cid:12) G, the following are equivalent (i) p is an n-truncated morphism (Def. 2.58); (ii) the homotopy fiber of p (over the essentially unique point of ∗ (cid:12) G) is an n-truncated object (Def. 2.57).Proof.
Let U ∈ Charts and consider homming it into the homotopy fiber sequence in question: X (pb) (cid:15) (cid:15) (cid:47) (cid:47) X p (cid:15) (cid:15) ∗ (cid:47) (cid:47) ∗ (cid:12) G ⇒ H ( U , X ) (pb) (cid:15) (cid:15) (cid:47) (cid:47) H ( U , X ) H ( U , p ) (cid:15) (cid:15) ∗ (cid:47) (cid:47) H ( U , ∗ (cid:12) G ) (cid:39) ∗ (cid:12) G H ( U , − ) preserves limits, the square on the right is again a homotopy pullback. Since U isa chart and G is discrete, we have the equivalence (140) shown on the bottom right. Since ∗ (cid:12) G has an essentiallyunique point, the square on the right exhibits the essentially unique homotopy fiber of the morphism H ( U , p ) .Since the charts U are generators of H (objects of an ∞ -site of definition), the morphism p is n -truncated (Def.2.58) precisely if for each chart U the homotopy fiber of H ( U , p ) is n -truncated. But the square on the right showsthat this homotopy fiber is H ( U , X ) , and hence this means, equivalently, that X is an n -truncated object (accordingto Def. 2.57). (cid:3) Examples of cohesive ∞ -toposes. We indicate some examples of cohesive ∞ -toposes (Def. 3.1), following [Sc13].For full details of the constructions see [SS20c]. Example 3.15 (Discrete cohesion) . The base ∞ -topos Groupoids ∞ is trivially a cohesive ∞ -topos (Def. 3.1) withall operations being identities: Groupoids ∞ × id ⊥ (cid:47) (cid:47) (cid:111) (cid:111) id ⊥ (cid:63) (cid:95) id ⊥ (cid:47) (cid:47) (cid:111) (cid:111) id (cid:63) (cid:95) Groupoids ∞ (144)For emphasis we also call this the ∞ -topos of geometrically discrete ∞ -groupoids . Definition 3.16 (Site for homotopical cohesion) . A small ∞ -site (70) is an ∞ -site for homotopical cohesion if (i) its Grothendieck topology is trivial and (ii) the underlying ∞ -category has finite products, i.e., has a terminal object and binary Cartesian products. Example 3.17 (Homotopical cohesion) . The ∞ -topos of ∞ -sheaves (Def. 2.42) over an ∞ -site C for homotopicalcohesion (Def. 3.16) is cohesive (Def. 3.1): H : = Sheaves ∞ ( C ) × lim −! ⊥ (cid:47) (cid:47) (cid:111) (cid:111) const ⊥ (cid:63) (cid:95) lim − ⊥ (cid:47) (cid:47) (cid:111) (cid:111) Chtc (cid:63) (cid:95)
Groupoids ∞ (145) (i) The operation Pnts (cid:39) lim − forms the limit of ∞ -presheaves regarded as ∞ -functors on C op (by Prop. 2.36); butsince C is assumed to have a terminal object, this is equivalently just the evaluation on that object:Pnts ( X ) (cid:39) X ( ∗ ) (cid:39) H ( ∗ , X ) , where on the right we used the ∞ -Yoneda lemma (Prop. 2.38). This makes manifest how Pnts ( X ) is the “underlying ∞ -groupoid of points of X ”. (ii) The operation Shp (cid:39) lim −! is the colimit of ∞ -presheaves regarded as ∞ -functors (by Prop. 2.36). Since thecolimit of any representable functor is the point (Lemma 2.40) C const ∗ (cid:49) (cid:49) (cid:31) (cid:127) y (cid:47) (cid:47) Sheaves ∞ ( C ) Shp (cid:47) (cid:47)
Groupoids ∞ , this means that C serves itself as a category of Charts in this case (Def. 3.9). Example 3.18 (Smooth cohesion) . The ∞ -sheaf ∞ -topos (Def. 2.42) over the site of SmoothManifolds (Def. 2.9,see [FSS12, App.]), which we call the ∞ -topos of smooth ∞ -groupoids SmoothGroupoids ∞ : = Sheaves ∞ ( SmoothManifolds ) , is cohesive (Def. 3.1): The adjoint quadruple (127) arises as in Example 3.17, which here happens to descend from ∞ -presheaves to ∞ -sheaves.In this case we have: 43 i) A category of Charts (Def. 3.9) is given (Prop. 2.10) by CartesianSpaces (Def. 2.5)CartesianSpaces (cid:31) (cid:127) y (cid:47) (cid:47) Sheaves ∞ ( CartesianSpaces ) (cid:39) (cid:47) (cid:47) SmoothGroupoids ∞ S ( y ( R n )) (cid:39) ∗ (146) (ii) The concrete 0-truncated objects (Def. 3.8) are equivalently the diffeological spaces (Def. 2.6), including the
D-topological spaces (Def. 2.2) as well as smooth and possibly infinite-dimensional Fr´echet manifolds (Prop.2.11) as further full subcategories (32):TopologicalSpaces Cdfflg (cid:47) (cid:47)
DTopologicalSpaces (cid:27) (cid:123) (cid:45) (cid:45)
DiffeologicalSpaces (cid:31) (cid:127) concrete0-truncatedobjects (cid:47) (cid:47)
SmoothGroupoids ∞ Fr´echetManifolds (cid:35) (cid:3) (cid:49) (cid:49) (147) (iii)
The concrete 1-truncated objects (Def. 3.8) form the ( , ) -category of diffeological groupoids with Morita/Hilsum-Skandalis morphisms (Remark 2.72) between them, which includes, by (147), the ( , ) -categories of D-topologicalgroupoids and of (possibly infinite-dimensional Fr´echet-)Lie groupoids: TopologicalGroupoids
Cdfflg (cid:47) (cid:47)
DTopologicalGroupoids (cid:28) (cid:124) (cid:45) (cid:45)
DiffeologicalGroupoids (cid:31) (cid:127) concrete1-truncatedobjcts (cid:47) (cid:47)
SmoothGroupoids ∞ Fr´echetLieGroupoids (cid:34) (cid:2) (cid:49) (cid:49) (148) (iv)
The cohesive shape (128) is given equivalently [Pav20][BEBP19] by the smooth ∞ -path ∞ -groupoid : S X (cid:39) lim −! Maps (cid:0) ∆ • smth , X (cid:1) ∈ SmoothGroupoids ∞ , hence Shp ( X ) (cid:39) lim −! X ( ∆ • smth ) ∈ Groupoids ∞ (149)where ∆ • smth is the simplicial smooth manifold of extended simplices (Def. 2.19) and Maps ( − , − ) denotes theinternal hom (56) in SmoothGroupoids ∞ . (v) The cohesive shape (128) of (a) any topological space and (b) any finite-dimensional smooth manifold regarded,respectively, as smooth ∞ -groupoids via (147) is equivalently (by (149) with Prop. 2.20, and by [Sc13, 4.3.29],respectively) its standard topological homotopy type Shp Top (36): (a)
TopologicalSpaces
Shp
Top ⇓ (cid:39) (cid:51) (cid:51) Cdfflg (cid:47) (cid:47)
DiffeologicalSpaces (cid:31) (cid:127) (cid:47) (cid:47)
SmoothGroupoids ∞ Shp (cid:47) (cid:47)
Groupoids ∞ (150) (b) SmoothManifolds
Shp
Top ◦ Dtplg ⇓ (cid:39) (cid:51) (cid:51) (cid:31) (cid:127) (cid:47) (cid:47) DiffeologicalSpaces (cid:31) (cid:127) (cid:47) (cid:47)
SmoothGroupoids ∞ Shp (cid:47) (cid:47)
Groupoids ∞ (151) (vi) The cohesive shape (128) of a topological groupoid, when regarded, via its coreflection (32), as a D-topologicalgroupoid and hence as a smooth ∞ -groupoid (148) is equivalently (by (150), and since S is left adjoint and hencepreserves homotopy colimits, Prop. 2.26) its simplicial-topological shape (Def. 42):TopologicalGroupoids Shp sTop ⇓ (cid:39) (cid:50) (cid:50) Cdfflg (cid:47) (cid:47)
DiffeologicalGroupoids (cid:31) (cid:127) (cid:47) (cid:47)
SmoothGroupoids ∞ Shp (cid:47) (cid:47)
Groupoids ∞ (152) Example 3.19 (Spectral cohesion) . Let H be a cohesive ∞ -topos (Def. 3.1). Then its tangent ∞ -topos T H = SpectralBundles ( H ) (Example 2.51) is cohesive [Sc13, 4.1.9] over the base tangent ∞ -topos (121): T H × T Shp ⊥ (cid:47) (cid:47) (cid:111) (cid:111) T Disc ⊥ (cid:63) (cid:95) T Pnts ⊥ (cid:47) (cid:47) (cid:111) (cid:111) T Chtc (cid:63) (cid:95) T Groupoids ∞ (153) These are the ∆ -generated spaces of [Sm][Dug03]; see Remark 2.3. emark 3.20 (Differential cohomology in cohesive ∞ -toposes) . The intrinsic cohomology theory (22) of a cohe-sive ∞ -topos (Def. 3.1) is differential cohomology [Sc13]. (i) In the case when H : = SmoothGroupoids ∞ (Example 3.18), this is a non-abelian differential cohomology theorygeneralizing the theory of Cartan-Ehresmann connections on smooth fiber bundles to ∞ -connections on smooth ∞ -bundles [SSS12][FSS12][NSS12a]. (ii) In the case when H : = T SmoothGroupoids ∞ is the cohesive tangent ∞ -topos (Example 3.19) to that of smooth ∞ -groupoids (Example 3.18), the intrinsic cohomology furthermore subsumes abelian Hopkins-Singer differentialcohomology theories and variants [BNV13], as well as the twisted versions of these (Remark 2.96), such as twisteddifferential KU-theory [GS19a] and twisted differential KO-theory [GS19b]. We present a formulation of differential geometry internal to ∞ -toposes which we call elastic [Sc13], adjoining tothe plain shape operation S of §3.1.1 a de Rham shape operation ℑ , in generalization of [Si96][ST97]. Definition 3.21 (Elastic ∞ -topos) .(i) An elastic ∞ -topos over B = Groupoids ∞ is an ∞ -topos H (Def. 2.30) whose base geometric morphism (Prop.2.43), to be denoted Pnts : H (cid:47) (cid:47) Groupoids ∞ , is equipped with a factorization as follows, having adjoints (Def.2.24) as shown: Shp :Pnts : H (cid:111) (cid:111) “reduced” Rdcd ⊥ (cid:63) (cid:95) “infinitesimal shape” Shp inf ⊥ (cid:47) (cid:47) (cid:111) (cid:111) “infinitesimally discrete” Disc inf ⊥ (cid:63) (cid:95) “infinitesimal points” Pnts inf (cid:47) (cid:47) (cid:111) (cid:111)
Chtc (cid:63) (cid:95) − − − ⊥ H ℜ × Shp ℜ ⊥ (cid:47) (cid:47) (cid:111) (cid:111) Disc ℜ ⊥ (cid:63) (cid:95) Pnts ℜ (cid:47) (cid:47) B : Disc elastic ∞ -topos reducedsub-topos discretesub-topos (154) (ii) Hence the elastic ∞ -topos H is, in particular, a cohesive ∞ -topos over B , according to Def. 3.1, and so is itssub- ∞ -topos H ℜ of reduced objects. (iii) We write (cid:0) ℜ : = Rdcd ◦ Shp inf (cid:1) “reduced” (cid:97) (cid:0) ℑ : = Disc inf ◦ Shp inf (cid:1) “´etale” (cid:97) (cid:0) L : = Disc inf ◦ Pnts inf (cid:1) “locally constant” : H −! H (155)for the further induced modalities (20) ( elastic modalities ), accompanying the cohesive modalities of (128). Examples of elastic ∞ -toposes. We indicate some examples of elastic ∞ -toposes (Def. 3.21), following [Sc13].For full details on the constructions, see [SS20c]. Definition 3.22 (Jets of Cartesian spaces) . Let k ∈ N . (i) We write k JetsOfCartesianSpaces (cid:31) (cid:127) C ∞ ( − ) (cid:47) (cid:47) CommutativeAlgebras op R R n × D W (cid:31) (cid:47) (cid:47) C ∞ ( R n ) ⊗ R ( R ⊕ W ) (156)for the full subcategory of that of commutative R -algebras on those which are tensor products of (a) the algebra ofreal-valued smooth functions on a Cartesian space R n , with (b) a finite-dimensional real algebra with a maximalideal W that is nilpotent of order k +
1, in that W k + = (ii) We write ∞ JetsOfCartesianSpaces : = (cid:83) k ∈ N k JetsOfCartesianSpaces (cid:31) (cid:127) C ∞ ( − ) (cid:47) (cid:47) CommutativeAlgebras op R R × D W (cid:31) (cid:47) (cid:47) C ∞ (cid:0) R n (cid:1) ⊗ R W (157)for the analogous full subcategory where each W is (finite dimensional and) nilpotent of some finite order. (iii) We regard these categories as equipped with the coverage (Grothendieck pre-topology) whose covers are thefamilies of morphisms of the form (cid:8) R n × D f i × id (cid:47) (cid:47) R n × D (cid:9) i ∈ I such that (cid:8) R n f i (cid:47) (cid:47) R n (cid:9) i ∈ I is a cover in CartesianSpaces (Def. 2.5).45 emma 3.23 (Coreflections of jets of Cartesian spaces) . Consider the k
JetsOfCartesianSpaces from Def. 3.22. (i)
For k = , this is equivalently the category of plain Cartesian spaces of Def. 2.5: (cid:39) CartesianSpaces . (ii) For any k ∈ N , the evident full inclusion of k JetsOfCartesianSpaces into ( k + ) JetsOfCartesianSpaces is co-reflective ∞ JetsOfCartesianSpaces (cid:111) (cid:111)
Rdcd ∞ (cid:63) (cid:95) Shp inf , ∞ ⊥ (cid:47) (cid:47) · · · (cid:111) (cid:111) Rdcd (cid:63) (cid:95) Shp inf , ⊥ (cid:47) (cid:47) (cid:111) (cid:111) Rdcd (cid:63) (cid:95) Shp inf , ⊥ (cid:47) (cid:47) (cid:111) (cid:111) Rdcd (cid:63) (cid:95)
Shp inf ⊥ (cid:47) (cid:47) CartesianSpaces with C ∞ (cid:0) Shp inf , k (cid:0) R n × D W (cid:1)(cid:1) (cid:39) C ∞ ( R n ) ⊗ R ( R ⊕ W ) / W k + . (158) Proof.
Statement (i) follows as a special case of the general fact, sometimes known as
Milnor’s exercise (since thekey idea is hinted at in [MSt74, Prob. 1-C]), that passage to their real algebras of smooth functions embeds smoothmanifolds fully faithfully into the opposite or real algebras (a general proof is in [KMS93, 35.10], see also [Gr05];for general perspective see [Nes03, 6]) :SmoothManifolds (cid:31) (cid:127) C ∞ ( − ) (cid:47) (cid:47) CommutativeAlgebras op R . Statement (ii) follows readily from the definition, using the fact that algebra homomorphisms preserve order ofnilpotency. (cid:3)
Example 3.24 (Jets of smooth ∞ -groupoids) . For k ∈ N (cid:116) { ∞ } , the ∞ -sheaf ∞ -topos (Def. 2.42) over the site of k -jets of Cartesian spaces (Def. 3.22) k JetsOfSmoothGroupoids ∞ : = Sheaves ∞ ( k JetsOfCartesianSpaces ) is elastic (Def. 3.21), with ( Rdcd (cid:97)
Shp inf ) in (154) given by Kan extension of the co-reflections of sites fromLemma 3.23: k JetsOfSmoothGroupoids ∞ (cid:111) (cid:111) Rdcd ⊥ (cid:63) (cid:95) Shp inf ⊥ (cid:47) (cid:47) (cid:111) (cid:111) Disc inf ⊥ (cid:63) (cid:95) Pnts inf (cid:47) (cid:47) (cid:111) (cid:111)
Chtc (cid:63) (cid:95)
SmoothGroupoids ∞ × Shp ℜ ⊥ (cid:47) (cid:47) (cid:111) (cid:111) Disc ℜ ⊥ (cid:63) (cid:95) Pnts ℜ (cid:47) (cid:47) Groupoids ∞ (i) Here for k = ∞ : = ∞ . (159) (ii) For the case k = ∞ , the underlying 1-topos is the “Cahiers topos” [Du79a][Ko86][KS17]. (iii) For any k , we have: (a) The full sub- ∞ -topos of reduced objects (154) is (by Lemma 3.23) that of smooth ∞ -groupoids from Example3.18 k JetsOfSmoothGroupoids ∞ (cid:111) (cid:111) Disc inf (cid:63) (cid:95)
SmoothGroupoids ∞ (160) (b) the 0-truncated concrete objects (Def. 3.8) are still equivalently the diffeological spaces (Def. 2.6) as wasthe case in (147)DTopologicalSpaces (cid:28) (cid:124) (cid:46) (cid:46) DiffeologicalSpaces (cid:31) (cid:127) (cid:47) (cid:47) k JetsOfSmoothGroupoids ∞ Fr´echetManifolds (cid:33) (cid:1) (cid:48) (cid:48) (161)and, more generally, the 1-truncated concrete objects are still the diffeological groupoids , as was the case in(148):DTopologicalGroupoids (cid:29) (cid:125) (cid:46) (cid:46)
DiffeologicalGroupoids (cid:31) (cid:127) (cid:47) (cid:47) k JetsOfSmoothGroupoids ∞ Fr´echetLieGroupoids (cid:33) (cid:1) (cid:48) (cid:48) (162) (c)
A category of charts (Def. 3.9) for JetsOfSmoothGroupoids ∞ is given by k JetsOfCartesianSpaces (Def. 3.22)itself. 46
Etale geometry.Definition 3.25 ( ´Etale-over- X modality) . Let H be an elastic ∞ -topos (Def. 3.21) and X ∈ H an object. We saythat the ´etale-over-X modality on the slice ∞ -topos over X (Def. 2.46) is the ∞ -functor H / X ℑ X (cid:47) (cid:47) H / X Y f (cid:15) (cid:15) Y × ℑ X ℑ Y ( η ℑ X ) ∗ ( ℑ f ) (cid:15) (cid:15) X X X η ℑ X (cid:40) (cid:40) f (cid:42) (cid:42) (cid:41) (cid:41) X × ℑ X ℑ Y (cid:15) (cid:15) (cid:47) (cid:47) (pb) ℑ X ℑ f (cid:15) (cid:15) Y η ℑ Y (cid:47) (cid:47) ℑ Y which sends any morphism f into X to the pullback of its image under the plain ´etale modality ℑ (155) along itsunit morphism (48), hence to the left vertical morphism in the Cartesian square shown on the right. Definition 3.26 (Local diffeomorphism) . Let H be an elastic ∞ -topos (Def. 3.21). We say that a morphism Y f ! X in H is a local diffeomorphism if it is ´etale-over- X (Def. 3.25) ℑ X ( f ) (cid:39) X , hence (see Prop. 3.32 for this implication) if the naturality square of the unit (48) of the ℑ -modality (155) is aCartesian square: Y f ´et (cid:15) (cid:15) ⇔ Y (pb) f (cid:15) (cid:15) η ℑ Y (cid:47) (cid:47) ℑ Y ℑ f (cid:15) (cid:15) X X η ℑ X (cid:47) (cid:47) ℑ X (163) Lemma 3.27 (Closure of class of local diffeomorphisms) . Let H be an elastic ∞ -topos (Def. 3.21). The class oflocal diffeomorphisms in H (Def. 3.26) (i) satisfies left-cancellation: given a pair of composable morphisms f , g where g is a local diffeomorphism, thenf is so precisely if the composite g ◦ f is:Z g ◦ f (cid:34) (cid:34) f (cid:47) (cid:47) Y ´et g (cid:124) (cid:124) X ⇒ (cid:16) f is a local diffeo ⇔ g ◦ f is a local diffeo (cid:17) . (164) (ii) is closed under pullbacks: if in a Cartesian square the right vertical morphism is a local diffeomorphism, thenso is the left morphism Y (cid:48) × X Y (pb) (cid:47) (cid:47) g ∗ f (cid:15) (cid:15) Y ´et f (cid:15) (cid:15) Y (cid:48) g (cid:47) (cid:47) X ⇒ g ∗ f is a local diffeo . Proof.
This is a routine argument: (i)
For two composable morphisms, consider the pasting of their η ℑ -naturalitysquares Z f (cid:15) (cid:15) η ℑ Z (cid:47) (cid:47) (pb) ℑ Z ℑ f (cid:15) (cid:15) Y g (cid:15) (cid:15) η ℑ Y (cid:47) (cid:47) (pb) ℑ Y ℑ g (cid:15) (cid:15) X η ℑ X (cid:47) (cid:47) ℑ X By the functoriality of ℑ , the total rectangle is the η ℑ -naturality square of g ◦ f . But, by the pasting law (Prop.2.23) and the assumption that the bottom square is Cartesian, the total rectangle is Cartesian precisely if so is thetop square. 47 ii) For two morphisms with the same codomain, consider the pasting of their pullback square with the η ℑ -naturality square of one of them, as shown on the left here: Y (cid:48) × X Y g ∗ f (cid:15) (cid:15) f ∗ g (cid:47) (cid:47) (pb) Y f (cid:15) (cid:15) η ℑ (cid:47) (cid:47) (pb) ℑ Y ℑ f (cid:15) (cid:15) Y (cid:48) g (cid:47) (cid:47) X η ℑ X (cid:47) (cid:47) ℑ X (cid:39) Y × X Y (cid:48) g ∗ f (cid:15) (cid:15) η ℑ ( Y × XY (cid:48) ) (cid:47) (cid:47) ℑ ( Y × X Y (cid:48) ) ℑ ( f ∗ f ) (cid:47) (cid:47) ℑ ( g ∗ f ) (cid:15) (cid:15) (pb) ℑ Y (cid:48) ℑ g (cid:15) (cid:15) Y (cid:48) η ℑ Y (cid:48) (cid:47) (cid:47) ℑ Y (cid:48) ℑ ( f (cid:48) ) (cid:47) (cid:47) ℑ X By the naturality of η ℑ , this pasting diagram on the left is equivalent to that shown on the right. Moreover, if f is alocal diffeomorphisms, it follows that three of the squares are pullbacks (the rightmost one by using that ℑ is rightadjoint and thus preserves pullbacks, Prop. 2.26), as shown. With that, the pasting law (Prop. 2.23) implies, first,that the total rectangle on the left is a pullback, hence also that on the left, and then that the remaining square onthe right is a pullback. This means that g ∗ f is a local diffeomorphism. (cid:3) Definition 3.28 (Local neighborhood) . Let H be an elastic ∞ -topos (Def. 3.21). For Y f −! X a morphism in H ,we say that the corresponding local neighborhood of Y in X is the purely ´etale aspect of f , hence is the object N f X ∈ H / X given by ℑ / X ( f ) (cid:39) ( η ℑ X ) ∗ ( ℑ f ) , hence given by the following homotopy pullback square: N f X (pb) (cid:47) (cid:47) ℑ / X ( f ) (cid:15) (cid:15) ℑ X ℑ f (cid:15) (cid:15) Y η ℑ X (cid:47) (cid:47) ℑ Y Definition 3.29 (Tangent bundle) . Let H be an elastic ∞ -topos (Def. 3.21). Then for X ∈ H any object, we say thatits infinitesimal tangent bundle is T X : = X × ℑ X X ∈ H / X , hence the left morphism in this Cartesian square: T X (cid:47) (cid:47) ( η ℑ X ) ∗ ( η ℑ X ) ! ( id X ) (cid:15) (cid:15) (pb) X η ℑ X (cid:15) (cid:15) X η ℑ X (cid:47) (cid:47) ℑ X (165) Example 3.30 (Local neighborhood of a point) . Let H be an elastic ∞ -topos (Def. 3.21). For X ∈ H any objectand ∗ x −! X any point, the homotopy fiber of the tangent bundle (Def. 3.29) over x is equivalent to the localneighborhood of x (Def. 3.28): T x X (cid:39) N x X . (166)This follows immediately from the definitions, by the pasting law (Prop. 2.23): N x X (cid:39) T x X (pb) (cid:15) (cid:15) (cid:47) (cid:47) T X (pb) (cid:47) (cid:47) (cid:15) (cid:15) X η ℑ X (cid:15) (cid:15) ∗ x (cid:47) (cid:47) X η ℑ X (cid:47) (cid:47) ℑ X Proposition 3.31 (Pullback along local diffeomorphisms preserves tangent bundles) . In an elastic ∞ -topos (Def.3.21), pullback along a local diffeomorphism Y f ´et (cid:47) (cid:47) X (Def. 3.26) preserves tangent bundles (Def. 3.29) in thatf ∗ ( T X ) (cid:39) TY via: TY (pb) (cid:15) (cid:15)
T f (cid:47) (cid:47)
T X (cid:15) (cid:15) Y f ´et (cid:47) (cid:47) X roof. Consider the pasting of the defining Cartesian squares, shown on the left here: f ∗ T X (pb) (cid:15) (cid:15) (cid:47) (cid:47)
T X (cid:15) (cid:15) (cid:47) (cid:47) (pb) X η ℑ X (cid:15) (cid:15) Y f ´et (cid:47) (cid:47) X η ℑ X (cid:47) (cid:47) ℑ X (cid:39) TY (cid:15) (cid:15) (cid:47) (cid:47) (pb) Y η ℑ X (cid:15) (cid:15) f (cid:47) (cid:47) (pb) X η ℑ X (cid:15) (cid:15) Y η ℑ X (cid:47) (cid:47) ℑ Y ℑ f (cid:47) (cid:47) ℑ X By the pasting law (Prop. 2.23), the total rectangle on the left is itself Cartesian. Moreover, the bottom compositemorphism on the left is equivalent to the bottom composite morphism on the right, by the naturality of η ℑ X . There-fore, using again the pasting law (Prop. 2.23) the total rectangle on the left is equivalent to the pasting of the twoconsecutive Cartesian squares shown on the right. These identify, in the top row, the middle object Y by (163) andthus the left object TY by (165). (cid:3) ´Etale toposes.Definition 3.32 ( ´Etale topos) . Let H be an elastic ∞ -topos (Def. 3.21) and X ∈ H . Then we say that the ´etale ∞ -topos of X , to be denoted ´Et X , is the full sub- ∞ -category (Def. 2.1) of the slice ∞ -topos over X (Prop. 2.46) onthose morphisms that are local diffeomorphisms (Def. 3.26): ´Et X : = (cid:0) H / X (cid:1) ℑ X (cid:31) (cid:127) (cid:47) (cid:47) H / X . (167) Proposition 3.33 (Reflections of ´etale toposes) . Let H be an elastic ∞ -topos (Def. 3.21) and X ∈ H an object.Then the ´etale topos ´Et X from Def. 3.32: (i) is indeed an ∞ -topos (Def. 2.30); (ii) its defining full inclusion (167) has both a left- and a right-adjoint (Def. 2.24): ´Et X (cid:111) (cid:111) Etl X ⊥ (cid:31) (cid:127) i X (cid:47) (cid:47) (cid:111) (cid:111) LcllCnstnt X ⊥ H / X (168) (iii) whose induced adjoint modality (20) (cid:0) ℑ X : = i X ◦ ´Etl X “´etale over X ” (cid:1) (cid:97) (cid:0) L X : = i X ◦ LcllCnstnt X “locally constant over X ” (cid:1) (169) is on the left that of Def. 3.25: ´Etl X : Y p (cid:15) (cid:15) X ( η ℑ X ) ∗ ( ℑ Y ) ( η ℑ X ) ∗ ( ℑ p ) (cid:15) (cid:15) X i.e.: ( η ℑ X ) ∗ ( ℑ Y ) ( η ℑ X ) ∗ ( ℑ p ) (cid:15) (cid:15) ( ℑ p ) ∗ ( η ℑ X ) (cid:47) (cid:47) (pb) ℑ Y ℑ p (cid:15) (cid:15) X η ℑ X (cid:47) (cid:47) ℑ X (170) Proof.
First to see that (170) is well-defined as a functor to ´Et X (this proceeds as in [CHM85, 3.3][CJKP97,3][CRi20, 7.3]): We need to check that ( η ℑ X ) ∗ ( ℑ p ) is a local diffeomorphism (Def. 3.26). For this, it is sufficientto have equivalences ℑ (cid:0) ( η ℑ X ) ∗ ( ℑ p ) (cid:1) (cid:39) ℑ p , (171)and ( ℑ p ) ∗ ( η ℑ X ) (cid:39) η ℑ X (172)because then the Cartesian square on the right of (170) exhibits this property.But (171) follows by applying ℑ to the square on the right of (170), by idempotency (Prop. 2.29) and sinceequivalences are preserved by pullback (Example 2.22). With this, (172) follows from the naturality of the ℑ -unit,by the universal factorization shown dashed in the following diagram:49 η ℑ X ) ∗ ( ℑ Y ) (cid:39) (cid:41) (cid:41) η ℑ ( η ℑ X ) ∗ ( ℑ Y ) (cid:42) (cid:42) ( η ℑ X ) ∗ ( ℑ p ) (cid:41) (cid:41) ( η ℑ X ) ∗ ( ℑ Y ) ( η ℑ X ) ∗ ( ℑ p ) (cid:15) (cid:15) ( ℑ p ) ∗ ( η ℑ X ) (cid:47) (cid:47) (pb) ℑ Y ℑ p (cid:15) (cid:15) X η ℑ X (cid:47) (cid:47) ℑ X (173)Notice that, similarly, there is a natural transformation Y p (cid:40) (cid:40) η Etl XY (cid:47) (cid:47) Etl X ( Y ) ´et (cid:117) (cid:117) X (174)induced as the universal factorization shown dashed in the following diagram: Y (cid:39) (cid:41) (cid:41) η ℑ Y (cid:42) (cid:42) p (cid:40) (cid:40) ( η ℑ X ) ∗ ( ℑ Y ) ( η ℑ X ) ∗ ( ℑ p ) (cid:15) (cid:15) ( ℑ p ) ∗ ( η ℑ X ) (cid:47) (cid:47) (pb) ℑ Y ℑ p (cid:15) (cid:15) X η ℑ X (cid:47) (cid:47) ℑ X (175)and notice that this in an ℑ -equivalence: ℑ (cid:0) η Etl X Y (cid:1) is an equivalence . (176)Condition (176) follows by applying ℑ to the whole left part of the diagram on the right of (177), using idempotency(Prop. 2.29) and that equivalences are preserved by pullback (Example 2.22).Second, to see that (170) defines a left adjoint to the inclusion: We need to check the corresponding hom-equivalence (47), shown on the left here:´Etl X ( Y ) ´Etl X ( p ) (cid:28) (cid:28) (cid:101) f (cid:47) (cid:47) Y ´et (cid:7) (cid:7) B ⇔ Y p (cid:23) (cid:23) f (cid:47) (cid:47) Y ´et (cid:7) (cid:7) B (cid:39) ℑ Y (cid:31) (cid:31) ℑ f (cid:47) (cid:47) ℑ Y (cid:127) (cid:127) Y p (cid:33) (cid:33) η ℑ Y (cid:48) (cid:48) η Etl XY (cid:47) (cid:47) ( η ℑ X ) ∗ ( ℑ X ) η ℑ ( η ℑ X ) ∗ ( ℑ X ) (cid:59) (cid:59) Etl X ( p ) (cid:15) (cid:15) (cid:101) f (cid:47) (cid:47) Y ´et (cid:123) (cid:123) η ℑ Y (cid:54) (cid:54) ℑ XX η ℑ X (cid:52) (cid:52) (177)On the right of (177) we show an induced factorization: The square sub-diagram on the right of (177) is Cartesianby the assumption that we are homming into a local diffeomorphism, while the square in the middle is Cartesianby (173). Thus, given f , the morphism (cid:101) f is induced by the universal property of the right Cartesian square.Conversely, given (cid:101) f , precomposition with the η Etl X Y (175) gives a morphism f . To see that this correspondence isan equivalence, we just need to observe that ℑ ( (cid:101) f ) (cid:39) ℑ f . This follows by (176).Thus we have established the existence of the left adjoint ´Etl X . With this, to see the right adjoint LcllCnst X aswell as the fact that ´Et is an ∞ -topos, it is now sufficient to show that ´Et X (cid:31) (cid:127) i X (cid:47) (cid:47) H / X preserves colimits: Because,by the reflection ´Etl X this implies, first, that ´Et X is a presentable ∞ -category, in fact an ∞ -topos (by Prop. 2.41,since it is then an accessibly embedded reflective subcategory of the slice H / X , which is an ∞ -topos by Prop. 2.46);and thus, second, the existence of the right adjoint by the adjoint ∞ -functor theorem (Prop. 2.27).50o to see that i X preserves colimits, consider any small I ∈ Categories ∞ and a diagram Y • : I (cid:47) (cid:47) ´Et X (cid:31) (cid:127) i X (cid:47) (cid:47) H / X . (178)Since i X is fully faithful by construction, it is sufficient to show that the colimit of this diagram formed in H / X is itself in the image of i X . This colimit, in turn, is computed in H (by Example 2.52) with its morphism q to X universally induced, and this we need to show to be a local diffeomorphism (Def. 3.26). Hence we need to showthat the following square on the left is Cartesian:lim −! Y • q (cid:15) (cid:15) η ℑ lim −! Y • (cid:47) (cid:47) (pb) ℑ (cid:0) lim −! Y • (cid:1) ℑ q (cid:15) (cid:15) X η ℑ X (cid:47) (cid:47) ℑ X ⇔ lim −! Y • q (cid:15) (cid:15) ( η ℑ Y • ) (cid:47) (cid:47) (pb) lim −! (cid:0) ℑ Y • (cid:1) ℑ q (cid:15) (cid:15) X η ℑ X (cid:47) (cid:47) ℑ X ⇔ ∀ i ∈ I Y iq i (cid:15) (cid:15) η ℑ Yi (cid:47) (cid:47) (pb) ℑ Y i ℑ q i (cid:15) (cid:15) X η ℑ X (cid:47) (cid:47) ℑ X But, since ℑ is a left adjoint and hence preserves colimits (Prop. 2.26), this is equivalent to the square on in middlebeing Cartesian. Finally, by universality of colimits (54) in the ∞ -topos H , this is equivalent to all the squares onthe right being Cartesian. This is the case, by the assumption (178). (cid:3) Remark 3.34 (Local and global ∞ -section functors.) . Let H be an elastic ∞ -topos (Def. 3.21) and X ∈ H . Then wemay think of the ´etale ∞ -topos ´Et X (Def. 3.32, Prop. 3.33) as the internal construction of the ∞ -topos of ∞ -sheavesover X . Under this interpretation: i) the ∞ -functor LcllCnst (168) has the interpretation of sending any ∞ -bundle E (cid:47) (cid:47) X (Notation 2.45) to its ∞ -sheaf of local sections E : = LcllCnst X ( E ) ; ii) the direct image of the base geometric morphism (71) has the interpretation of sending any ∞ -sheaf to its ∞ -groupoid of global sections: ∞ -bundlesover X H / X (cid:111) (cid:111) i X (cid:63) (cid:95) ( − ) : = LcllCnstnt X form ∞ -sheaf of local sections ⊥ (cid:47) (cid:47) Γ X (cid:51) (cid:51) ∞ -sheaveson X ´Et X (cid:111) (cid:111) ∆ X Γ X form ∞ -groupoid of global sections ⊥ (cid:47) (cid:47) Groupoids ∞ (179)Notice that the global sections of the ∞ -sheaf of local sections of an ∞ -bundle E is the global sections of that ∞ -bundle (as in Remark 2.94): Γ X (cid:0) E (cid:1) (cid:39) Γ X ( E ) (by the essential uniqueness of the base geometric morphism (Prop. 2.43) and the fact that the base geometricmorphism on ∞ -bundles forms global sections, Remark 2.95). ´Etale groupoids.Definition 3.35 ( ´Etale groupoid) . Let H be an elastic ∞ -topos (Def. 3.21). (i) We say that X • ∈ Groupoids ( H ) (Def. 2.68) is an ´etale groupoid if all its face maps are local diffeomorphisms(Def. 3.26): X • is ´etale groupoid ⇔ ∀ n ∈ N ≤ i ≤ n X n + d i ´et (cid:47) (cid:47) X n . (ii) We write ´EtaleGroupoids ( H ) (cid:31) (cid:127) (cid:47) (cid:47) Groupoids ( H ) ∈ Categories ∞ (180)for the full sub- ∞ -category of that of all groupoids (97) on those that are ´etale groupoids.As a variant of Prop. 2.70 we have: 51 roposition 3.36 ( ´Etale groupoids are equivalent to stacks with ´etale atlases) . Let H be an elastic ∞ -topos (Def. 3.21) and X • ∈ Groupoids ( H ) (Def. 2.68). Then the following con-ditions are equivalent: (i) The groupoid X • is an ´etale groupoid (Def. 3.35). (ii) The associated atlas X a (cid:47) (cid:47) (cid:47) (cid:47) X (via Prop.2.70) is a local diffeomorphism (Def. 3.26). (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) X × X X (cid:39) ´et (cid:15) (cid:15) (cid:79) (cid:79) ´et ´et (cid:15) (cid:15) X s (cid:15) (cid:15) (cid:79) (cid:79) e t (cid:15) (cid:15) “´etale groupoid” X a ´et (cid:15) (cid:15) (cid:15) (cid:15) X (cid:15) (cid:15) (cid:15) (cid:15) “´etale atlas” X (cid:39) lim −! X • “´etale stack” (181) Proof.
By definition of local diffeomorphisms, we need to demonstrate the logical equivalence shown on the left: ∀ n φ ! n X n η ℑ Xn (cid:47) (cid:47) X φ (cid:15) (cid:15) (pb) ℑ X n ℑ X φ (cid:15) (cid:15) X n η ℑ Xn (cid:47) (cid:47) ℑ X n ⇔ X η ℑ X (cid:47) (cid:47) a (cid:15) (cid:15) (pb) ℑ X ℑ a (cid:15) (cid:15) lim −! X • η ℑ lim −! X • (cid:47) (cid:47) ℑ lim −! X • ⇔ X η ℑ X (cid:47) (cid:47) a (cid:15) (cid:15) (pb) ℑ X ℑ a (cid:15) (cid:15) lim −! X • lim −! η ℑ X • (cid:47) (cid:47) lim −! ℑ X • (182)But since ℑ preserves all limits and colimits (being a left and a right adjoint, Prop. 2.26), we have (a) also thelogical equivalence shown on the right of (182); and (b) that ℑ X • is itself a groupoid with atlas ℑ a , and that X • η ℑ X • (cid:47) (cid:47) ℑ X • is a morphism in Groupoids ( H ) (97). By (a) , it is now sufficient to prove the composite logicalequivalence in (182). By (b) , this follows with Prop. 2.73. (cid:3) Proposition 3.37 (Tangent stacks) . Let H be an elastic ∞ -topos (Def. 3.21) and X • ∈ ´EtaleGroupoids ( H ) (Def.3.35) with ´etale atlas X ´et (cid:47) (cid:47) X (via Prop. 3.36). Then: (i) the system of tangent bundles T X • (Def. 3.29) is itself an ´etale groupoid (Def. 3.35), the tangent groupoid; (ii) its atlas (under Prop. 3.36) is the differential T X Ta (cid:47) (cid:47) T X of the given atlas, hence the tangent stack is:T X (cid:39) lim −! T X • (183) Proof. (i)
That
T X • is itself a groupoid (Def. 2.68) follows because both the tangent bundle construction T ( − ) (165) as well as the groupoid Segal conditions (96) are pullback constructions, hence limits, which commute overeach other. To see that T X • is an ´etale groupoid, consider the following diagram: (cid:116) (cid:116) (cid:52) (cid:52) (cid:116) (cid:116) (cid:52) (cid:52) (cid:116) (cid:116) (cid:116) (cid:116) (cid:52) (cid:52) (cid:116) (cid:116) (cid:52) (cid:52) (cid:116) (cid:116) T (cid:0) X × X X (cid:1) (cid:15) (cid:15) (cid:47) (cid:47) (cid:116) (cid:116) (cid:52) (cid:52) (cid:116) (cid:116) X × X X (cid:15) (cid:15) (cid:116) (cid:116) (cid:52) (cid:52) (cid:116) (cid:116) T X (cid:47) (cid:47) (cid:15) (cid:15) (cid:118) (cid:118) X (cid:15) (cid:15) (cid:118) (cid:118) lim −! T X • (cid:47) (cid:47) (cid:15) (cid:15) X (cid:15) (cid:15) (cid:116) (cid:116) (cid:52) (cid:52) (cid:116) (cid:116) (cid:52) (cid:52) (cid:116) (cid:116) (cid:116) (cid:116) (cid:52) (cid:52) (cid:116) (cid:116) (cid:52) (cid:52) (cid:116) (cid:116) X × X X (cid:47) (cid:47) (cid:116) (cid:116) (cid:52) (cid:52) (cid:116) (cid:116) ℑ (cid:0) X × X X (cid:1) (cid:116) (cid:116) (cid:52) (cid:52) (cid:116) (cid:116) X (cid:47) (cid:47) (cid:117) (cid:117) ℑ X (cid:117) (cid:117) X (cid:47) (cid:47) ℑ X (184)Here the simplicial sub-diagram in the top right consists of local diffeomorphism by the assumption that X • is ´etale.But this implies that all the horizontal squares in the top of (184) are Cartesian, by Prop 3.31, hence that also allmorphisms of the simplicial sub-diagram in the top left are local diffeomorphisms, by Lemma 3.27. (ii) To see (183) we need to show that the front square in (184) is Cartesian. Observe: (a)
All horizontal squares in (184) are Cartesian: the top ones by the above argument for (i) , the bottom ones bythe assumption that X • is ´etale. (b) All solid vertical squares in (184) are also Cartesian, by definition (165) of tangent bundles.52 c) The object X in the bottom front left of (184) is not just the colimit of the simplicial sub-diagram in thebottom left, but in fact of the full left sub-diagram (because of the colimit of the top left sub-diagram in thefront top left). Similarly, the object ℑ X is in fact the colimit over the full right sub-diagram in (184) (usingthat ℑ preserves colimits, being a left adjoint, Prop. 2.26).Now (a) and (b) verify the assumption of Prop. 2.32 applied to the diagram (184), regarded as a natural transfor-mation from its left part to its right part; and with (c) , the conclusion of Prop. 2.32 says that the front square in(184) is Cartesian. (cid:3) Lemma 3.38 (Degreewise local diffeomorphisms of ´etale groupoids) . Let H be an elastic ∞ -topos (Def. 3.21) andX • , Y • ∈ ´EtaleGroupoids ( H ) (Def. 3.35). If a morphism X • f • (cid:47) (cid:47) Y • is such that for all n ∈ N , the componentX n f n (cid:47) (cid:47) Y n is a local diffeomorphism (Def. 3.26), then induced morphism on stacks X lim −! f • (cid:47) (cid:47) Y is also alocal diffeomorphism (Def. 3.36).Proof. Consider the following diagram: (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) X (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:47) (cid:47) η ℑ X (cid:39) (cid:39) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) Y (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) η ℑ Y (cid:38) (cid:38) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) ℑ X (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:47) (cid:47) ℑ Y (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) X (cid:47) (cid:47) (cid:15) (cid:15) η ℑ X (cid:39) (cid:39) Y (cid:15) (cid:15) η ℑ Y (cid:38) (cid:38) ℑ X (cid:15) (cid:15) (cid:47) (cid:47) ℑ Y (cid:15) (cid:15) X (cid:47) (cid:47) η ℑ X (cid:39) (cid:39) Y η ℑ Y (cid:39) (cid:39) ℑ X (cid:47) (cid:47) ℑ Y Observe that: (a) all solid η ℑ -naturality squares in this diagram are Cartesian, by the assumption that the rear part of thediagram is a degreewise local diffeomorphism of ´etale groupoids. (b) Y is not just the colimit of the partial diagram Y • in the rear right, but in fact is also the colimit of the fullnon-dashed rear part of the diagram (using that X is the colimit of the rear left part). Similarly, ℑ Y is thecolimit of the non-dashed front part of the diagram (using that ℑ preserves limits and colimits, being a leftand a right adjoint, Prop. 2.26).Hence if we regard the diagram as a natural transformation from its rear to its front part, then Prop. 2.32 appliesand says that also the bottom dashed square is Cartesian, and hence that X ! Y is a local diffeomorphism. (cid:3) Definition 3.39 ( ´Etalification of groupoids) . Let H be an elastic ∞ -topos (Def. 3.21) and X • ∈ Groupoids ( H ) (Def.2.68). Notice that, by Prop. 2.70 for all n ∈ N we have for all 0 ≤ i ≤ n that all face maps X n + d i (cid:47) (cid:47) X n arein fact equivalent to each other, being related by an automorphism of X n + given by permutation of fiber productfactors (98) X • (cid:39) X (cid:39) (cid:32) (cid:32) X (cid:39) (cid:32) (cid:32) X (cid:39) (cid:32) (cid:32) X (cid:111) (cid:111) d (cid:118) (cid:118) d X (cid:111) (cid:111) d (cid:117) (cid:117) d (cid:1) (cid:1) d X (cid:111) (cid:111) (cid:119) (cid:119) (cid:3) (cid:3) (185)(and similarly for the degeneracy maps). Therefore, we may regard X • as a diagram in the slice H X . and apply L X (169) to this diagram (185) to obtain X ´et • (cid:39) L X X (cid:39) (cid:38) (cid:38) L X X (cid:39) (cid:38) (cid:38) L X X (cid:39) (cid:38) (cid:38) X (cid:111) (cid:111) ´et (cid:117) (cid:117) ´et L X X (cid:111) (cid:111) ´et (cid:115) (cid:115) ´et (cid:125) (cid:125) ´et L X X (cid:111) (cid:111) (cid:118) (cid:118) (cid:1) (cid:1) (186)Observe that: 53 a) the simplicial diagram (186) is again a groupoid, since the right adjoint functor L X preserves the character-izing fiber products (96) (by Prop. 2.26); (b) this groupoid is ´etale (Def. 3.35), since the morphisms of the form L X X n −! X in (186) are local diffeo-morphisms by construction, whence all other morphisms L X X n −! L X X n are local diffeomorphisms bythe left-cancellation property (164).Hence we say that: (i) The simplicial diagram (186) is the ´etalification of the groupoid X • . X ´et • ∈ ´EtaleGroupoids ( H ) . (187) (ii) If the corresponding atlas of X • (via Prop. 2.70) is denoted X (cid:47) (cid:47) (cid:47) (cid:47) X , then we write X ´et (cid:47) (cid:47) (cid:47) (cid:47) X ´et (188)for the corresponding ´etale atlas (via Prop. 3.36) of the ´etalified groupoid (187). We present a formulation of super-geometry internal to ∞ -toposes which we call solid [Sc13]. Super-geometry.Definition 3.40 (Solid ∞ -topos) .(i) An ∞ -topos H (Def. 2.30) over B = Groupoids ∞ is a solid ∞ -topos if its base geometric morphism (Prop. 2.24),to be called Pnts : H (cid:47) (cid:47) B , is equipped with a factorization as follows, with adjoints (Def. 2.24) as shown: Shp : Γ : H × “even” Evn ⊥ (cid:47) (cid:47) (cid:111) (cid:111) “bosonic” Bsnc ⊥ (cid:63) (cid:95) “super shape” Shp sup ⊥ (cid:47) (cid:47) (cid:111) (cid:111) “super discrete” Disc sup (cid:63) (cid:95)
Pnts inf ⊥ (cid:47) (cid:47) (cid:111) (cid:111) Chtc ⊥ (cid:63) (cid:95) H (cid:32) (cid:111) (cid:111) Rdcd ⊥ (cid:63) (cid:95) Shp inf ⊥ (cid:47) (cid:47) (cid:111) (cid:111) Disc inf (cid:63) (cid:95) − − − H ℜ × Shp ℜ ⊥ (cid:47) (cid:47) (cid:111) (cid:111) Disc ℜ ⊥ (cid:63) (cid:95) Pnts ℜ (cid:47) (cid:47) B : Disc solid ∞ -topos bosonicsub-topos reducedsub-topos discretesub-topos (189) (ii) In particular, a solid ∞ -topos is also an elastic ∞ -topos (Def. 3.21), as its is sub- ∞ -topos H (cid:32) of bosonic objects. (iii) We write (cid:0) ⇒ : = Bsn ◦ Evn (cid:1) “even” (cid:97) (cid:0) (cid:32) : = Bsn ◦ Shp sup (cid:1) “bosonic” (cid:97) (cid:0)
Rh : = Disc sup ◦ Shp sup (cid:1) “rheonomic” : H −! H (190)for the further induced modalities (20) ( solid modalities ) accompanying the elastic modalities (155) and the cohe-sive modalities (128). Examples of solid ∞ -toposes. We indicate an example of a solid ∞ -topos (Def. 3.40). For full details on theconstruction see [SS20c]. In generalization of Def. 3.22 we have the following: Definition 3.41 ( ∞ -Jets of super Cartesian spaces) .(i) Write ∞ JetsOfSuperCartesianSpaces (cid:31) (cid:127) C ∞ ( − ) (cid:47) (cid:47) CommutativeSuperAlgebras op R R n | q × D W (cid:31) (cid:47) (cid:47) C ∞ (cid:0) R n (cid:1) ⊗ R ∧ • R (cid:0) R q (cid:1) ⊗ R ( R ⊕ W ) (191)for (as in [KS97][KS00]) the full subcategory of the opposite of super-commutative super-algebras over the realnumbers on those which are tensor products of (a) algebras C ∞ ( R n ) of smooth functions on a Cartesian space R n , for d ∈ N ; (b) Grassmann algebras ∧ • R R q on q ∈ N generators in odd degree;54 c) finite dimensional R ⊕ W ∈ CommutativeAlgebras with a single nilpotent maximal ideal W . (ii) We regard this as a site via the the coverage (i.e., a Grothendieck pre-topology) whose covers are of the form (cid:110) R n × R | q (cid:124) (cid:123)(cid:122) (cid:125) R n | q × D f i × id × id (cid:47) (cid:47) R n × R | q × D (cid:111) i ∈ I such that (cid:110) R n f i (cid:47) (cid:47) R n (cid:111) i ∈ I is a cover in CartesianSpaces (Def. 2.5). Lemma 3.42 (Reflections of super-commutative algebras into commutative algebras) . The canonical inclusion of ∞ JetsOfCartesianSpaces (Def. 3.22) into ∞ JetsOfSuperCartesianSpaces (Def. 3.41) has a left and a right adjoint(Def. 2.24) ∞ JetsOfSuperCartesianSpaces
Evn (cid:47) (cid:47) (cid:111) (cid:111)
Bsnc ⊥⊥ (cid:63) (cid:95) Shp sup (cid:47) (cid:47) ∞ JetsOfCartesianSpaces (192) where: (i)
The left adjoint
Evn in (192) is given in terms of super-algebras of smooth functions (191) by passage to thesub-algebra of even-graded elements:C ∞ (cid:16) Evn (cid:0) R n | q × D (cid:1)(cid:17) (cid:39) C ∞ (cid:0) R n | q × D (cid:1) even (cid:39) C ∞ (cid:0) R n × D (cid:1) ⊗ R C ∞ (cid:0) R | q (cid:1) even . (193) (ii) The right adjoint
Shp sup in (192) is given in terms of super-algebras of smooth functions (191) by passage tothe quotient algebra by the ideal of odd-graded elements:C ∞ (cid:16) Shp sup (cid:0) R n | q × D (cid:1)(cid:17) (cid:39) C ∞ (cid:0) R n | q × D (cid:1) / C ∞ (cid:0) R n | q × D (cid:1) odd (cid:39) C ∞ (cid:0) R n × D (cid:1) ⊗ R C ∞ (cid:0) R | q (cid:1) / C ∞ (cid:0) R | q (cid:1) odd (cid:124) (cid:123)(cid:122) (cid:125) (cid:39) R (cid:39) C ∞ (cid:0) R n × D (cid:1) (194) and hence directly by Shp sup (cid:0) R n | q × D (cid:1) (cid:39) R n × D . (195) Proof.
By regarding the situation under the defining embedding as being in CommutativeSuperAlgebras R (Def.3.41), it is equivalent to the statement that the canonical inclusion of commutative algebras into super-commutativesuper-algebras has a right and a left adjoint given by passage to the even sub-algebra and to the quotient by the oddideal, respectively:CommutativeSuperAlgebras R A A / A odd (cid:47) (cid:47) (cid:111) (cid:111) ⊥⊥ (cid:63) (cid:95) A A even (cid:47) (cid:47) CommutativeAlgebras R . This follows readily by inspection from the fact that homomorphisms of super-algebras preserve super-degree, bydefinition. One place where this adjoint triple has been made explicit before is [CR12, below Example 3.18]. (cid:3)
Example 3.43 (Jets of super-geometric ∞ -groupoids) . The ∞ -category of ∞ -sheaves (Def. 2.42) ∞ JetsOfSupergeometricGroupoids ∞ : = Sheaves ∞ (cid:0) ∞ JetsOfSuperCartesianSpaces (cid:1) over the site from Def. 3.41 is a solid ∞ -topos (Def. 3.40).55 i) Its bosonic (190) sub- ∞ -topos is that of ∞ JetsOfSmoothGroupoids (Example 3.24) and its reduced (154) sub- ∞ -topos that of SmoothGroupoids ∞ (Example 3.18): ∞ JetsOfSupergeometricGroupoids ∞ Evn (cid:47) (cid:47) (cid:111) (cid:111)
Bsnc ⊥ (cid:63) (cid:95) Shp sup ⊥ (cid:47) (cid:47) (cid:111) (cid:111) Disc sup ⊥ (cid:63) (cid:95) ∞ JetsOfSmoothGroupoids ∞ (cid:111) (cid:111) Disc inf (cid:63) (cid:95)
SmoothGroupoids ∞ (cid:111) (cid:111) Disc (cid:63) (cid:95)
Gropoids ∞ ... (cid:63)(cid:31) (cid:79) (cid:79) ∞ (cid:63)(cid:31) (cid:79) (cid:79) JetsOfSmoothGroupoids ∞ (cid:63)(cid:31) (cid:79) (cid:79) (cid:116) (cid:116) (cid:113)(cid:81) where the adjoint triple (cid:0) Evn (cid:97)
Bsnc (cid:97)
Shp sup (cid:1) arises by left Kan extension from that of Lemma 3.42. (ii)
The full inclusion of SmoothManifolds, inherited from (147), extends to a full inclusion of super-manifolds (asin [CCF11, 4.6][HKST11, 2]):
SmoothManifolds (cid:31) (cid:127)
Disc sup (cid:47) (cid:47)
SuperManifolds (cid:31) (cid:127) (cid:47) (cid:47) ∞ JetsOfSupergeometricGroupoids ∞ (196) (iii) Accordingly, super-Lie groups (e.g. [Ya93][CCF11, 7]) embed faithfully into all group objects (Prop. 2.74):
Groups (cid:0)
SmoothManifolds (cid:1)
Lie groups (cid:31) (cid:127)
Disc sup (cid:47) (cid:47)
Groups (cid:0)
SuperManifolds (cid:1) super Lie groups (cid:31) (cid:127) (cid:47) (cid:47)
Groups (cid:0) ∞ JetsOfSupergeometricGroupoids ∞ (cid:1) (197) (iv) In particular, for d ∈ N and N ∈ Spin ( d , ) Representations R , the corresponding supersymmetry groups, i.e., the super-Poincar´e group and its underlying translational super-Minkowski group (e.g. [Fr99, §3]) are group objects R d , | N super-Minkowskisuper Lie group (cid:31) (cid:127) (cid:47) (cid:47) Iso (cid:0) R d , | N (cid:1) super-Poincar´esuper Lie group (cid:47) (cid:47) (cid:47) (cid:47) Spin ( d , ) ∈ Groups (cid:0) ∞ JetsOfSupergeometricGroupoids (cid:1) . (198) Remark 3.44 (Superspace cohomology theory in solid ∞ -toposes) . The intrinsic cohomology (22) in the solid ∞ -topos of ∞ JetsOfSupergeometricGroupoids ∞ (Example 3.43) (i) includes the super-rational cohomology of super-Minkowski spacetimes (198) that governs the fundamental( κ -symmetric) super p -brane sigma-models of string/M-theory [FSS13b][FSS16a][FSS16b], review in [FSS19a]. (ii) Its enhancement to twisted super-rational cohomology of super-Minkowski spacetimes (198), which happens(by Remark 2.96) in the intrinsic cohomology of the tangent ∞ -topos T (cid:0) ∞ JetsOfSupergeometricGroupoids ∞ (cid:1) (Ex-ample 2.51), encodes the double dimensional reduction from fundamental M-branes to D-branes [BSS18]. (iii) Its enhancement to proper equivariant super-rational cohomology of super-Minkowski spacetimes (198),which happens (by Remark 5.4 and Theorem 5.9 below) in the intrinsic cohomology of the singular-solid ∞ -topos Singular ∞ JetsOfSupergeometricGroupoids ∞ (Example 3.2 below), encodes also the black (solitonic) super p -branes [HSS18]. Lemma 3.45 (In super-geometric groupoids ´etale implies bosonic) . In the solid ∞ -topos of ∞ JetsOfSupergeometricGroupoids (Example 3.41) we have a natural equivalence (cid:32) ◦ ℑ (cid:39) ℑ (199) saying that ℑ -modal objects (155) are bosonic (190) .Proof. Observe that on ∞ JetsOfSuperCartesianSpaces y (cid:44) −! ∞ JetsOfSupergeometricGroupoids ∞ (Def. 3.41), wehave a natural equivalence ℜ ◦ ⇒ (cid:39) ℜ (200)56aying that the reduction (155) of the even aspect (190) of the space is equivalently the reduced aspect.To see this, consider R n | q × D W ∈ ∞ JetsOfSuperCartesianSpaces and use, by Example 3.43 with Lemma 3.42,the operation ℜ ◦ ⇒ is given in terms of the defining super-algebras of functions (3.41) by passage to the reducedalgebra of the even subalgebra: C ∞ (cid:16) ℜ ◦ ⇒ (cid:0) R n | q × D W (cid:1)(cid:17) (cid:39) (cid:16)(cid:0) C ∞ (cid:0) R n (cid:1) ⊗ R (cid:0) ∧ • R R q (cid:1) ⊗ R ( R ⊕ W ) (cid:1) even (cid:17) red (cid:39) (cid:16) C ∞ (cid:0) R n (cid:1) ⊗ R (cid:0) ∧ • R R q (cid:1) even (cid:124) (cid:123)(cid:122) (cid:125) (cid:39) R ⊕∧ R q ⊕∧ R q ⊕··· ⊗ R ( R ⊕ W ) (cid:17) red (cid:39) (cid:16) C ∞ ( R n ) ⊗ R (cid:0) R ⊕ ( W ⊕ ∧ R q ⊕ ∧ R q ⊕ · · · ) (cid:1)(cid:17) red (cid:39) C ∞ ( R n ) ⊗ R (cid:0) R ⊕ ( W ⊕ ∧ R q ⊕ ∧ R q ⊕ · · · ) (cid:1) red (cid:124) (cid:123)(cid:122) (cid:125) (cid:39) R (cid:39) C ∞ ( R n ) . Here in the last step we used that every non-unit element in the Grassmann algebra is nilpotent. But, by (194) and(158), we also have C ∞ (cid:0) ℜ ( R n | q × D W ) (cid:1) (cid:39) C ∞ (cid:0) Shp inf ◦ Shp sup ( R n | q × D W ) (cid:1) (cid:39) C ∞ (cid:0) Shp inf ( R n × D W ) (cid:1) (cid:39) C ∞ (cid:0) R n (cid:1) , where in the first step we used the elastic structure (189) ℜ : = Bsnc ◦ Rdcd ◦ Shp inf ◦ Shp sup leaving the two fullembeddings on the left notationally implicit. Since all these equivalences are natural, this implies (200). With this,we have the following sequence of natural equivalences for general X ∈ H : = ∞ JetsOfSupergeometricGroupoids ∞ : H (cid:0) R n | q × D , (cid:32) ◦ ℑ ( X ) (cid:1) (cid:39) H (cid:16) ℜ ◦ ⇒ (cid:0) R n | q × D (cid:1) , X (cid:17) (cid:39) H (cid:16) ℜ (cid:0) R n | q × D (cid:1) , X (cid:17) (cid:39) H (cid:0) R n | q × D , ℑ X (cid:1) , where the first and the last steps are the defining hom-equivalences (47) while the middle step is (200). Thus thestatement (199) follows, by the ∞ -Yoneda lemma (Prop. 2.38). (cid:3) Given a cohesive ∞ -topos H ⊂ as in §3.1.1, we construct here a new ∞ -topos H (Def. 3.48 below), to be called singular-cohesive , with the following properties:1. H contains ((213) below) for each finite group G , an object ≺ G ∈ H , to be thought of as the generic G -orbi-singularity ( Figure D ).2. H carries (Prop. 3.50 below) an adjoint triple of modalities (20) to be read as follows < “singular” (cid:97) ⊂ “smooth” (cid:97) ≺ “orbi-singular” with H ⊂ being the full sub- ∞ -category of smooth objects in H ,3. such that (Prop. 3.62 below): < (cid:0) ≺ G (cid:1) (cid:39) ∗ “The purely singular aspect of an orbi-singularity is the quotient of a point, hence a point.” ⊂ (cid:0) ≺ G (cid:1) (cid:39) ∗ (cid:12) G “The purely smooth aspect of an orbi-singularity is the homotopy quotient of a point.” ≺ (cid:0) ≺ G (cid:1) (cid:39) ≺ G “An orbi-singularity is purely orbi-singular.”57ssentially this list of conditions might completely characterize H to be as in Def. 3.48 below. Here weleave a fully axiomatic characterization of singular cohesion as an open problem and are content with making thefollowing definitions: Singular cohesive geometry.Definition 3.46 (The 2-site of singularities) .(i)
We write Singularities : = Groupoids ≤ , cn , fin (cid:31) (cid:127) (cid:47) (cid:47) Groupoids ∞ (201)for the full sub- ∞ -category of ∞ -groupoids on the connected 1-truncated objects whose π is finite. (ii) A skeleton of this ( , ) -category has, of course, as objects the delooping groupoids (Example 2.14) ∗ (cid:12) G thatare presented by a single object and a finite group G of automorphisms of that object. (iii) When regarded as objects of Singularities in (201), we will denote these by “ ≺ ” attached to the symbol for thegroup: ≺ G (cid:95) (cid:15) (cid:15) ∈ Singularities (cid:127) (cid:95) (cid:15) (cid:15) ∗ (cid:12) G ∈ Groupoids ∞ (202) (iv) The hom- ∞ -groupoids between these singularities are, equivalently, the action groupoids (Example 2.15)whose objects are group homomorphisms and whose morphisms are conjugation actions on these:Singularities (cid:0) ≺ G , ≺ G (cid:1) : = Groupoids ∞ (cid:0) ∗ (cid:12) G , ∗ (cid:12) G (cid:1) (cid:39) Groups ( G , G ) (cid:12) conj G (203) (v) We regard Singularities as an ∞ -site with trivial Grothendieck topology, so that ∞ -sheaves on Singularities are ∞ -presheaves (65). Remark 3.47 (The global orbit category) . The category Singularities in Def. 3.46 is sometimes known in theliterature as the “global orbit category” (though at other times this term is used for its wide but non-full subcategoryon the faithful morphisms). It has elsewhere been denoted: “Orb case 1 (cid:13) ” (in [HG07, 4.1]), “Glob” (in [Re14, 2.2]),“Orb” (in [K¨o16, 2.1][Ju20, 3.2]) and (up to equivalence) “ O gl ” (in [Schw17][K¨o16, 2.2]). The terminology inDef. 3.46 is meant to be more suggestive of the role this category plays in the theory, from the perspective ofcohesive homotopy theory. In fact, the (global) orbit category is often taken to contain not just all finite groups,but all compact Lie groups, with the hom-spaces then being the geometric realization of the topological mappinggroupoids. We restrict to discrete groups (hence finite if compact) for reasons explained in Remark 3.64 below.This restriction is also amplified in [DHLPS19]. Definition 3.48 (Singular-cohesive ∞ -topos) . Consider a cohesive ∞ -topos (Def. 3.1), now to be denoted with“ ⊂ ”-subscripts H ⊂ × Shp ⊥ (cid:47) (cid:47) (cid:111) (cid:111) Dsc ⊥ (cid:63) (cid:95) Pnts ⊥ (cid:47) (cid:47) (cid:111) (cid:111) coDsc (cid:63) (cid:95) B ⊂ : = Groupoids ∞ (204)and assumed to have a site of Charts (Def. 3.9). The corresponding singular-cohesive ∞ -topos is that of H ⊂ -valued ∞ -sheaves (65) over the site of Singularities (Def. 3.46): H : = Sheaves ∞ (cid:0) Singularities , H ⊂ (cid:1) (cid:111) (cid:111) NnOrbSnglrSmth ⊥ (cid:47) (cid:47) (cid:79) (cid:79) Disc Pnts (cid:97) (cid:15) (cid:15) H ⊂ (cid:79) (cid:79) Disc Pnts (cid:97) (cid:15) (cid:15) B : = Sheaves ∞ (cid:0) Singularities , B ⊂ (cid:1) (cid:111) (cid:111) NnOrbSnglr (cid:63) (cid:95)
Smth ⊥ (cid:47) (cid:47) B ⊂ (205)58here horizontally we are showing the base geometric morphisms (Prop. 2.43) of sheaves over the site Singularities,while vertically we are showing the base geometric morphism (127) of H ⊂ over B ⊂ extended objectwise overSingularities, by functoriality. Lemma 3.49 (Singularities is 2-site for homotopical cohesion) . The 2-site
Singularities (Def. 3.46) is an ∞ -sitefor homotopical cohesion, in the sense of Def. 3.16.Proof. It is immediately checked that1. the terminal object is given by the trivial group: ∗ (cid:39) ≺ (206)2. Cartesian product is direct product of groups: ≺ G × ≺ G (cid:39) ≺ G × G . (cid:3) Proposition 3.50 (Singular cohesion) . A singular-cohesive ∞ -topos (Def. 3.48) H Pnts (cid:43) (cid:43)
Smth (cid:114) (cid:114) H ⊂ Pnts (cid:43) (cid:43) B Smth (cid:115) (cid:115) B ⊂ is itself cohesive (Def. 3.1) in two ways: (i) over the singular-base ∞ -topos B by the cohesion of H ⊂ ! B ⊂ (127) applied object-wise over all Singularities H × “shape” Shp ⊥ (cid:47) (cid:47) (cid:111) (cid:111) “discrete” Disc ⊥ (cid:63) (cid:95) “points” Pnts ⊥ (cid:47) (cid:47) (cid:111) (cid:111) “chaotic” Chtc (cid:63) (cid:95) B ; (207) (ii) over the non-singular cohesive base ∞ -topos H ⊂ (Def. 3.1) in that the global section geometric morphism H Smth −! H ⊂ of (205) is part of a cohesive adjoint quadruple, to be denoted H × “singular” Snglr ⊥ (cid:47) (cid:47) (cid:111) (cid:111) “not orbi-singular” NnOrbSnglr ⊥ (cid:63) (cid:95) “smooth” Smth ⊥ (cid:47) (cid:47) (cid:111) (cid:111) “orbi-singular” OrbSnglr (cid:63) (cid:95) H ⊂ . (208) Proof.
The first statement is immediate. The second statement follows via Lemma 3.49 by Example 3.17. (cid:3)
Notation 3.51 (Singular-elastic/solid ∞ -topos) . Let H be a singular-cohesive ∞ -topos (Def. 3.48) with underlyingsmooth cohesive ∞ -topos H ⊂ (cid:44) ! H . Then (i) if H ⊂ is in fact an elastic ∞ -topos (Def. 3.21), we say that H is a singular-elastic ∞ -topos ; (ii) if H ⊂ is in fact a solid ∞ -topos (Def. 3.40), we say that H is a singular-solid ∞ -topos . Definition 3.52 (Singular-cohesive modalities) . Given a singular cohesive ∞ -topos (Def. 3.48), with its singularcohesion from Prop. 3.50, we write (cid:0) < : = NnOrbSnglr ◦ Snglr (cid:1) “singular” (cid:97) (cid:0) ⊂ : = NnOrbSnglr ◦ Smth (cid:1) “smooth” (cid:97) (cid:0) ≺ : = OrbSnglr ◦ Smth (cid:1) “orbi-singular” (209)for the adjoint triple of modalities H ! H induced (20) via (208); accompanying the cohesive modalities (128)induced via (207).The above terminology reflects the difference (see Figure D) between a plain singularity < (singular but not orbi-singular) as opposed to its enhancement to an actual orbifold singularity ≺ . We record the following elementarybut important consequence: 59 roposition 3.53 (Smooth orbi-singular is smooth) . The singularity modalities (Def. 3.52) satisfy: < ◦ ⊂ (cid:39) ⊂ and ⊂ ◦ ≺ (cid:39) ⊂ . Proof.
As in Prop. 3.2. (cid:3)
Lemma 3.54 (Objectwise application of singularity modalities) . The singular-modalities in (208) are computedobjectwise over
Charts , as in Example 3.17, followed by ∞ -sheafification L Charts (69) : Sheaves ∞ (cid:0) Singularities × Charts (cid:1)
Snglr (cid:43) (cid:43) (cid:107) (cid:107)
NnOrbSnglr (cid:31) (cid:127) (cid:47) (cid:47)
PreSheaves ∞ (cid:0) Singularities × Charts (cid:1) lim −! Singularities (cid:47) (cid:47) (cid:111) (cid:111) ⊥ const Singularities (cid:63) (cid:95)
PreSheaves ∞ (cid:0) Charts (cid:1) L Charts (cid:47) (cid:47) (cid:111) (cid:111) ⊥ (cid:63) (cid:95) Sheaves ∞ (cid:0) Charts (cid:1)
Proof.
By essential uniqueness of adjoints (47). (cid:3)
Examples of singular-cohesive ∞ -toposes.Example 3.55 (Singular ∞ -groupoids) . For H ⊂ : = Groupoids ∞ the base ∞ -topos of plain ∞ -groupoids (35), thesingular-cohesive ∞ -topos from Def. 3.48SingularGroupoids ∞ : = Sheaves ∞ (cid:0) Singularities , Groupoids ∞ (cid:1) is that of traditional unstable global homotopy theory [Schw18, §1s], as discussed in this form in [Re14, §4.1](here with evaluation on all finite groups instead of all compact Lie groups). Example 3.56 (Singular-smooth ∞ -groupoids) . (i) We call the singular-cohesive ∞ -topos (Def. 3.48) over thoseof smooth ∞ -groupoids (Example 3.18) the ∞ -topos of singular-smooth ∞ -groupoids :SingularSmoothGroupoids ∞ : = Sheaves ∞ (cid:0) Singularities , SmoothGroupoids ∞ (cid:1) (cid:39) Sheaves ∞ (cid:0) CartesianSpaces × Singularities (cid:1) . (210) (ii) We call the singular-elastic ∞ -topos (Def. 3.51) over JetsOfSmoothGroupoids ∞ (Example 3.24)SingularJetsOfSmoothGroupoids ∞ : = Sheaves ∞ (cid:0) Singularities , JetsOfSmoothGroupoids ∞ (cid:1) (cid:39) Sheaves ∞ (cid:0) JetsOfCartesianSpaces × Singularities (cid:1) . (211) (iii) We call the singular-solid ∞ -topos (Def. 3.51) over ∞ JetsOfSupergeometricGroupoids ∞ (Example 3.43)Singular ∞ JetsOfSupergeometricGroupoids ∞ : = Sheaves ∞ (cid:0) Singularities , ∞ JetsOfSupergeometricGroupoids ∞ (cid:1) (cid:39) Sheaves ∞ (cid:0) ∞ JetsOfSuperCartesianSpaces × Singularities (cid:1) . (212)For the second lines of (211), (211), and (212), see Lemma 3.60. Basic properties of singular cohesion.Definition 3.57 (Orbi-singularities) . Let H be singular-cohesive ∞ -topos (Def. 3.48). (i) We regard the objects ≺ G ∈ Singularities (202) as objects of H under the ∞ -Yoneda-embedding (Prop. 2.37)and the inclusion (205) of discrete objects: ≺ G ∈ Singularities (cid:31) (cid:127) y (cid:47) (cid:47) Sheaves ∞ (cid:0) Singularities , B ⊂ (cid:1) (cid:31) (cid:127) Disc (cid:47) (cid:47)
Sheaves ∞ (cid:0) Singularities , H ⊂ (cid:1) = H . (213) (ii) More generally, for G ∈ Groups ( Groupoids ∞ ) Groups ( Disc ) (cid:47) (cid:47) Groups ( H ⊂ ) any discrete ∞ -group (207), we also write ≺ G : = ≺ ( B G ) ∈ H (214)for the orbi-singularization (208) of its delooping (101).60emma 3.61 shows that the two notations in Def. 3.57 are consistent with each other. Remark 3.58 (Finite groups in singular cohesion) . Given a singular-cohesive ∞ -topos (Def. 3.48), the images ofa finite group G under the following sequence of inclusions are naturally all denoted by the same symbol:Groups fin (cid:31) (cid:127) (cid:47) (cid:47) Groups ( Set ) (cid:31) (cid:127) (cid:47) (cid:47) Groups ( Groupoids ∞ ) (cid:31) (cid:127) Groups ( Disc ) (cid:47) (cid:47) Groups (cid:0) H ⊂ (cid:1) (cid:31) (cid:127) Grp ( NnOrbSnglr ) (cid:47) (cid:47) Groups (cid:0) H (cid:1) G (cid:31) (cid:47) (cid:47) G (cid:31) (cid:47) (cid:47) G (cid:31) (cid:47) (cid:47) G (cid:31) (cid:47) (cid:47) G (215)With this understood, we also have identifications as follows (where now the ambient ∞ -categories are implicitfrom the context): ∗ (cid:12) G (cid:39) Disc ( ∗ (cid:12) G ) and ≺ G (cid:39) Disc (cid:0) ≺ G (cid:1) (216)where on the right we are recalling the definition (213).Similarly: Remark 3.59 (Smooth charts in singular cohesion) . Consider a singular-cohesive ∞ -topos (Def. 3.48) with an ∞ -site Charts of charts (Def. 3.9). Then images of the charts U ∈ Charts under the ∞ -Yoneda embedding (Prop.2.37), and further under NnOrbSnglr (205), are naturally denoted by the same symbol: S y (cid:47) (cid:47) H ⊂ NnOrbSnglr (cid:47) (cid:47) H U (cid:31) (cid:47) (cid:47) U (cid:31) (cid:47) (cid:47) U (217) Lemma 3.60 ( ∞ -Yoneda on product site) . Consider a singular-cohesive ∞ -topos H (Def. 3.48) with an ∞ -site Charts of cohesive charts (Def. 3.9) for H ⊂ . (i) Then a site (Def. 2.42) for the full singular-cohesive H is the Cartesian product site SingularCharts : = Charts × Singularities (218) in that H (cid:39) Sheaves ∞ (cid:0) Charts × Singularities (cid:1) . (219) (ii) Under the ∞ -Yoneda embedding (Prop. 2.37) objects in the product site map to the Cartesian product of theirprolonged Yoneda embeddings (in the sense of Remark 3.58 and Remark 3.59): Charts × Singularities y (cid:47) (cid:47) H (cid:0) U , ≺ G (cid:1) (cid:31) (cid:47) (cid:47) U × ≺ G , where on the right we are using the abbreviated notation from (213) and (217) .Proof. On the one hand, we have a natural equivalence H (cid:16) y (cid:0) U , ≺ G (cid:1) , y (cid:0) U , ≺ G (cid:1)(cid:17) (cid:39) Charts ( U , U ) × Singularities (cid:0) ∗ (cid:12) G , ∗ (cid:12) G (cid:1) (220)by fully-faithfulness of the ∞ -Yoneda embedding (Prop. 2.37) and by the definition of product sites. On the otherhand, we have a sequence of natural equivalences H (cid:16) y (cid:0) U , ≺ G (cid:1) , U × ≺ G (cid:17) = H (cid:16) y (cid:0) U , ≺ G (cid:1) , NnOrbSnglr ( U ) × Disc (cid:0) ≺ G (cid:1)(cid:17) (cid:39) H (cid:16) y (cid:0) U , ≺ G (cid:1) , NnOrbSnglr ( U ) (cid:17) × H (cid:16) y (cid:0) U , ≺ G (cid:1) , Disc (cid:0) ≺ G (cid:1)(cid:17) (cid:39) H ⊂ (cid:16) Snglr (cid:0) y (cid:0) U , ≺ G (cid:1)(cid:1) , U (cid:17) × B (cid:16) S (cid:0) y (cid:0) U , ≺ G (cid:1)(cid:1) , ≺ G (cid:17) (cid:39) H ⊂ (cid:0) U , U (cid:1) × B (cid:0) ≺ G , ≺ G (cid:1) (cid:39) Charts (cid:0) U , U (cid:1) × Singularities (cid:0) ≺ G , ≺ G (cid:1) . (221)Here the first step is by definition, the second step is the universal property of the Cartesian product, and the thirdstep is the hom-equivalence (47) of the adjunctions Snglr (cid:97) NnOrbSnglr and S (cid:97) Disc, respectively. In the fourthstep, we use (136) and (223), respectively. The last step is the fully-faithfulness of the ∞ -Yoneda embedding(Prop. 2.37). Since both (220) and (221) are natural in (cid:0) U (cid:48) , ( ∗ (cid:12) G ) ≺ (cid:1) , and since their right hand sides coincide,it follows by the ∞ -Yoneda embedding (Prop. 2.37) that also the representatives of the left hand sides coincide: y (cid:0) U , ≺ G (cid:1) (cid:39) U × ≺ G . (cid:3) emma 3.61 (Images and pre-images of orbi-singularities) . Let H be a singular-cohesive ∞ -topos (Def. 3.48).Then the images and pre-images of the generic singularities ≺ G (213) under the functors (208) exhibiting thesingular cohesion are as follows (see Figure D): ≺ G (cid:38) Snglr (cid:115) (cid:115) (cid:24)
Smth (cid:43) (cid:43) (cid:107) (cid:107)
OrbSnglr (cid:24) ∗ (cid:12) G ∈ H ∗ = ∗ / G ∗ (cid:12) G (cid:38) NnOrbSnglr (cid:50) (cid:50) ∈ H ⊂ (222) Proof.
By the singular cohesion established in the proof of Prop. 3.50 we have that:1. the functor Snglr (cid:39) lim −! is the colimit functor (Prop. 2.36),2. the functor Smth (cid:39) Singularities (cid:0) ≺ , − (cid:1) is the hom-functor (25) out of the terminal object (206).Using this, we deduce the claim:1. Since colimits of representable ∞ -functors are equivalent to the point (Lemma 2.40) we haveSnglr (cid:0) ≺ G (cid:1) (cid:39) ∗ (cid:39) ∗ / G . (223)2. Observing that (203) reduces to Singularities (cid:0) ≺ , ≺ G (cid:1) (cid:39) ∗ (cid:12) G we haveSmth (cid:0) ≺ G (cid:1) (cid:39) ∗ (cid:12) G .
3. With this and by the various adjunctions we have, for U ∈ H ⊂ any geometically contractible generator (136)and K ∈ Groups fin any finite group, the following sequence of natural equivalences: H (cid:16) U × ≺ K , OrbSnglr ( ∗ (cid:12) G ) (cid:17) (cid:39) H ⊂ (cid:16) Smth (cid:0) U × ≺ K (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) (cid:39) U × Smth (cid:0) ≺ K (cid:1) , ∗ (cid:12) G (cid:17) (cid:39) H ⊂ (cid:0) U × ( ∗ (cid:12) K ) , ∗ (cid:12) G (cid:124)(cid:123)(cid:122)(cid:125) (cid:39) Disc ( ∗ (cid:12) G ) (cid:1) (cid:39) Groupoids ∞ (cid:0) Shp ( U ) (cid:124) (cid:123)(cid:122) (cid:125) (cid:39)∗ × ( ∗ (cid:12) K ) , ∗ (cid:12) G (cid:1) (cid:39) Singularities (cid:0) ≺ K , ≺ G (cid:1) (cid:39) B (cid:16) Shp ( U ) (cid:124) (cid:123)(cid:122) (cid:125) (cid:39)∗ × ≺ K , ≺ G (cid:17) (cid:39) B (cid:16) Shp (cid:0) U × ≺ K ) , ≺ G (cid:17) (cid:39) H (cid:16) U × ≺ K , Disc (cid:0) ≺ G (cid:1)(cid:17) (cid:39) H (cid:16) U × ≺ K , ≺ G (cid:17) , where in several steps we recognized geometric discreteness, by (216) in Remark 3.58.But, by Lemma 3.60, this chain of natural equivalences in total a natural equivalence of the form H (cid:16) y (cid:0) U , ≺ K (cid:1) , OrbSnglr (cid:0) ∗ (cid:12) G (cid:1)(cid:17) (cid:39) H (cid:16) y (cid:0) U , ≺ K (cid:1) , ≺ G (cid:17) . From this, the ∞ -Yoneda embedding (Prop. 2.37) implies that OrbSnglr (cid:0) ∗ (cid:12) G (cid:1) (cid:39) ≺ G . (cid:3) It is useful to re-express this in terms of the modalities:
Proposition 3.62 (Orbi-singularities are orbi-singular) . Let H be a singular-cohesive ∞ -topos (Def. 3.48) andconsider a finite group G ∈ Groups fin (215) . Then the images of the generic orbi-singularity ≺ G ∈ H (213) underthe modalities (209) are (see Figure D): < (cid:0) ≺ G (cid:1) (cid:39) ∗ , ⊂ (cid:0) ≺ G (cid:1) (cid:39) ∗ (cid:12) G , ≺ (cid:0) ≺ G (cid:1) (cid:39) ≺ G . (224) Proof.
This follows directly with Lemma 3.61 and the definition (209). For example: ≺ (cid:0) ≺ G (cid:1) (cid:39) OrbSnglr ◦ Smth (cid:0) ≺ G (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) (cid:39)∗ (cid:12) G (cid:124) (cid:123)(cid:122) (cid:125) (cid:39) ≺ G (cid:3)
62n the same vein, we also have the following immediate but important property:
Proposition 3.63 (Orbi-singularities are geometrically discrete) . Let H be a singular-cohesive ∞ -topos (Def. 3.48)and consider a finite group G ∈ Groups fin (215) . (i) Then the basic orbi-singularity ≺ G ∈ H (213) is geometrically discrete (128) and thus also pure shape: (cid:91) ≺ G (cid:39) ≺ G , S ≺ G (cid:39) ≺ G . (225) (ii) The same is true for
Smth ( ∗ (cid:12) G ) ≺ (cid:39) ∗ (cid:12) G: (cid:91) ( ∗ (cid:12) G ) (cid:39) ∗ (cid:12) G , S ( ∗ (cid:12) G ) (cid:39) ∗ (cid:12) G . (226) Proof.
Both statements follow immediately from the definitions and the fact that G is finite and hence geometricallydiscrete (215). (cid:3) Remark 3.64 (Need for discrete/finite groups in Singularities) . It is to make Lemma 3.61 and hence Prop. 3.62true that Def. 3.46 requires the global orbit category Singularities to consist of finite groups, instead of moregeneral compact Lie groups (Remark 3.47): If Singularities were to contain non-discrete compact Lie groups G ,then the same argument as in Lemma 3.61 would give in (224) the following more general formula: ⊂ ≺ G (cid:39) ∗ (cid:12) (cid:91) G (where on the right we think of the Lie group G as being cohesive via (147)). Since the condition G (cid:39) (cid:91) G char-acterizes discrete groups, this would break Prop. 4.2 below, in that then the shape of the orbi-singularization of atopological groupoid would take non-traditional values on non-discrete groups in the global orbit category.The following lemma further illustrates the nature of orbi-singular cohesion: Lemma 3.65 (Smooth 0-truncated objects are orbi-singular) . Let H be a singular-cohesive ∞ -topos (Def. 3.48).Then if X ∈ H ⊂ , is smooth (209) and 0-truncated (Def. 2.57), it is also orbi-singular (209) : τ ( X ) (cid:39) X and ⊂ ( X ) (cid:39) X ⇒ ≺ ( X ) (cid:39) X . (227) Proof.
Since X is smooth, there exists X ⊂ ∈ H ⊂ such that X (cid:39) Smth ( X ⊂ ) . Observe that X being 0-truncated impliesthat X ⊂ is 0-truncated, (by using in Def. 2.57 the hom-equivalence (47) of the right adjoint Smth).Now let S be any site (69) for H ⊂ . Then, for U ∈ S (cid:44) ! H ⊂ and G ∈ Groups fin , we have the following sequenceof natural equivalences, using the various adjoint functors, their idempotency and respect for products: ( ≺ X ) (cid:0) Smth ( U ) × ≺ G (cid:1) (cid:39) ( ≺ Smth ( X ⊂ )) (cid:0) Smth ( U ) × ≺ G (cid:1) (cid:39) H (cid:0) Smth ( U ) × ≺ G , ≺ Smth ( X ⊂ ) (cid:1) (cid:39) H (cid:0) Smth ( U ) × ⊂ ( ≺ G ) , Smth ( X ⊂ ) (cid:1) (cid:39) H (cid:0) Smth ( U × ( ∗ (cid:12) G )) , Smth ( X ⊂ ) (cid:1) (cid:39) H ⊂ (cid:0) U × ( ∗ (cid:12) G ) , X ⊂ (cid:1) (cid:39) Groupoids ∞ (cid:16) ∗ (cid:12) G , H ⊂ (cid:0) U , X ⊂ (cid:1)(cid:17) (cid:39) Groupoids ∞ (cid:16) ∗ , H ⊂ (cid:0) U , X ⊂ (cid:1)(cid:17) (cid:39) H ⊂ (cid:0) U , X ⊂ (cid:1) (cid:39) H (cid:0) Smth ( U ) , Smth ( X ⊂ ) (cid:1) (cid:39) H (cid:0) Smth ( U ) × ≺ G , Smth ( X ⊂ ) (cid:1) (cid:39) H (cid:0) Smth ( U ) × ≺ G , X (cid:1) (cid:39) X (cid:0) Smth ( U ) × ≺ G (cid:1) . ∞ -Yoneda embedding (Prop. 2.37), while the middle step uses the fact that X ⊂ is 0-truncated, hence that H ⊂ ( U , X ⊂ ) is 0-truncated (i.e. a set), to find that there is in fact no dependency on G .Hence the claim follows by the ∞ -Yoneda embedding (Prop. 2.37), in view of Lemma 3.60. (cid:3) Remark 3.66 (Degenerate case of orbi-singular) . The natural language statement of Lemma 3.65 shows that themodality ≺ “orbi-singular” (208) really means: “All singularities that are present are orbi-singularities.”, whichbecomes a trivially satisfied condition when there are no singularities, such as for smooth and 0-truncated objects. Interplay between geometric and singular cohesion.Lemma 3.67 (Smooth commutes with shape) . In a singular-cohesive ∞ -topos (Def. 3.48) the smooth-modality (209) commutes with all three cohesive modalities (128) (as per Prop. 3.50): ⊂ ◦ S (cid:39) S ◦ ⊂ , ⊂ ◦ (cid:91) (cid:39) (cid:91) ◦ ⊂ , ⊂ ◦ (cid:93) (cid:39) (cid:93) ◦ ⊂ . Proof.
Under the defining identification H (cid:39) Sheaves ∞ (cid:0) Singularities , H ⊂ (cid:1) , let X ∈ H be any object regarded as a H ⊂ -valued ∞ -presheaf on Singularities: X : ≺ K X ( ≺ K ) ∈ H ⊂ . Observe then (by Example 3.17 via Lemma 3.49) that ⊂ turns such a presheaf into the constant presheaf on itsvalue at the terminal object ≺ : (cid:0) ⊂ X (cid:1) : ≺ K X ( ≺ ) . On the other hand, the geometric modalities operate objectwise over Singularities (Remark 3.54): (cid:0) S X (cid:1) : ≺ K S (cid:0) X ( ≺ K ) (cid:1) . With this, we have the following sequence of natural equivalences for X ∈ H and ≺ K ∈ Singularities: (cid:0) ⊂ S X (cid:1) ( ≺ K ) (cid:39) (cid:0) S X (cid:1) ( ≺ ) (cid:39) S (cid:0) X ( ≺ ) (cid:1) (cid:39) S (cid:0) ( ⊂ X )( ≺ K ) (cid:1) (cid:39) (cid:0) S ⊂ X (cid:1) ( ≺ K ) . Hence the claim follows by the ∞ -Yoneda embedding (Prop. 2.37). The argument for (cid:91) and (cid:93) is analogous. (cid:3) Remark 3.68 (Dichotomy between naive and proper orbifold cohomology via singular-cohesion) . In contrast toLemma 3.67, the orbi-singular modality ≺ (209) does not commute with the cohesive shape modality S (128),in general. This phenomenon is the very source of the proper equivariant structure seen in singular-cohesive ∞ -toposes, reflected in the following dichotomy between geometric- and homotopy fixed points of an orbi-space andin the distinction between proper- and Borel-equivariant cohomology: ≺ ◦ S S ◦ ≺ Def. 3.69 (i)
Homotopyfixed-points Geometricfixed-points Def. 3.69 (ii)
Def. 5.1 Borel-equivariantcohomology Proper equivariantcohomology Def. 5.2Def. 5.13 Tangentially twistedcohomology Tangentially twistedproper orbifold cohomology Def. 5.15
Definition 3.69 (Geometric- and homotopy-fixed points) . Let H be a singular-cohesive ∞ -topos (Def. 3.48), G ∈ Groups ( H ) (Prop. 2.74) being discrete G (cid:39) (cid:91) G and 0-truncated G (cid:39) τ G , and ( X , ρ ) ∈ G Actions ( H ) (Prop. 2.79)with smooth X (cid:39) ⊂ X , hence X ∈ H ⊂ (cid:31) (cid:127) NnOrbSinglr (cid:47) (cid:47) H . For any subgroup K ⊂ G , the ∞ -groupoid of ≺ K -points in the slice (Prop. 2.46) over ≺ G (214)...64 i) ...of the orbi-singularization (208) of the shape (207) of X (cid:12) G is the homotopy fixed point space of X HmtpFxdPntSpc K ( X ) : = H (cid:14) ≺ G (cid:16) ≺ K , ≺ S ( X (cid:12) G ) (cid:17) . (228) (ii) ...of the shape (207) of the orbi-singularization (208) of X (cid:12) G is the geometric fixed point space of X GmtrcFxdPntSpc K ( X ) : = H (cid:14) ≺ G (cid:16) ≺ K , S ≺ ( X (cid:12) G ) (cid:17) . (229)On the right we are using Prop. 3.62 and Prop. 3.63 to see that both expressions indeed live in the slice over ≺ G . Proposition 3.70 (Homotopy-fixed point spaces are fixed loci in shapes) . The homotopy-fixed point spaces (228) of the G-space X in Def. 3.69 are, equivalently, the fixed-loci (Def. 2.97) of the shape
Shp ( X ) ∈ Groupoids ∞ (127) of X : HmtpFxdPntSpcs K ( X ) (cid:39) (cid:0) Shp ( X ) (cid:1) K ∈ Groupoids ∞ (230) with respect to the induced G (cid:39) S G-action (using Prop. 3.4, discreteness of G and cohesion in the form of Prop.3.2).Proof.
We claim a sequence of natural equivalences as follows:HmtpFxdPntSpc K ( X ) : = H (cid:14) ≺ G (cid:16) ≺ K , ≺ S ( X (cid:12) G ) (cid:17) (cid:39) H (cid:14) ≺ G (cid:16) ≺ K , ≺ ( S X ) (cid:12) G (cid:17) (cid:39) H (cid:14) OrbSnglr ( ∗ (cid:12) G ) (cid:16) OrbSinglr (cid:0) ∗ (cid:12) K (cid:1) , OrbSnglr (cid:0) ( S X ) (cid:12) G (cid:1)(cid:17) (cid:39) (cid:0) H ⊂ (cid:1)(cid:14) ∗ (cid:12) G (cid:0) ∗ (cid:12) K , ( S X ) (cid:12) G (cid:1) (cid:39) (cid:0) Groupoids ∞ (cid:1)(cid:14) ∗ (cid:12) G (cid:0) ∗ (cid:12) K , Shp ( X ) (cid:12) G (cid:1) (cid:39) (cid:0) Groupoids ∞ (cid:1)(cid:14) ∗ (cid:12) K (cid:0) ∗ (cid:12) K , Shp ( X ) (cid:12) K (cid:1) (cid:39) (cid:0) Shp ( X )) K (231)Here the first step is the definition (228), and the second step uses Prop. 3.4, discreteness of G and cohesion in theform of Prop. 3.2. In the third step we observe with ≺ K (cid:39) ≺ ( ∗ (cid:12) K ) (Lemma 3.61) and ≺ : = OrbSinglr ◦ Smth(209) that all objects and morphisms are in the image of OrbSnglr, and in the fourth step we use that this functoris fully faithful, by Prop. 3.50. In the fifth step, we similarly observe that all objects and morphisms are, in fact,furthermore in the image of Disc (by assumption on G and by definition of S : = Disc ◦ Shp (128)), which is fullyfaithful by the axioms of cohesion (127). The sixth step observes the universal factorization through the pullback ∗ (cid:12) K (cid:38) (cid:38) (cid:47) (cid:47) Shp ( X ) (cid:12) G (cid:118) (cid:118) ∗ (cid:12) G (cid:39) ∗ (cid:12) K (cid:47) (cid:47) Shp ( X ) (cid:12) K (cid:42) (cid:42) (cid:117) (cid:117) (pb) ∗ (cid:12) K (cid:41) (cid:41) Shp ( X ) (cid:12) G (cid:116) (cid:116) ∗ (cid:12) G The pullback, in turn, is the homotopy quotient of the restricted action, as shown, by Prop. 2.85. With this, thelast step follows by Example 2.99. In summary, the composite of the sequence of equivalences (231) gives thestatement (230). (cid:3)
Example 3.71 (Geometric fixed points generally differ from homotopy fixed points) . As in Example 3.56, let H : = SingularSmoothGroupoids ∞ . For n ∈ N , n ≥
1, consider the Cartesian space R n ∈ SmoothManifolds (cid:44) −! H ,via (147), and regard it as equipped with the additive translation action of Z n induced from the left action of theadditive group ( R n , +) on itself, under the canonical inclusion ( Z n , +) (cid:44) ! ( R n , +) :65 R n , ρ (cid:96) ) ∈ Z n Actions ( Hs ) . (232)So the quotient of this action R n (cid:12) Z n (cid:39) R n / Z n (cid:39) T n ∈ SmoothManifolds (cid:44) −! H is the standard n -torus. We thenhave for the two notions of fixed-point spaces from Def. 3.69: (i) The
Homotopy-fixed point space (228) of the action (232) is equivalently the point (by Prop. 3.70 and (146)):HmtpFxdPntSpc Z n ( R n ) (cid:39) (cid:0) S R n (cid:124)(cid:123)(cid:122)(cid:125) (cid:39)∗ (cid:1) Z n (cid:39) ∗ (ii) The geometric fixed point space (229) of the action (232) is emptyGmtrcFxdPntSpc Z n ( R n ) (cid:39) (cid:0) R n (cid:1) Z n (cid:39) ∅ This follows by Lemma 4.7, using that no element of the set underlying R n is fixed by the action of Z n .66 Orbifold geometry
Within an ambient context of singular-cohesive homotopy theory (§3), we now formulate the two geometric aspectsof orbifolds:- §4.1 – as cohesive spaces with orbi-singularities,- §4.2 – as cohesive spaces locally equivalent to a given model space.In the end, we combine both aspects to form the proper ∞ -categories of orbifolds : this is Def. 4.58 below. We observe (Prop. 4.2) that the shape of the orbi-singularization of a topological groupoid, regarded in singular-smooth homotopy theory (Example 3.56), is the corresponding orbispace in global equivariant homotopy theory.
Remark 4.1 (Orbispaces in topology and in global equivariant homotopy theory) .(i) Orbispaces in topology.
The term orbispace was originally introduced [Hae90] to mean the topologicalversion of orbifolds, i.e., Satake’s original concept [Sa56] but disregarding any differentiable structure. Fromthe perspective of ´etale groupoids/stacks, this means to consider topological groupoids/stacks instead of Liegroupoids/differentiable stacks. So this usage of the term “orbispace” serves to complete the following table:Smooth manifold Topological manifoldorbifold orbispace ( geometric sense ) Lie groupoid topological groupoiddifferentiable stack topological stackIn this sense, orbispaces have been discussed, e.g., in [Hae84][Hae91, §5][Ch01][He01]. (ii) Orbispaces in global equivariant homotopy theory.
In [HG07] it was suggested to change perspective andto instead regard these topological groupoids X top via the systems of homotopy types of all their geometric fixedpoint spaces, by the following formula [HG07, 4.2] (beware the differing conventions, as per Remark 3.47): G homotopy type of (fat) geometric realization of (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Maps (cid:0) B G , X top (cid:1) topological mapping groupoid (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) orbispace ( equivariant homotopical sense ) (233)This is a global-equivariant version of how topological G -spaces are incarnated in G -equivariant homotopy theoryvia Elmendorf’s theorem (recalled as Prop. B.10), and has served to motivate the development of global equivarianthomotopy theory [Schw18].In the course of this development, homotopy theorists adopted the term “orbispace” to refer not to the topo-logical groupoid X top (as [Hae90] originally did) but rather to the global equivariant homotopy type that is rep-resented via (233). Usage of the term orbispace in this sense of global homotopy theory is, after [HG07], in[Re14][K¨o16][Schw17][Lu19, 3][Ju20]. In [Ju20, 3.15] formula 233 is used (following suggestions in [Schw17,Introd.][Schw18, p. ix-x]) to define (abelian, non-geometric) cohomology of orbifolds with coefficients in globalequivariant spectra.Our Prop. 4.2 below shows that these two different meanings of the term “orbispace” in the literature aredisentangled as well as unified by the notion of singular cohesion (Def. 3.48), in that orbispaces in the sense (ii) are the shape S (127) of the orbi-singularization ≺ (209) of the topological groupoids in (i) :TopologicalGroupoids S ◦ ≺ (cid:47) (cid:47) Orbispaces X top (cid:47) (cid:47) (cid:16) ≺ G (cid:13)(cid:13) Maps (cid:0) B G , X top (cid:1)(cid:13)(cid:13) (cid:17) (234)Hence Prop. 4.2 below means that, before passing to their pure shape, we may think of the orbi-singularizations ofobjects in singular-cohesive ∞ -toposes as cohesive orbispaces , lifting the concept of plain orbispaces in the sense (ii) from plain homotopy theory to geometric (differential, ´etale) homotopy theory, hence back to sense (i) andbeyond. 67he crucial fact underlying the phenomenon (234), both in Prop. 4.2 and in Lemma 4.7 below, is that the probeof an orbi-singular object ≺ X ⊂ by a generic orbi-singularity ≺ K (202) is, by adjunction (209), equivalently theprobe of the underlying smooth object by the smooth aspect of ≺ K , hence is, by (224) in Prop. 3.62, the geometric G -fixed locus in X ⊂ : ≺ G (cid:47) (cid:47) ≺ X ⊂ (209) ⇔ ⊂ ≺ G (cid:47) (cid:47) X ⊂ (224) ⇔ ∗ (cid:12) G (cid:47) (cid:47) X ⊂ . (235)Equivalently, since ≺ G (cid:39) ≺ ( ∗ (cid:12) G ) (Lemma 3.61) the composite corespondence (235) is fully-faithfulness of ≺ . Example: Topological groupoids as cohesive orbispaces.Proposition 4.2 (Shape of orbi-singularized topological groupoid is orbispace) . Let H : = SingularSmoothGroupoids ∞ (Example 3.56), and let TopologicalGroupoids
Cdfflg (cid:47) (cid:47)
SmoothGroupoids ∞ NnOrbSnglr (cid:47) (cid:47) H X top (cid:31) (cid:47) (cid:47) X ⊂ be a topological groupoid, regarded via the embeddings (148) and (208) . If X ⊂ is such that both its space ofobjects and of morphisms are retracts of cell complexes (for instance: both are CW-complexes (27) ) then the shape (207) of its orbi-singularization (209) is, as an ∞ -presheaf (205) of ∞ -groupoids on Singularities (3.46) (i.e., onthe global orbit category, Remark 3.47) S ≺ X ⊂ ∈ Sh ∞ (cid:0) Singularities (cid:1) (cid:31) (cid:127)
Disc (cid:47) (cid:47) H given by the assignment (235) S ≺ X ⊂ : ≺ G (cid:13)(cid:13) Maps (cid:0) B G , X top (cid:1)(cid:13)(cid:13) , (236) where on the right we have the fat geometric realization of the topological functor groupoid [Se74] (see [HG07,2.3]), with B G (cid:39) ∗ (cid:12) G (Example 2.14) regarded as a finite topological groupoid.Proof.
Recall from (146) in Example 3.18 that Charts : = CartesianSpaces (Def. 2.5) is a site of cohesive charts(Def. 3.9) for SmoothGroupoids ∞ . We claim that for R n ∈ CartesianSpaces and ≺ G ∈ Singularities (Def. 3.46),hence R n × ≺ G ∈ CartesianSpaces × Singularities (Lemma 3.60), we have the following sequence of natural equiv-alences: H (cid:0) R n × ≺ G , ≺ X ⊂ (cid:1) = H (cid:0) R n × ≺ G , OrbSnglr (cid:0) X ⊂ (cid:1)(cid:1) (cid:39) H ⊂ (cid:0) Smth (cid:0) R n × ≺ G (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) (cid:39) R n × B G , X ⊂ (cid:1) (cid:39) H ⊂ (cid:16) R n , Maps (cid:0) B G , X ⊂ (cid:1)(cid:17) (cid:39) H ⊂ (cid:16) R n , Cdfflg
Maps (cid:0) B G , X top (cid:1)(cid:17) . (237)Here the first step is (240), the second is the hom-equivalence (47) of the adjunction Smth (cid:97) OrbSnglr (208)and using under the brace that Smth preserves products (by Prop. 2.26), that R n is already smooth, and thatSmth (cid:0) ≺ G (cid:1) (cid:39) ( ∗ (cid:12) G ) by (222). The third step is Lemma A.5.Since also the composite of all these natural equivalences is thus natural, the ∞ -Yoneda lemma (Prop. 2.38) impliesthat ≺ X ⊂ : ≺ K Cdfflg Maps (cid:0) B G , X ⊂ (cid:1) . Now, since S acts objectwise over ≺ K (207), we find from this that S ≺ X ⊂ : ≺ K S Cdfflg
Maps (cid:0) B G , X top (cid:1) (cid:39) Shp sTop (cid:16)
Maps (cid:0) B G , X top (cid:1)(cid:17) (cid:39) (cid:13)(cid:13) Maps (cid:0) B G , X top (cid:1)(cid:13)(cid:13) . Here the first step is (152) and the last step follow by Prop. 2.18. (cid:3) ohesive G -orbispaces. We now discuss in more detail the analogue of Prop. 4.2 in (a) the special case of globalquotient stacks X ⊂ (cid:39) X (cid:12) G by a discrete group G , but (b) in the full generality of X being any 0-truncated cohesivespace (not necessarily a topological space, but for instance a smooth manifold or diffeological space (147) or evena non-concrete object). Remark 4.3 (Good orbifolds and good cohesive orbispaces) . The traditional orbifolds that arise as global quotients X ⊂ (cid:39) X (cid:12) G of a smooth manifold X by the action of a discrete group G are called good orbifolds (e.g. [Ka08, 6]).Therefore, the cohesive G -orbispaces discussed now (Def. 4.4) could be called (after forgetting their slicing over ≺ G ) the good cohesive orbispaces . Definition 4.4 (Cohesive G -orbispace) . Let H be a singular-cohesive ∞ -topos (Def. 3.48) and G ∈ Groups ( H ) (Prop. 2.74) discrete G (cid:39) (cid:91) G . We say that a cohesive G-orbispace is an object X p (cid:15) (cid:15) ≺ G ∈ H / ≺ G in the slice over the G -orbi-singularity (214) that is: (a) orbi-singular: ≺ ( p ) (cid:39) p (Def. 3.52) , (b) ( τ ) / ≺ G ( p ) (cid:39) p (Def. 2.57) . (238) Definition 4.5 (Universal covering space of a G -orbi-singular space) . Given a Cohesive G -orbispace X ∈ H / ≺ G (Def. 4.4), we say that its universal covering space X ∈ H the homotopy fiber of the defining morphism to ≺ G overits essentially unique point: X fib ( p ) (cid:47) (cid:47) X p (cid:15) (cid:15) ≺ G (239) Proposition 4.6 (Properties of universal covering spaces) . Let H be a singular-cohesive ∞ -topos (Def. 3.48).Given a G-orbi-singular space X ∈ H / ≺ G (Def. 4.4), its universal covering space X (Def. 4.5) (i) is: (a) τ ( X ) (cid:39) X (Def. 2.57) , (b) smooth: ⊂ ( X ) (cid:39) X (Def. 3.1) , (ii) and is equipped with a G-action (Prop. 2.79) such that X is the orbi-singularization (209) of the corre-sponding homotopy quotient: X (cid:39) ≺ (cid:0) X (cid:12) G (cid:1) . (240) Proof. (i)
That X is (a) p is 0-truncated and using Lemma 3.14. To seethat X is (b) smooth, observe that by the defining assumption (238) that p is orbi-singular, it is the image underOrbSnglr (208) of a morphism p ⊂ in H ⊂ : X fib ( p ) (cid:47) (cid:47) X p (cid:15) (cid:15) ≺ G (cid:39) OrbSnglr X ⊂ fib ( p ⊂ ) (cid:47) (cid:47) X ⊂ p ⊂ (cid:15) (cid:15) ∗ (cid:12) G . (241)We claim that in fact X (cid:39) NnOrbSinglr ( X ⊂ ) , whence X (cid:39) ⊂ ( X ) : First, since OrbSnglr is a right adjoint it preserveshomotopy fibers (Prop. 2.26), fib ( p ) (cid:39) OrbSnglr (cid:0) fib ( p ⊂ ) (cid:1) , hence we have X (cid:39) OrbSnglr ( X ⊂ ) . It follows, inparticular, that X ⊂ is 0-truncated, since X (cid:39) OrbSnglr ( X ⊂ ) is 0-truncated by part (a) , and using that OrbSnglr isfully faithful. From this it follows that OrbSnglr ( X ⊂ ) (cid:39) NnOrbSinglr ( X ⊂ ) , by Lemma 3.65. Together this gives theclaim (b) .With this, part (ii) now follows by comparison with (107). (cid:3) hape of Cohesive G -orbispaces. We derive the following formula (242) in Prop. 5.6 which generalizes theembedding of G -spaces into global equivariant homotopy theory, discussed in [Re14, p. 7][Lu19, 3.2.17], fromtopological G -spaces to general cohesive G -spaces. Below in §5.1 this serves to prove that the intrinsic cohomologyof good cohesive orbispaces subsumes proper equivariant cohomology (Theorem 5.9). Lemma 4.7 (Shape of Cohesive G -orbispaces) . Let H be a singular-cohesive ∞ -topos (Def. 3.48). (35) , G ∈ Groups (cid:0) H (cid:1) (Prop. 2.74) be a 0-truncated G (cid:39) τ G and discrete G (cid:39) (cid:91)
G and let X ∈ H be smooth X (cid:39) ⊂ X and0-truncated X (cid:39) τ X and equipped with a G-action ( X , ρ ) ∈ G Actions ( H ) (Prop. 2.79). (i) Then the orbi-singularization (208) of the corresponding homotopy quotient (107) X : = ≺ (cid:0) X (cid:12) G (cid:1) ∈ H : = Sheaves ∞ (cid:0) Singularities , H ⊂ (cid:1) , when regarded as an H ⊂ -valued ∞ -presheaf on Singularities (205) , assigns to a singularity ≺ K (213) the disjointunion of fixed loci X φ ( K ) (Def. 2.97) of the smooth covering space X (Def. 4.5) for all group homomorphisms φ : K ! G homotopy-quotiented (108) by the residual G-action (Prop. 4.6): X : ≺ K (cid:18) (cid:70) φ ∈ Groups ( K , G ) X φ ( K ) (cid:19) (cid:12) G . (242) (ii) Moreover, its shape (207)Shp (cid:16) ≺ (cid:0) X (cid:12) G (cid:1)(cid:17) ∈ SingularGroupoids : = Sheaves ∞ (cid:0) Singularities (cid:1) assigns to a singularity ≺ K (213) the cohesive shape (127) of these disjoint unions of fixed loci (Def. 2.97) of thesmooth covering space X (Def. 4.5) homotopy-quotiented by its G-action (Prop. 4.6): Shp (cid:0) X (cid:1) : ≺ K Shp (cid:18) (cid:70) φ ∈ Groups ( K , G ) X φ ( K ) (cid:19) (cid:12) G . (243) Proof.
We claim that for U ∈ Charts (Def. 3.9) and ≺ K ∈ Singularities (Def. 3.46), hence U × ≺ K ∈ Charts × Singularities (Lemma 3.60), we have the following sequence of natural equivalences: H (cid:0) U × ≺ K , X (cid:1) = H (cid:0) U × ≺ K , OrbSnglr (cid:0) X (cid:12) G (cid:1)(cid:1) (cid:39) H ⊂ (cid:0) Smth (cid:0) U × ≺ K (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) (cid:39) U × ( ∗ (cid:12) K ) , X (cid:12) G (cid:1) (cid:39) Groupoids ∞ (cid:0) ( ∗ (cid:12) K ) , H ⊂ (cid:0) U , X (cid:12) G (cid:1)(cid:1) (cid:39) Groupoids (cid:0) ( ∗ (cid:12) K ) , H ⊂ (cid:0) U , X (cid:1) (cid:12) G (cid:1) (cid:39) (cid:18) (cid:70) φ ∈ Groups ( K , G ) H ⊂ (cid:0) U , X (cid:1) φ ( K ) (cid:19) (cid:12) G (cid:39) (cid:18) (cid:70) φ ∈ Groups ( K , G ) H ⊂ (cid:0) U , X φ ( K ) (cid:1)(cid:19) (cid:12) G (cid:39) (cid:18) H ⊂ (cid:18) U , (cid:70) φ ∈ Groups ( K , G ) X φ ( K ) (cid:19)(cid:19) (cid:12) G (cid:39) H ⊂ (cid:18) U , (cid:18) (cid:70) φ ∈ Groups ( K , G ) X φ ( K ) (cid:19) (cid:12) G (cid:19) . (244)Here the first step is (240), the second is the adjunction Smth (cid:97) OrbSnglr (208) and using under the brace thatSmth preserves products (by Prop. 2.26), that U is already smooth by assumption, and that Smth (cid:0) ≺ K (cid:1) (cid:39) ( ∗ (cid:12) K ) by (222). The third step is the tensoring of H over ∞ -groupoids (Prop. 2.34) (using the geometric discreteness ( ∗ (cid:12) K ) (cid:39) Disc ( ∗ (cid:12) K ) by Remark 3.58) The fourth step uses the geometric contractibility of U and the discretenessof G to identify H ⊂ ( U , X (cid:12) G ) (cid:39) H ⊂ ( U , X ) (cid:12) G (Lemma 3.12). The fifth is the general observation of Example 2.16about hom-groupoids between quotient groupoids of sets. The sixth step uses Prop. 3.13 to find that the fixedpoints in the set of maps are the maps into the fixed point locus. After this key step, we just re-organize term: Theseventh step uses the connectedness of U (Lemma 3.10) to find that a coproduct of homs out of U is a hom into thecoproduct. Finally, the eighth step uses again Lemma 3.12.70 i) The composite equivalence (244) implies the first claim (242) by the ∞ -Yoneda embedding (Prop. 2.37),using Lemma 3.60. (ii) From this, the second claim (243) follows, using that Shp acts objectwise over Singularities (207), and pre-serves homotopy quotients by discrete groups (Prop. 3.4). (cid:3)
Remark 4.8 (Relevance of 0-truncated orbi-singular spaces) .(i)
The crucial assumption that makes the proof of Lemma 4.7 work is, (a) that G is discrete and (b) that X is0-truncated. This is what yields 1-groupoidal homs in the middle step of (244) and thus the form of the expressionin the next step, as on the right hand side of (40). (ii) Without the assumption of X being 0-truncated over ∗ (cid:12) G , the proof of Lemma 4.7 would proceed verbatimup to that middle step, but then would break as the nontrivial morphisms present in X would then mix with thoseof the action by G . (iii) Lemma 4.7 shows that this subltety is closely related to the cohesive nature of the problem: We either havea space which is 0-truncated but carries cohesive (i.e. geometric) structure, or we turn it into its cohesive shapewhich is un-truncated but geometrically discrete.
Singular quotient of Cohesive G -orbispaces.Proposition 4.9 (Singular quotient of G -orbi-singular space) . Let H be a singular-cohesive ∞ -topos (Def. 3.48),G ∈ Groups ( H ) being discrete G (cid:39) (cid:91) G and 0-truncated G (cid:39) τ G. For X be a G-orbi-singular space (Def. 4.4)with universal covering space X ∈ H ⊂ , (cid:44) ! H equipped with its induced G-action (Def. 4.5, Prop. 4.6). Then thesingularization (208) of X is the plain G-quotient of X Snglr (cid:0) X (cid:1) (cid:39) X / G ∈ H ⊂ , (cid:44) −! H ⊂ (i.e., the quotient of the G-action formed in the 1-topos H ⊂ , of 0-truncated objects).Proof. For U ∈ Charts, write H ⊂ (cid:0) U , X (cid:1) (cid:12) G (cid:39) (cid:71) c (cid:0) ∗ (cid:12) H c (cid:1) ∈ Groupoids (245)for the essentially unique decomposition of the groupoid on the left into its connected components c ∈ π (cid:16) H ⊂ (cid:0) U , X (cid:1) (cid:12) G (cid:17) (cid:39) H ⊂ (cid:0) U , X (cid:1) / G ∈ Set (246)each of which is equivalent to the delooping groupoid (Example 2.14) of its fundamental group H c : = π (cid:16) H ⊂ (cid:0) U , X (cid:1) (cid:12) G , c (cid:17) ∈ Groups . Now, by Lemma 4.7 and re-instantiating the last few manipulations in (244), we have that over each U ∈ Chartsthe incarnation of the G -orbi-singular space X as an ∞ -presheaf on Singularities is given by: X ( U ) : ≺ K Groupoids (cid:16) ∗ (cid:12) K , H ⊂ (cid:0) U , X (cid:1) (cid:12) G (cid:17) (cid:39) Groupoids (cid:16) ∗ (cid:12) K , (cid:71) c (cid:0) ∗ (cid:12) H c (cid:1)(cid:17) (cid:39) (cid:71) c Groupoids (cid:16) ∗ (cid:12) K , ∗ (cid:12) H c (cid:17) (cid:39) (cid:71) c Singularities (cid:16) ≺ K , ≺ H c (cid:17) . (247)Here the first step is (245), the second step uses that the delooping groupoids ∗ (cid:12) K are connected and the last stepobserves the definition of Singularities (Def. 3.46). By the ∞ -Yoneda embedding (Prop. 2.37) over the site ofSingularities (201) this means that X ( U ) (cid:39) (cid:71) c ≺ H c ∈ Sheaves ∞ (cid:0) Singularities (cid:1) . (248)With this, we find that Snglr ( X ) ∈ PreSheaves ∞ ( Charts ) is given by71nglr (cid:0) X (cid:1) : U Snglr (cid:0) X ( U ) (cid:1) (cid:39) Snglr (cid:16) (cid:70) c ≺ H c (cid:17) (cid:39) (cid:71) c Snglr (cid:16) ≺ H c (cid:17) (cid:39) (cid:71) c ∗(cid:39) π (cid:16) H ⊂ (cid:0) U , X (cid:1) (cid:12) G (cid:17) (cid:39) H ⊂ (cid:0) U , X (cid:1) / G . Here the first line is the object-wise application of Snglr (Remark 3.54), while the next line is (248). From therewe use that Snglr, being a left adjoint, preserves coproducts (Prop. 2.26) and then that it takes the elementarysingularies to points, by Lemma 3.61. Finally, we identify (246). But this resulting assignment is just that of X / G ∈ PreSheaves ( Charts ) : X / G : U H ( U , X ) / G and hence the claim follows. (cid:3) Examples of Cohesive G -orbispaces. We make explicit two classes of examples of cohesive G -orbispaces (Def.4.4): Fr´echet-smooth orbispaces and topological orbispaces. Example 4.10 (Fr´echet smooth G -orbispaces) . Consider X ∈ Fr´echetManifolds (cid:31) (cid:127) (cid:47) (cid:47)
SmoothGroupoids ∞ a (possibly infinite-dimensional Fr´echet-)smooth manifold regarded as a 0-truncated concrete smooth ∞ -groupoid(147). Given a G ∈ Groups ( H ) (215) being discrete G (cid:39) (cid:91) G , a smooth action ρ of G on X is equivalently ahomotopy fiber sequence in SmoothGroupoids ∞ of this form (Prop. 2.79): X fib ( ρ ) (cid:47) (cid:47) X (cid:12) G ρ (cid:15) (cid:15) ∗ (cid:12) G . Here the homotopy quotient (107) X (cid:12) G ∈ LieGroupoids (cid:31) (cid:127) (cid:47) (cid:47)
SmoothGroupoids ∞ is the corresponding (possibly infinite-dimensional Fr´echet-)Lie groupoid, regarded as a smooth ∞ -groupoid viathe embedding (148). Its orbi-singularization (208) is a G -orbi-singular space, in the sense of Def. 4.4, in the ∞ -topos SingularSmoothGroupoids ∞ (211): X (cid:15) (cid:15) ≺ G : = OrbSnglr X (cid:12) G (cid:15) (cid:15) ∗ (cid:12) G . (249)This orbi-singular smooth groupoid (249) what we suggest is the proper incarnation of the quotient orbifold that ispresented by the smooth manifold X with its G -action. Notice that (see Figure G ): (i) its purely smooth aspect is the Lie groupoid ⊂ (cid:0) X (cid:1) (cid:39) X (cid:12) G ∈ LieGroupoids (cid:31) (cid:127) (cid:47) (cid:47)
SingularSmoothGroupoids ∞ , (by Prop. 4.6) which is the incarnation of this orbifold, according to [MP97][PS10] (ii) its purely singular aspect is the diffeological space < (cid:0) X (cid:1) (cid:39) X / G ∈ DiffeologicalSpaces (cid:31) (cid:127) (cid:47) (cid:47)
SingularSmoothGroupoid ∞ (by Prop. 4.9) which is the incarnation of this orbifold, according to [IKZ10].However, it is only the full orbi-singular object X which is structured enough to have proper (Bredon-)equivariantcohomology. This is the content of Theorem 5.9 below. 72 xample 4.11 (Topological G -orbispaces) . For G a finite group, let G (cid:121) X top be a topological G -space (Def B.1)with Borel construction X top (cid:47) (cid:47) X × G EG (cid:15) (cid:15) BG Via its continuous diffeology (32), this is equivalently a 0-truncated (and concrete) object in H ⊂ : = SmoothGroupoids ∞ (Example 3.18) X : = Cdfflg ( X top ) ∈ H ⊂ , equipped with a smooth G -action (Prop. 2.79) X (cid:47) (cid:47) X (cid:12) G (cid:15) (cid:15) ∗ (cid:12) G . The orbi-singularization (208) of the corresponding homotopy quotient is a G -orbi-singular space (Def. 4.4) X (cid:15) (cid:15) ≺ G : = OrbSnglr
Cdfflg ( X top ) (cid:12) G (cid:15) (cid:15) ∗ (cid:12) G . Proposition 4.12 (Shape of good orbifolds) . Consider a finite-dimensional smooth G-orbifold, as in Example 4.10(a good orbifold, Remark 4.3) X : = OrbSnglr (cid:0) X (cid:12) G (cid:1) . Then its cohesive shape (208) Shp (cid:0) X (cid:1) ∈ Sheaves ∞ (cid:0) Singularities (cid:1) is, over any singularity ≺ K (202) , the topolog-ical shape (36) of the G-Borel construction on the disjoint union of all K-fixed subspaces X φ ( K ) top ⊂ X top (349) in theunderlying (32) D-topological G-space (Def. B.1):
Shp (cid:0) X (cid:1) : ≺ K Shp
Top (cid:18)(cid:18) (cid:70) φ ∈ Groups ( K , G ) ( Dtplg ( X )) φ ( K ) (cid:19) × G EG (cid:19) . (250) Proof.
With Lemma 4.7, the task is reduced to showing that, for φ ( K ) ⊂ G any specified subgroup, we have anequivalence Shp (cid:0) X φ ( K ) (cid:1) (cid:39) Shp
Top (cid:0) ( Dtplg ( X )) φ ( K ) (cid:1) ∈ Groupoids ∞ between the cohesive shape (127) of the orbi-singular homotopy quotient of X by G and the ordinary topologicalshape (36) of the D-topological space underlying X . But this is (151) in Example 3.18, given by [Sc13, 4.3.29]. (cid:3) Proposition 4.13 (Shape of topological G -orbi spaces) . Consider the topological G-orbi-singular space, as inExample 4.11, X : = OrbSnglr (cid:0)
Cdfflg ( X top ) (cid:12) G (cid:1) . Then its cohesive shape (208) Shp (cid:0) X (cid:1) ∈ Sheaves ∞ (cid:0) Singularities (cid:1) is, over any singularity ≺ K (202) , the topo-logical space (36) of the G-Borel construction on the disjoint union of all K-fixed subspaces X φ ( K ) top ⊂ X top (349) : Shp (cid:0) X (cid:1) : ≺ K Shp
Top (cid:18)(cid:18) (cid:70) φ ∈ Groups ( K , G ) X φ ( K ) top (cid:19) × G EG (cid:19) . (251) Proof.
With Lemma 4.7, the task is reduced to showing that, for φ ( K ) ⊂ G any specified subgroup, we have anequivalence Shp (cid:0) Cdfflg ( X top ) φ ( K ) (cid:1) (cid:39) Shp
Top (cid:0) X φ ( K ) top (cid:1) ∈ Groupoids ∞ between the cohesive shape (127) of the orbi-singular homotopy quotient by G of the continuous-diffeologicalspace and the ordinary topological shape (36) But this is item (150) in Example 3.18, given by combining theresult (149) of [BEBP19] with Prop. 2.20 from [CW14]. (cid:3) .2 Orbifolds We introduce a general theory of orbi-singular spaces, whose underlying smooth cohesive groupoid is locallydiffeomorphic to a fixed local model space V . Since, for V = R n ∈ JetsOfSmoothGroupoids ∞ , these are ordinary n -folds (i.e., ordinary n -dimensional manifolds for any n , see Example 4.17), or, more generally, ´etale ∞ -groupoidswith atlases by n -folds (Example 4.18), including ordinary orbifolds, we generally speak of V -folds , with a hattip to [Sa56]. Externally these are V -´etale ∞ -stacks (Remark 4.15) but their theory internal to the ambient elastic ∞ -topos (such as the construction of their frame bundles in Prop. 4.26) is elegant and finitary and lends itself tofull formalization in homotopy type theory [We18] (see p. 5). The proper incarnation (see Remark 4.60) of these V -folds as orbifolds is via their orbi-singularization (Def. 4.58, Remark 4.60). V -folds and V -´etale groupoids.Definition 4.14 ( V -folds) . Let H be an elastic ∞ -topos H (Def. 3.21). (i) Given V ∈ Groups ( H ) (Prop. 2.74), we say that an object X ∈ H is a V -fold if there exists a correspondencebetween V and X U ´et (cid:117) (cid:117) ´et (cid:41) (cid:41) (cid:41) (cid:41)
V X (252)such that (a) both morphisms are local diffeomorphisms (Def. 3.26) and (b) the right one is, in addition, an effective epimorphism (Def. 2.63), then called a
V -atlas of X (100). (ii)
We write V Folds ( H ) ⊂ H (253)for the full sub- ∞ -category of V -folds in H and we write V Folds ( H ) ´et ⊂ H (254)for its wide subcategory on those morphisms which are local diffeomorphisms (Def. 3.26). Remark 4.15 ( V -folds and V -´etale groupoids) . By Prop. 3.36, a V -fold (Def. 4.14) is a stack (100) whose choiceof V -atlas (252) realizes it as an ´etale groupoid (Def. 3.35) with space of objects locally diffeomorphic over V : (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) U × X U (cid:39) pr (cid:15) (cid:15) (cid:79) (cid:79) ∆ pr (cid:15) (cid:15) U s (cid:15) (cid:15) (cid:79) (cid:79) e t (cid:15) (cid:15) “ V -´etale groupoid” V (cid:111) (cid:111) ´et U a ´et (cid:15) (cid:15) (cid:15) (cid:15) U (cid:15) (cid:15) (cid:15) (cid:15) “ V -atlas” X (cid:39) lim −! U • “ V -fold” (255) Example 4.16 ( V is a V -fold) . Let H be an elastic ∞ -topos H (Def. 3.21) and V ∈ Groups ( H ) (Prop. 2.74). Thenthe underlying object V ∈ H itself is a V -fold (Def. 4.14): A V -atlas (252) is given by the identity morphisms V id ´et (cid:117) (cid:117) id ´et (cid:41) (cid:41) (cid:41) (cid:41) V V . (256) Example 4.17 (Smooth manifolds are R n -folds) . For k ∈ N with k ≥
1, let H = k JetsOfSmoothGroupoids ∞ (Ex-ample 3.24). Then, for every n ∈ N , the object V : = R n ∈ CartesianSpaces (cid:31) (cid:127) (cid:47) (cid:47) k JetsOfSmoothGroupoids ∞ (257)canonically carries the structure of a group object ( R n , +) ∈ Groups ( H ) , via addition in R n regarded as a vectorspace. Now every smooth manifold 74 ∈ SmoothManifolds (cid:31) (cid:127) (cid:47) (cid:47) k JetsOfSmoothGroupoids ∞ of dimension n is a V -fold, hence an R n -fold in the sense of Def. 4.14: For any choice of atlas in the traditionalsense of manifold theory, namely an open cover (cid:8) U j φ i (cid:47) (cid:47) X (cid:9) j ∈ J by local diffeomorphisms φ i from open subsets of Cartesian space U j (cid:31) (cid:127) ι j (cid:47) (cid:47) R n , a V -atlas (252) is obtained by setting: (cid:116) j ∈ J U j ´et ( ι j ) j ∈ J (cid:118) (cid:118) ´et ( φ j ) j ∈ J (cid:40) (cid:40) (cid:40) (cid:40) R n X (258) Example 4.18 (Differentiable ´etale stacks are R n -folds [Sc13, Prop. 4.5.56]) . Let H = JetsOfSmoothGroupoids ∞ (Example 3.24) and take V = ( R n , +) as in Example 4.17. Then a diffeological groupoid X ∈ H (161) is a V -fold(Def. 4.14) for V = R n (259) if it is an n -dimensional differentiable ´etale stack in that: (i) it admits an atlas (effective epimorphism) X (cid:47) (cid:47) (cid:47) (cid:47) X from a smooth n -manifold X (via (147) and (160)) (ii) its source and target morphisms with respect to this atlas are local diffeomorphisms.Generally, a smooth ∞ -groupoid presented by a Kan simplicial smooth manifold is an R n -fold in the sense ofDef. 4.14 if it presents an ´etale ∞ -groupoid in that all its simplicial face maps are local diffeomorphisms. Examples 4.19 (Super-manifolds are R n | q -folds) . Let H = ∞ JetsOfSupergeometricGroupoids ∞ (Example 3.43).Then, for every n , q ∈ N , the super-Cartesian space (Def. 3.41) V : = R n | q ∈ ∞ JetsOfSuperCartesianSpaces (cid:31) (cid:127) (cid:47) (cid:47) ∞ JetsOfSupergeometricGroupoids ∞ (259)carries the structure of a group object, whose bosonic aspect (189) is (259). The corresponding V -folds (Def. 4.14)are the ( n | q ) -dimensional supermanifolds (196). Example 4.20 (General super ´etale ∞ -stacks) . Let H = ∞ JetsOfSupergeometricGroupoids ∞ (Example 3.43). Thenfor any V ∈ Groups ( H ) the corresponding V -´etale ∞ -stacks (Remark 4.15) realize a flavor of super ´etale ∞ -stacks ,locally modeled on V . Lemma 3.45 implies that, generally, the bosonic part (cid:32) X of a super ´etale ∞ -stack is a bosonic´etale ∞ -stack locally modeled on the bosonic part (cid:32) V of V : V Folds ( H ) (cid:32) (cid:47) (cid:47) (cid:32) V Folds ( H ) supergeometric´etale ∞ -stack X (cid:32) X underlying bosonic´etale ∞ -stack Quotients of V -folds.Proposition 4.21 (Orbifolding of a V -fold is a V -fold) . Let H be an elastic ∞ -topos H (Def. 3.21), V , G ∈ Groups ( H ) (Prop. 2.74) with G (cid:39) (cid:91) G discrete, and ( X , ρ ) ∈ G Actions ( H ) (Prop. 2.79). Then if X is a V -fold(Def. 4.14) so is its homotopy quotient X (cid:12) G (108) . Specifically, if U ´et (cid:47) (cid:47) (cid:47) (cid:47) X is a V -atlas for X (252) , then aV -atlas for V (cid:12)
G is given by composition with the homotopy fiber inclusion map fib ( ρ ) (107) :U ´et (cid:118) (cid:118) ´et (cid:40) (cid:40) (cid:40) (cid:40) V X fib ( ρ ) (cid:47) (cid:47) X (cid:12) G . (260) Proof.
We need to show that the composite morphism on the right of (260) is (a) an effective epimorphism and (b) a local diffeomorphism. Since both of these classes of morphisms are closed under composition (Lemma 2.65 andLemma 3.27), it is sufficient to show that fib ( ρ ) itself has these two properties.75or (a) observe that, by definition of homotopy fibers (107), we have a Cartesian square X (cid:15) (cid:15) fib ( ρ ) (cid:47) (cid:47) (pb) X (cid:12) G ρ (cid:15) (cid:15) ∗ (cid:47) (cid:47) (cid:47) (cid:47) B G (261)Here the bottom morphism is an effective epimorphism (Example 2.75). Since these are preserved by homotopypullback, also fib ( ρ ) is an effective epimorphism.For (b) consider the image of this square (261) under ℑ . Since ℑ is both a right and a left adjoint it preservesCartesian squares and homotopy quotients (by Prop. 2.26), while it preserves discrete objects by elasticity (154)and idempotency (Prop. 2.28, Prop. 2.29). Therefore ℑ X (cid:15) (cid:15) ℑ fib ( ρ ) (cid:39) fib ( ℑ ρ ) (cid:47) (cid:47) (pb) (cid:0) ℑ X (cid:1) (cid:12) G ℑ ρ (cid:15) (cid:15) ∗ (cid:47) (cid:47) (cid:47) (cid:47) B G (262)is Cartesian. Consider finally the pasting composite of this second square (262) with the naturality square of η ℑ on fib ( ρ ) : X η ℑ X (cid:15) (cid:15) fib ( ρ ) (cid:47) (cid:47) X (cid:12) G η ℑ X (cid:12) G (cid:15) (cid:15) ρ (cid:121) (cid:121) ℑ X (cid:15) (cid:15) (cid:47) (cid:47) (pb) (cid:0) ℑ X (cid:1) (cid:12) G ℑ ρ (cid:15) (cid:15) ∗ (cid:47) (cid:47) B G (263)Here the composite morphism on the right is equivalent to ρ , as shown, by the naturality of η ℑ and using thatthe object B G , being discrete, is ℑ -modal. Therefore, the total outer rectangle of (263) is Cartesian by (261).Moreover, the bottom square of (263) is Cartesian by (262). Therefore the pasting law (Prop. 2.23) implies thatthe top square of (263) is Cartesian. But this means (163) that fib ( ρ ) is a local diffeomorphism. (cid:3) Proposition 4.22 (Induced G -action on the tangent bundle) . Let H be an elastic ∞ -topos H (Def. 3.21), V , G ∈ Groups ( H ) (Prop. 2.74), with G (cid:39) (cid:91) G discrete, ( X , ρ ) ∈ G Actions ( H ) (Prop. 2.79) and X ∈ V Folds ( X ) (Def.4.14). Then the tangent bundle T X (Def. 3.29) carries an essentially unique G-action T ρ such that: (i) the defining projection T X ! X is G-equivariant (Def. 2.83); (ii) the homotopy quotient of T X is the tangent bundle of the orbifolded V -fold X (cid:12)
G (Prop. 4.21): ( T X ) (cid:12) G (cid:39) T ( X (cid:12) G ) ∈ H (cid:14) X (cid:12) G . (264) Proof.
Consider the following diagram:
T X (cid:42) (cid:42) (cid:15) (cid:15) (cid:47) (cid:47) X (cid:15) (cid:15) fib ( ρ ) (cid:42) (cid:42) T ( X (cid:12) G ) (cid:47) (cid:47) (cid:15) (cid:15) T ρ (cid:6) (cid:6) X (cid:12) G η ℑ X (cid:12) G (cid:39) η ℑ X (cid:12) G (cid:15) (cid:15) X (cid:15) (cid:15) fib ( ρ ) (cid:41) (cid:41) η ℑ X (cid:47) (cid:47) ℑ X ℑ fib ( ρ ) (cid:41) (cid:41) X (cid:12) G ρ (cid:15) (cid:15) η ℑ X (cid:12) G (cid:47) (cid:47) ( ℑ X ) (cid:12) G ∗ (cid:42) (cid:42) B G (265)Here the bottom left square is that characterizing the G -action on X , by (107); while the bottom and right squaresare both the naturality square of η ℑ on the morphism fib ( ρ ) (where we use that ℑ commutes with taking thehomotopy quotient by the discrete group G ). Now observe that:76 a) The bottom and right squares are pullback squares since fib ( ρ ) is a local diffeomorphism (Def. 3.26) byProp. 4.21. (b) The front and back squares are pullback squares by the definition of tangent bundles (Def. 3.29).In particular, the solid part of the diagram is homotopy-commutative, so that, by the universal property of the frontpullback square, the dashed morphism exists, essentially uniquely, such as to make the top and the top left squarehomotopy-commutative. Further observe, by repeatedly applying the pasting law (Prop. 2.23), that: (c)
The top left square is a homotopy pullback since the back, right and front squares are pullbacks by (a) and(b). (d)
The total left rectangle is a pullback, since the top one is so, by (c), and the bottom one is so, by the actionproperty (107).Thus, again by the action property (107), the total left rectangle exhibits a G -action on T X whose homotopyquotient is as claimed (264), and its factorization into two pullback squares as shown exhibits the projection
T X ! X as a homomorphism of G -actions, hence as being G -equivariant (Def. 2.83). (cid:3) Proposition 4.23 (Induced G -action on local neighborhood of fixed point) . Let H be an elastic ∞ -topos H (Def.3.21), V , G ∈ Groups ( H ) (Prop. 2.74), with G (cid:39) (cid:91) G discrete, ( X , ρ ) ∈ G Actions ( H ) (Prop. 2.79) with X ∈ V Folds ( X ) (Def. 4.14) and ∗ x (cid:47) (cid:47) X a homotopy fixed point (Def. 2.97). Then the induced G-action T ρ onthe tangent bundle T X , from Prop. 4.22, restricts to a G-action T x ρ on the local neighborhood T x X (Example 3.30)of the homotopy fixed point x.Proof.
Consider the following diagram: T x X (cid:47) (cid:47) (cid:15) (cid:15) (cid:42) (cid:42) T X (cid:41) (cid:41) (cid:15) (cid:15) ( T x X ) (cid:12) G (cid:47) (cid:47) T x ρ (cid:15) (cid:15) ( T X ) (cid:12) G (cid:15) (cid:15) T ρ (cid:5) (cid:5) ∗ x (cid:47) (cid:47) (cid:42) (cid:42) X (cid:15) (cid:15) fib ( ρ ) (cid:42) (cid:42) B G x (cid:12) G (cid:47) (cid:47) X (cid:12) G ρ (cid:15) (cid:15) ∗ (cid:42) (cid:42) B G (266)Here the squares on the right are from (265) and are thus both homtopy Cartesian. The rear square is the homotopypullback square defining the tangent fiber, and we define the front square to be a homotopy pullback, giving us theobject denoted ( T x X ) (cid:12) G . We need to show that this object really is the homotopy quotient of the restricted action.But the bottom horizontal square homotopy-commutes, exhibiting the homotopy fixed point by (122), so that, byapplying the pasting law (Prop. 2.23) to the top vertical squares, it follows that also the top left square is Cartesian.This already identifies ( T x X ) (cid:12) G as the homotopy quotient of some G -action on T x X , by Prop. 2.79. To see that thisis indeed the restricted action, observe that the front triangle commutes, again by (122), so that the total diagramexhibits the fiber inclusion T x X ! T X as being a homomorphism G -actions T x ρ ! T ρ (by Prop. 2.79). (cid:3) Frame bundles.Definition 4.24 (Structure group of V -folds) . Let H be an elastic ∞ -topos (Def. 3.21) and V ∈ Groups ( H ) (Prop.2.74), to be regarded as the local model space of V -folds (Def. 4.14). (i) Then we say that the automorphism group (Def. 2.86) of the local neighborhood (Example 3.30) of the neutralelement ∗ e (cid:47) (cid:47) V (Example 2.76) Aut ( T e V ) ∈ Groups ( T e V ) (267)is the structure group of V -folds . (ii) We write ( T e V , ρ Aut ) ∈ Aut ( T e V ) Actions ( H ) for its canonical action (114). 77 xample 4.25 (Ordinary general linear group) . Let H = JetsOfSmoothGroupoids ∞ (Example 3.24) and let V : = ( R n , +) ∈ Groups ( SmoothManifolds ) (cid:31) (cid:127) (cid:47) (cid:47) Groups ( H ) via the full inclusion (161), with R n regarded as a group under addition of tuples of real numbers. Then thestructure group of R n -folds, according to Def. 4.24, is the traditional general linear group, regarded as a Lie group:Aut ( T R n ) (cid:39) GL ( n ) . Proposition 4.26 (Frame bundle) . Let H be an elastic ∞ -topos H (Def. 3.21), V ∈ Groups ( H ) (Prop. 2.74) andX ∈ H a V -fold (Def. 4.14). Then the tangent bundle of X (Def. 3.29) is a fiber bundle (Def. 2.91) with typicalfiber the local neighborhood T e V (Def. 3.28) of the neutral element ∗ e −! V , hence is the associated bundle of an
Aut ( T e V ) -principal (267) bundle (Prop. 2.88), to be called the frame bundle of X : tangent bundle T X (cid:47) (cid:47) (cid:15) (cid:15) (pb) (cid:0) T e V (cid:1) (cid:12) Aut (cid:0) T e V (cid:1) (cid:15) (cid:15) X (cid:96) Frames ( X ) (cid:47) (cid:47) B Aut ( T e V ) frame bundle Frames ( X ) (cid:15) (cid:15) (cid:47) (cid:47) (pb) ∗ (cid:15) (cid:15) X (cid:96) Frames ( X ) (cid:47) (cid:47) B Aut ( T e V ) structuregroup (268) Proof.
By Prop. 3.31 the tangent bundles over any V -atlas (252) for X form two Cartesian squares as follows: TU (cid:117) (cid:117) (cid:41) (cid:41) (cid:41) (cid:41) (cid:15) (cid:15) (pb) V × T e V (cid:39) TV (pb) (cid:15) (cid:15) T X (cid:15) (cid:15) U ´et (cid:117) (cid:117) ´et (cid:41) (cid:41) (cid:41) (cid:41) V X (269)Moreover, by Prop. 4.32 the tangent bundle of V is trivial, as shown on the left. Since Cartesian products arepreserved by homotopy pullback, the left square implies that also TU (cid:39) U × T e V is trivial. But with this theexistence of the right square is the defining characterization for T X being a T e V -fiber bundle. (cid:3) Remark 4.27 (Frame bundles are well-defined) . The frame bundle (Def. 4.26) of a V -fold (Def. 4.14) is inde-pendent, up to a contractible space of equivalences, of the choice of V -atlas (252) in the construction (269): Thisfollows as a special case of the essential independence of classifying maps of fiber bundles from the choice oftrivializing cover, as in Prop. 2.92, using that not only the class of effective epimorphisms but also that of localdiffeomorphisms is closed under pullback and composition (Lemma 3.27). Proposition 4.28 ( V -fold is Aut ( T e V ) -quotient of its frame bundle) . Let H be an elastic ∞ -topos H (Def. 3.21),V ∈ Groups ( H ) (Prop. 2.74) and X ∈ V Folds ( H ) (Def. 4.14). Then X is equivalent to the homotopy quotient (108) of its own frame bundle (Prop. 4.26) by Aut ( T e V ) :X (cid:39) Frames ( X ) (cid:12) Aut ( T e V ) . Proof.
This is immediate from the equivalence between principal bundles and homotopy qotient projections (Re-mark 2.89) applied to the frame bundle (268). (cid:3)
Example 4.29 (Frame bundles on smooth manifolds) . Let H = JetsOfSmoothGroupoids ∞ (Example 3.24) and X ∈ SmoothManifolds (cid:44) ! H a smooth manifold (161) regarded as an R n -fold according to Example 4.17. (i) Then its frame bundle, according to Prop. 4.26, is the GL ( n ) -principal bundle on X which is the frame bundlein the traditional sense of differential geometry. (ii) For the same manifold but regarded in H = k JetsOfSmoothGroupoids ∞ with k ≥ ramed V -folds.Definition 4.30 (Framing) . Let H be an elastic ∞ -topos H (Def. 3.21). A framing of an objext X ∈ H is atrivialization of its tangent bundle Def. 3.29, hence an equivalence T X (cid:39) X × T x X ∈ H / X for ∗ x (cid:47) (cid:47) X any point. Remark 4.31 (Framing on a V -fold) . If X is a V -fold (Def. 4.14) then a framing on V in the sense of Def. 4.30is equivalent, by Prop. 4.26, to a trivialization of the frame bundle, hence to a trivialization of its classifying map(268): T X (cid:32) (cid:32) fr (cid:39) (cid:47) (cid:47) X × T e V (cid:123) (cid:123) X ⇔ ∗ (cid:43) (cid:43) X (cid:54) (cid:54) (cid:96) Frames ( X ) (cid:54) (cid:54) B Aut ( T e V ) . (cid:76) (cid:84) (cid:39) (cid:96) fr (270) Proposition 4.32 (Groups carry canonical framings by left-translation) . In an elastic ∞ -topos H (Def. 3.21) everygroup object V ∈ Groups ( H ) (Prop. 2.74) carries a canonical framing (Def. 4.30), which we call the framing byleft translation: TV fr (cid:96) (cid:39) (cid:47) (cid:47) V × T e V ∈ H / V . (271) Proof.
Since ℑ preserves group structure (as in Prop. 3.4), the defining homotopy fiber product of the tangentbundle of V (165) sits in a Mayer-Vietoris sequence (Prop. 2.78) as shown on the left of the following: TV (cid:47) (cid:47) (cid:15) (cid:15) (pb) ∗ (cid:96) e (cid:15) (cid:15) V × V ( η ℑ V , η ℑ V )= η ℑ V × V (cid:47) (cid:47) ℑ V × ℑ V ( − ) · ( − ) − (cid:47) (cid:47) ℑ V (cid:39) TV (pb) (cid:15) (cid:15) (cid:47) (cid:47) T e V (cid:15) (cid:15) (cid:47) (cid:47) (pb) ∗ (cid:96) e (cid:15) (cid:15) V × V ( − ) · ( − ) − (cid:47) (cid:47) V η ℑ V (cid:47) (cid:47) ℑ V (272)Using that ℑ preserves products (by Prop. 2.26) and using the naturality of its unit transformation η ℑ (48), thisCartesian square on the left is equivalent to the total rectangle shown on the right. By the pasting law (Prop. 2.23),this is the pasting of two Cartesian squares, the right one of which exhibits the local neighborhood T e V (Def. 3.28)as shown. To see what the Cartesian property of the left square on the right says, consider pasting to it the topsquare appearing in the diagram (105) which exhibits the group division ( − ) · ( − ) − in Example 2.77: TV (cid:47) (cid:47) (pb) (cid:15) (cid:15) T e V (cid:15) (cid:15) V × V pr (cid:15) (cid:15) ( − ) · ( − ) − (cid:47) (cid:47) (pb) V (cid:15) (cid:15) V (cid:47) (cid:47) ∗ (273)Since both squares are Cartesian, the pasting law (Prop. 2.23) says that the total rectangle is Cartesian. This is theequivalence (271). (cid:3) Proposition 4.33 (Canonical framing on group is equivariant under group automorphisms) . Consider an elastic ∞ -topos H (Def. 3.21), V , G ∈ Groups ( H ) (Prop. 2.74). with 0-truncated V (cid:39) τ V and ( V , ρ G ) ∈ G Actions ( H ) (Prop. 2.79) acting by group-automorphisms (Prop. 2.102) hence by restriction ρ G = B i ∗ ρ Aut
Grp (Prop. 2.85) alonga group homomorphism G i (cid:47) (cid:47) Aut e ( V ) , to the group-automorphism group Aut
Grp ( V ) (Def. 2.101). Then thecanonical framing fr (cid:96) on V from Prop. 4.32 is G-equivariant (Def. 2.83), in that it lifts to a morphism of G-actions(Prop. 2.79) of the form ( TV , T ρ ) fr (cid:96) (cid:47) (cid:47) ( V , ρ ) × ( T e V , T e ρ ) ∈ G Actions ( H ) , where T ρ is the induced action on TV from Prop. 4.22, and T e ρ is the induced action on T e V from Prop. 4.23(which exists since group-automorphisms of V are in particular pointed automorphisms of V (Def. 2.100). roof. Consider the following diagram: TV (cid:47) (cid:47) (cid:47) (cid:47) (cid:24) (cid:24) ( TV ) (cid:12) G φ (cid:42) (cid:42) φ (cid:32) (cid:32) (cid:32) (cid:32) TV (cid:39) (cid:39) (cid:15) (cid:15) φ (cid:47) (cid:47) ( V (cid:12) G ) × ∗ (cid:12) G (cid:0) ( T e V ) (cid:12) G (cid:1) (cid:42) (cid:42) (cid:15) (cid:15) T e V (cid:127) (cid:95) (cid:15) (cid:15) (cid:47) (cid:47) ( T e V ) (cid:12) G (cid:127) (cid:95) (cid:15) (cid:15) V × V ( − ) · ( − ) − (cid:39) (cid:39) pr (cid:15) (cid:15) (cid:47) (cid:47) ( V (cid:12) G ) × ∗ (cid:12) G ( V (cid:12) G ) (cid:42) (cid:42) (cid:15) (cid:15) V (cid:47) (cid:47) (cid:15) (cid:15) V (cid:12) G (cid:15) (cid:15) V (cid:40) (cid:40) fib ( ρ ) (cid:47) (cid:47) V (cid:12) G (cid:43) (cid:43) ∗ (cid:47) (cid:47) ∗ (cid:12) G (274)Here• the bottom square is the Cartesian square (107) which exhibits the action on V ,• the middle horizontal square is the Cartesian square which exhibits the equivariance under group-automorphismsof the group division operatoin (Prop. 2.103),• the total left rectangle is the Cartesian square from (273) which exhibits the canonical framing,• the total front face is the pasting of – on the bottom: the Cartesian square (107) which exhibits the action on V , – on the top: the Cartesian square which is the pasting of the top and the top-right squares in (266)equibiting the action on T e V and hence is itself Cartesian,• the bottom and the total right squares are the defining Cartesian squares of the fiber products, and hence, bythe pasting law, also their pasting to the total right square is Cartesian,• the total vertical rear square, with the dashed morphism φ on top, is the one thus induced from the universalproperty of the fiber product, and is itself Cartesian, by the pasting law (Prop. 2.23), (using, by the aboveitems, that the left, right and front squares are Cartesian and that the diagram of squares commutes)• the slanted square in the rear is the pasting of the Cartesian square on the left of (265), that exhibits theinduced G -action on TV , with the diagonal square on fib ( ρ ) .Now observe that inside this big diagram (274) we find the following solid homotopy-commutative sub-diagram TV (cid:15) (cid:15) (cid:15) (cid:15) (cid:47) (cid:47) ( T e V ) (cid:12) G (cid:127) (cid:95) (cid:15) (cid:15) ( TV ) (cid:12) G (cid:47) (cid:47) φ (cid:54) (cid:54) V (cid:12) G . Here the left morphism is an effective epimorphism (by Lemma 2.80) and the right morphism is (-1)-truncated bythe assumption that V is 0-truncated (Lemma xyz). Therefore, the connected/truncated factorization system (Prop.2.66) implies an essentially unique lift φ , as shown. This, in turn, implies the morphism φ in (274), again by theuniversal property of the homotopy fiber product.Now, since both the slanted as well as the vertical total rear squares are Cartesian, the diagram (274) showsthat the contravariant base change (Prop. 2.49) of φ along fib ( ρ ) is an equivalence. But since fib ( ρ ) is an effectiveepimorphism (Lemma 2.55) , base change along it is conservative (Prop. 2.55), and hence it follows that φ itselfis already an equivalence.With that identification, the total cube in (274) exhibits the G -equivariance of the framing. (cid:3) roposition 4.34 (Orbifolding of framed V -folds) . Let H be an elastic ∞ -topos H (Def. 3.21), V , G ∈ Groups ( H ) (Prop. 2.74) with G (cid:39) (cid:91) G discrete, and ( X , ρ X ) , ( T e V , ρ T e V ) ∈ G Actions ( H ) (Prop. 2.79) for X a V -fold (Def. 4.14)equipped with a framing fr (Def. 4.30). Then the following are equivalent: (i) The framing is G-equivariant (Def. 2.83) with respect to the induced action on T X (from Prop. 4.22) andthe product action ρ X × ρ T e V on X × T e V , hence lifts to a morphism ( T X , ρ T X ) fr (cid:39) (cid:47) (cid:47) ( X , ρ X ) × ( T e V , ρ T e V ) ∈ G Actions ( H ) (275) (ii) The classifying map (268) of the frame bundle (Def. 4.26) of the orbifolded V -fold X (cid:12)
G (Prop. 4.21) factorsthrough B G as X (cid:12) G (cid:96) Frames ( X (cid:12) G ) (cid:40) (cid:40) ρ X (cid:47) (cid:47) B G (cid:96) ρ TeV (cid:47) (cid:47) B Aut ( T e V ) (cid:11) (cid:19) (276) Proof.
Consider the following diagram:
T X (cid:24) (cid:24) fr (cid:36) (cid:36) (cid:47) (cid:47) T ( X (cid:12) G ) (cid:31) (cid:31) fr (cid:12) G (cid:41) (cid:41) (cid:47) (cid:47) ( T e V ) (cid:12) Aut ( T e V ) (cid:34) (cid:34) X × T e V (cid:15) (cid:15) (cid:47) (cid:47) X (cid:12) G × B G ( T e V ) (cid:12) G (cid:15) (cid:15) (cid:47) (cid:47) ( T e V ) (cid:12) G (cid:47) (cid:47) ρ TeV (cid:15) (cid:15) ( T e V ) (cid:12) Aut (cid:0) T e V (cid:1) (cid:15) (cid:15) X fib ( ρ X ) (cid:47) (cid:47) (cid:45) (cid:45) (cid:96) Frames ( X ) (cid:52) (cid:52) X (cid:12) G ρ X (cid:47) (cid:47) (cid:96) Frames ( X (cid:12) G ) (cid:44) (cid:44) B G (cid:96) ρ TeV (cid:47) (cid:47) B Aut (cid:0) T e V (cid:1) ∗ (cid:57) (cid:57) (cid:96) fr (cid:81) (cid:89) (cid:96) fr (cid:12) G (cid:11) (cid:19) Note that here: (a)
The total outer part of the diagram exhibits the given framing fr via its classifying homotopy (cid:96) fr, accordingto Remark 4.31. (b)
The front squares in the middle and on the right are the pullback squares that defines the diagonal G -actionand the classification of the ρ T e V -action respectively. Hence also their pasting composite is a pullback, bythe pasting law (Prop. 2.23). (i) First to see that G -equivariance of fr implies the factorization (276): By the characterization of G -actions (107) G -equivariance of fr means, equivalently, that fr is the morphism on homotopy fibers over B G induced from anequivalence fr (cid:12) G on homotopy quotients. But, by (b) and Prop. 4.22, such an equivalence is classified by ahomotopy of the form (276). (ii) Now to see that, conversely, the existence of a homotopy “ (cid:96) fr (cid:12) G ” of the form (276) implies the existence ofa G -equivariant framing fr (quotation marks now since we yet have to show that the two are related in this way).For this, we have to show that the morphism on homotopy fibers induced by fr (cid:12) G is a framing fr. But, by thenature of the G -action on T X from Prop. 4.22, the nature of the diagonal G -action exhibited by the middle frontsquare, and using the pasting law (Prop. 2.23), this means to show that the left front and rear squares are homotopypullbacks. For the front left square this follows by the factorization of ρ X ◦ fib ( ρ X ) through the point, using (a) , (b) and the pasting law (Prop. 2.23). For the rear left square, this follows by Prop. 3.31, since fib ( ρ ) is a localdiffeomorphism by Prop. 4.21. (cid:3) G -Structures.Definition 4.35 ( G -Structure coefficients) . Let H be an elastic ∞ -topos (Def. 3.21) and V ∈ Groups ( H ) (Prop.2.74). Then a coefficient for G-structure ( G , φ ) ∈ Groups ( H ) / Aut ( T e V ) is a group G equipped with a homomorphism of groups G φ (cid:47) (cid:47) Aut ( T e V ) to the structure group (Def. 4.24) of V -folds. Under delooping (101) this is equivalently a morphism in H of the form B G B φ (cid:47) (cid:47) B Aut ( T e V ) .81 efinition 4.36 (G-structures on V -folds) . Let H be an elastic ∞ -topos (Def. 3.21), V ∈ Groups ( H ) (Prop. 2.74), ( G , φ ) ∈ Groups ( H ) / Aut ( T e V ) (Def. 4.35) and X ∈ V Folds ( H ) (Def. 4.14). (i) We say that• a ( G , φ ) -structure on X (often just G-structure if φ is understood),• or ( G , φ ) -structure on its frame bundle (Def. 4.26),• or reduction of the structure group (4.24) along φ is a lift ( τ , g ) of the frame bundle classifying map (268) through B φ : B G B φ (cid:15) (cid:15) V -fold X (cid:96) Frames ( X ) (cid:47) (cid:47) G -structure τ (cid:52) (cid:52) B Aut ( T e V ) structure groupof frame bundle (cid:68) (cid:76) g (277) (ii) We say that the
G-frame bundle G
Frames ( X ) of a V -fold X equipped with such a ( G , φ ) -structure is the G -principal bundle which is classified (via Prop. 2.88): by τ , hence the object in the following diagram: G Frames ( X , τ ) (cid:15) (cid:15) (cid:47) (cid:47) (pb) ∗ (cid:15) (cid:15) Frames ( X ) (cid:15) (cid:15) (cid:47) (cid:47) (pb) ∗ (cid:15) (cid:15) X τ (cid:47) (cid:47) (cid:96) Frames ( X ) (cid:51) (cid:51) B G B φ (cid:47) (cid:47) B Aut ( T e V ) (278) (iii) We write ( G , φ ) Structures X ( H ) : = H / B Aut ( TeV ) (cid:0) (cid:96) Frames ( X ) , B φ (cid:1) ∈ Groupoids ∞ (279)for the ∞ -groupoid of ( G , φ ) -structures on the V -fold X .In direct generalization of Prop. 4.28 we have: Proposition 4.37 ( G -structured V -fold is G -quotient of its G -frame bundle) . Let H be an elastic ∞ -topos (Def.3.21), V ∈ Groups ( H ) (Prop. 2.74), ( G , φ ) ∈ Groups ( H ) / Aut ( T e V ) (Def. 4.35), X ∈ V Folds ( H ) (Def. 4.14) and ( τ , g ) ∈ ( G , φ ) Structures X ( H ) (Def. 4.36). Then (i) X is equivalently the homotopy quotient (108) of its G-frame bundle (278) by G:X (cid:39) G Frames ( X , τ ) (cid:12) G . (ii) the classifying map of the G-frame bundle on X exhibits the action of G on G Frames ( X , τ ) according to (107) .Proof. This is immediate from the equivalence between principal bundles and homotopy quotient projections (Re-mark 2.89) applied to the G -frame bundle (278): G Frames ( X , τ ) fib ( ρ ) (cid:39) fib ( τ ) (cid:15) (cid:15) G Frames ( X , τ ) (cid:12) G ρ G (cid:52) (cid:52) (cid:39) X τ (cid:47) (cid:47) B G (cid:3) Example 4.38 ( G -structure induced from framing) . Let H be an elastic ∞ -topos (Def. 3.21), V ∈ Groups ( H ) (Prop.2.74) and X ∈ V Folds ( H ) (Def. 4.14). Then a framing on X (Def. 4.30) induces a ( G , φ ) -structure (Def. 4.36) forany ( G , φ ) ∈ Groups ( H ) / Aut ( T e V ) , given by the pasting ∗ (cid:47) (cid:47) (cid:38) (cid:38) B G B φ (cid:15) (cid:15) X (cid:61) (cid:61) (cid:96) Frames ( X ) (cid:47) (cid:47) B Aut ( T e V ) (cid:72) (cid:80) (cid:96) fr (cid:46) (cid:54) (280)of the homotopy (cid:96) fr (270) which classifies the framing (Remark 4.31) with the homotopy that exhibits the grouphomomorphism φ as a morphism of pointed objects (Prop. 2.74).82 xample 4.39 (Canonical G -structure) . Let H be an elastic ∞ -topos H (Def. 3.21), and V ∈ Groups ( H ) (Prop.2.74). Then V itself, regarded as a V -fold by Example 4.16, carries a ( G , φ ) -structure (Def. 4.36) for any ( G , φ ) ∈ Groups ( H ) / Aut ( T e V ) , induced via Example 4.38 from its canonical framing fr (cid:96) (271) via left-translation (Prop. 4.32).We call this the canonical ( G , φ ) -structure on V : B G B φ (cid:15) (cid:15) V τ V (cid:54) (cid:54) (cid:47) (cid:47) B Aut ( T e V ) g V (cid:62) (cid:70) : = ∗ (cid:47) (cid:47) (cid:36) (cid:36) B G B φ (cid:15) (cid:15) V (cid:64) (cid:64) (cid:96) Frames ( V ) (cid:47) (cid:47) B Aut ( T e V ) (cid:72) (cid:80) (cid:96) fr (cid:96) (cid:47) (cid:55) (281) Local isometriesLemma 4.40 ( G -structures pull back along local diffeomorphisms) . Let H be an elastic ∞ -topos (Def. 3.21),V ∈ Groups ( H ) (Prop. 2.74) and ( G , φ ) ∈ Groups ( H ) / Aut ( T e V ) (Prop. 2.74, Def. 2.86, Example 3.30). Thenpre-composition constitutes a contravariant ∞ -functor (“pullback of ( G , φ ) -structures”) (cid:0) V Folds ( H ) ´et (cid:1) op (cid:47) (cid:47) Groupoids ∞ X f ´et (cid:15) (cid:15) ( G , φ ) Structures X ( H ) (cid:79) (cid:79) f ∗ ∈ τ ◦ f (cid:79) (cid:79) (cid:95) X ( G , φ ) Structures X ( H ) ∈ τ (282) from the ∞ -category (254) of V -folds and local diffeomorphisms, which assigns to any V -fold its ∞ -groupoid (279) of ( G , φ ) -structres (Def. 4.36).Proof. We need to show that for ( τ , g ) a ( G , φ ) -structure on X , the composite B G B φ (cid:15) (cid:15) X f ´et (cid:47) (cid:47) X τ (cid:51) (cid:51) (cid:96) Frames ( X ) (cid:47) (cid:47) B Aut ( T e V ) g (cid:49) (cid:57) (283)is a ( G , φ ) -structure on X . For this we need to exhibit a natural equivalence (cid:0) (cid:96) Frames ( X ) (cid:1) ◦ f (cid:39) (cid:96) Frames ( X ) so that X f ´et (cid:47) (cid:47) (cid:96) Frames ( X ) (cid:41) (cid:41) X (cid:96) Frames ( X ) (cid:15) (cid:15) τ (cid:47) (cid:47) B G B φ (cid:117) (cid:117) B Aut ( T e V ) (cid:39) (cid:47) (cid:55) g (cid:46) (cid:54) But this exists by Prop. 3.31. (cid:3)
Definition 4.41 (Local isometries between G -structured V -folds) . Let H be an elastic ∞ -topos (Def. 3.21), V ∈ Groups ( H ) (Prop. 2.74) and ( G , φ ) ∈ Groups ( H ) / Aut ( T e V ) (Prop. 2.74, Def. 2.86, Example 3.30). (i) For X , X ∈ V Folds (Def. 4.14) and ( τ i , g i ) ∈ ( G , φ ) Structures X i ( H ) (279), we say a local isometry , to bedenoted (cid:0) X , ( τ , g ) (cid:1) met ( f , σ ) (cid:47) (cid:47) (cid:0) X , ( τ , g ) (cid:1) is a pair X f ´et (cid:47) (cid:47) X , f ∗ ( τ , g ) (cid:39) σ (cid:47) (cid:47) ( τ , g ) , (284)consisting of a local diffeomorphism (Def. 3.26) and an equivalence of ( G , φ ) -structures (279) between that on itsdomain V -fold and the pullback (283) of the ( G , φ ) -structure on its codomain V -fold. (ii) Equivalently, by (283), a local isometry (284) is a morphism between ( G , φ ) -structured V -folds regarded asobjects in the iterated slice ∞ -topos (Example 2.47) (a) over B Aut ( T e V ) via their classifying maps of their frame bundles (268) (b) over (cid:0) B G , B φ (cid:1) via their ( G , φ ) -structure (277)of this form: 83 (cid:35) (cid:35) (cid:96) F r a m e s ( X ) τ (cid:46) (cid:46) f ´et (cid:47) (cid:47) X τ (cid:37) (cid:37) (cid:15) (cid:15) B G B Aut ( T e V ) (cid:117) (cid:117) B φ g (cid:50) (cid:58) σ (cid:106) (cid:114) g (cid:39) (cid:47) (cid:55) (cid:63) ∈ (cid:0) H / B Aut ( T e V ) (cid:1) / ( B G , B φ ) (cid:16)(cid:0) X , ( τ , g ) (cid:1) , (cid:0) X , ( τ , g ) (cid:1)(cid:17) . (285) (iii) Hence we write ( G , φ ) Structured V Folds ( H ) −! (cid:0) H / B Aut ( T e V ) (cid:1) / B G ∈ Categories ∞ (286)for the sub- ∞ -category of this iterated slice on 1-morphisms of the form (285). Integrability of G -structures.Definition 4.42 (Integrable G -structure) . Let H be an elastic ∞ -topos (Def. 3.21), V ∈ Groups ( H ) (Prop. 2.74), ( G , φ ) ∈ Groups ( H ) / Aut ( T e V ) (Def. 4.35). (i) Given ( X , ( τ X , g X )) ∈ ( G , φ ) Structured V Folds ( H ) (Def. 4.41), we say that ( τ , g ) is an integrable ( G , φ ) -structure on the V -fold X if there exists a correspondence of local isometries (284) between V equipped withits canonical ( G , φ ) -structure ( τ V , g V ) (Def. 4.39) to ( X , ( τ X , g X )) : (cid:0) U , ( τ U , g U ) (cid:1) met (cid:114) (cid:114) met (cid:44) (cid:44) (cid:44) (cid:44) (cid:0) V , ( τ V , g V ) (cid:1) (cid:0) X , ( τ X , g X ) (cid:1) (287)such that the right left is, in addition, an effective epimorphism (Def. 2.63), then called a ( V , ( τ V , g V )) -atlas of ( X , ( τ X , g X )) (100). (Underlying this, forgetting the ( G , φ ) -structures, is a V -atlas (252).) (ii) We writeIntegrably ( G , φ ) Structured V Folds ( H ) (cid:31) (cid:127) (cid:47) (cid:47) ( G , φ ) Structured V Folds ( H ) ∈ Categories ∞ (288)for the full sub- ∞ -category of that of ( G , φ ) -structured V -folds (286) on those that are integrable. Definition 4.43 (Locally integrable G -structure) . Let H be an elastic ∞ -topos (Def. 3.21), V ∈ Groups ( H ) (Prop.2.74), ( G , φ ) ∈ Groups ( H ) / Aut ( T e V ) (Def. 4.35), X ∈ V Folds ( H ) (Def. 4.14) and ( τ , g ) ∈ ( G , φ ) Structures X ( H ) (Def. 4.36). We say that ( τ , g ) is a locally integrable ( G , φ ) -structure if, for each point ∗ x (cid:47) (cid:47) X , there is a localdiffeomorphism φ x of the local neighborhood (Def. 3.28) of ∗ e (cid:47) (cid:47) V onto a local neighborhood of x such thatthe restriction of ( τ , g ) along φ is equivalent to the canonical ( G , φ ) -structure (Def. 4.39) on T e V : ∀ ∗ x (cid:47) (cid:47) X ∃ T e V φ x ´et (cid:47) (cid:47) X ∗ x (cid:65) (cid:65) e (cid:96) (cid:96) : φ ∗ x ( τ , g ) (cid:39) ( τ T e V , g T e V ) . Another way to say this: We have a correspondence of local isometries as in (287), but with the right legrequired to be an effective epimorphism only under (cid:91) . Example 4.44 ( G -Structures on smooth manifolds and orbifolds) .(i) Let H = JetsOfSmoothGroupoids ∞ (Example 3.24) G ∈ LieGroups (cid:44) −! Groups (cid:0) H (cid:1) (see (161)) and X ∈ SmoothManifolds (cid:44) −! H regarded as an R n -fold according to Example 4.17. In this case, the structure groupof X (Def. 4.24) is the ordinary general linear group GL R ( n ) (Example 4.25). Therefore, a G -structure on X inthe sense of Def. 4.36 is (by Example 4.29) a G -structure in the traditional sense of differential geometry [St64,VII][Kob72][Mol77]; and it is integrable according to Def. 4.42 if it is “flat” in the traditional sense of [Gu65]and locally integrable according to Def. 4.43 precisely if it is “uniformly 1-flat” in the traditional sense of [Gu65],namely if it is torsion-free (review in [Lot01]). Examples include:84 φ (cid:47) (cid:47) GL R ( n ) ( G , φ ) -structure Locally integrable Integrable seeSp R ( n ) (cid:31) (cid:127) (cid:47) (cid:47) GL R ( n ) almostsymplectic symplectic symplectic [St64, VII.2] GL C ( n / ) (cid:31) (cid:127) (cid:47) (cid:47) GL R ( n ) almostcomplex complex complexO ( n ) (cid:31) (cid:127) (cid:47) (cid:47) GL R ( n ) Riemannian torsion-freeRiemannian flatRiemannianO ( n − , ) (cid:31) (cid:127) (cid:47) (cid:47) GL R ( n ) Lorentzian torsion-freeLorentzian flatLorentzian [LPZ13] O ( n ) × R (cid:31) (cid:127) (cid:47) (cid:47) GL R ( n ) CO ( n ) -structure conformal flatconformal [AG98] CR ( n / − ) (cid:31) (cid:127) (cid:47) (cid:47) GL R ( n ) CR ( n ) -structure Cauchy-Riemann flatCauchy-Riemann [DT06] GL H ( n / ) (cid:31) (cid:127) (cid:47) (cid:47) GL R ( n ) GL H ( n / ) -structure hypercomplex flathypercomplex [Jo95] U ( n / ) (cid:31) (cid:127) (cid:47) (cid:47) GL R ( n ) hermitianalmost complex K¨ahler K¨ahler [Mor07, 11.1] SU ( n / ) (cid:31) (cid:127) (cid:47) (cid:47) GL R ( n ) SU ( n ) -structure Calabi-Yau Calabi-Yau [Pri15, 1.3] Sp ( n / ) · Sp ( ) (cid:31) (cid:127) (cid:47) (cid:47) GL R ( n ) almost unimodularquaternionic quaternionic K¨ahler flatquaternionic K¨ahler [AM93a][AM93b] Sp ( n / ) (cid:31) (cid:127) (cid:47) (cid:47) GL R ( n ) almostHyperk¨ahler Hyper¨ahler flatHyperk¨ahler G (cid:31) (cid:127) (cid:47) (cid:47) GL R ( ) G -structure torsion-free G -structure flat/interable G -structure [Br05] Spin ( ) (cid:31) (cid:127) (cid:47) (cid:47) GL R ( ) Spin ( ) -structure torsion-freeSpin ( ) -structure flatSpin ( ) -structure [Br87][Jo01] (ii) For k > H = k JetsOfSmoothGroupoids ∞ (Def. 3.24) the local integrability condition of Def. 4.43 is ofthe form of the “uniformly k -flatness”-condition of [Gu65]. But beware that, according to Def. 4.36 but in contrastto [Gu65], in this case the G -structure itself is not on the plain frame bundle but on the order- k jet frame bundle(by Example 4.29). Haefliger groupoids.Definition 4.45 (Haefliger groupoid) . Let H be an elastic ∞ -topos (Def. 3.21) and V ∈ Groups ( H ) (Prop. 2.74). (i) With no further structure, (a)
The V - Haefliger groupoid is the ´etale groupoid (Def. 3.35)Haef • ( V ) ∈ ´EtaleGroupoids ( H ) which is the ´etalification (Def. 3.39) of the Atiyah groupoid (Def. 2.90) of the frame bundle (Def. 4.24) of V regarded as a V -fold (Example 4.16):Haef • ( V ) : = At ´et • (cid:0) Frames ( V ) (cid:1) . (289) (b) The
V -Haefliger stack of V is the corresponding V -fold (according to Remark 4.15): H aef ( V ) : = A t ´et (cid:0) Frames ( V ) (cid:1) ∈ V Folds . (290) (ii) Given, in addition, ( G , φ ) ∈ Groups ( H ) / Aut ( T e V ) (Def. 4.35), with G Frames ( V ) ! V denoting the G -framebundle (278) corresponding to the canonical ( G , φ ) -structure on V (Example 4.39), we say (a) the (cid:0) V , ( G , φ ) (cid:1) - Haefliger groupoid is the ´etale groupoid (Def. 3.35)Haef • (cid:0) V , ( G , φ ) (cid:1) ∈ ´EtaleGroupoids ( H ) which is the ´etalification (Def. 3.39) of the Atiyah groupoid (Def. 2.90) of the G -frame bundle (278):Haef • (cid:0) V , ( G , φ ) (cid:1) : = At ´et • (cid:0) G Frames ( V ) (cid:1) . (291) (b) The (cid:0) V , ( G , φ ) (cid:1) -Haefliger stack of V is the corresponding V -fold (according to Remark 4.15): H aef (cid:0) V , ( G , φ ) (cid:1) : = A t ´et (cid:0) G Frames ( V ) (cid:1) ∈ V Folds . (292)85 roposition 4.46 (Haefliger stack represents V -fold structure) . Let H be an elastic ∞ -topos (Def. 3.21) V ∈ Groups ( H ) (Prop. 2.74) and X ∈ H . Then the following are equivalent: (i) X is a V -fold (Def. 4.14); (ii)
X admits a local diffeomorphism to the V -Haefliger stack (Def. 4.45).Proof.
First consider the implication (i) ⇒ (ii) : Assuming X is a V -fold, consider a V -atlas (252) V (cid:111) (cid:111) ´et U ´et (cid:47) (cid:47) (cid:47) (cid:47) X . By Prop. 3.31 (and as in the proof of Prop. 4.26) the pullbacks of the frame bundles of V and of X along this V -atlasto U coincide there, which means that we have a homotopy-commutative square of their classifying maps (268) asshown on the bottom left of the following diagram: (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) U × X U ´et (cid:15) (cid:15) (cid:79) (cid:79) ´et ´et (cid:15) (cid:15) (cid:47) (cid:47) At (cid:0) Frames ( V ) (cid:1) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) U ´et (cid:15) (cid:15) (cid:15) (cid:15) ´et (cid:47) (cid:47) V (cid:96) Frames ( V ) (cid:15) (cid:15) (cid:15) (cid:15) X (cid:96) Frames ( X ) (cid:47) (cid:47) B Aut ( T e V ) ⇔ (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) U × X U ´et (cid:15) (cid:15) (cid:79) (cid:79) ´et ´et (cid:15) (cid:15) ´et (cid:47) (cid:47) At ´et (cid:0) Frames ( V ) (cid:1) ´et (cid:15) (cid:15) (cid:79) (cid:79) ´et ´et (cid:15) (cid:15) U ´et (cid:15) (cid:15) (cid:15) (cid:15) ´et (cid:47) (cid:47) V ´et (cid:15) (cid:15) (cid:15) (cid:15) X (cid:47) (cid:47) H aef ( V ) (293)By passing to nerves (Example 2.69) of the vertical morphisms, this induces a morphism of groupoids as shownon the top left. But U • is an ´etale groupoid (by Prop. 3.36), and U −! V is a local diffeomorphism by definitionof V -atlases, so that the top left part of the left diagram in (293) is in the ´etale slice over V (Def. 3.32). Therefore,the adjunction (168) of Prop. 3.33 implies that the top part of the diagram on the left of (293) factors through the´etalification (Def. 3.39) as shown in the top part on the right. With this we get the dashed morphism on the rightby passing to colimits over the vertical simplicial diagrams (as in Prop. 3.36).It only remains to see that the dashed morphism on the right is itself a local diffeomorphism. For this observethat al the horizontal morphisms are local diffeomorphisms, using the assumptions and then left-cancellability(Lemma 3.27). Therefore the statement follows with Lemma 3.38.For the converse implication (ii) ⇒ (i) : Given a local diffeomorphism as shown dashed on the right of (293),we need to produce a V -atlas for X . So now define the bottom square on the right of (293) to be the pullback ofthe ´etale atlas of the Haefliger stack along the griven morphism. This does make the top left span of the square a V -atlas by the fact that the classes of local diffeomorphisms and of effective epimorphisms are both closed underpullback (by Lemma 2.65 and Lemma 3.27). (cid:3) Proposition 4.47 ( G -Structured Haefliger stack represents integrable G -structure) . Let H be an elastic ∞ -topos(Def. 3.21), V ∈ Groups ( H ) (Prop. 2.74), ( G , φ ) ∈ Groups ( H ) / Aut ( T e V ) (Def. 4.35). The (cid:0) V , ( G , φ ) (cid:1) -Haefligergroupoid (Def. 4.45), carries a canonical integrable ( G , φ ) -structure (Def. 4.42) ( τ H , g H ) ∈ ( G , φ ) Structures H Haef ( V ) ( H ) (294) such that the operation of pullback of (283) along local diffeomorphism (Lemma 4.40) constitutes a natural bijec-tion π Integrably ( G , φ ) Structured V Folds ( H ) (cid:39) π ´Et H aef ( V , ( G , φ )) (cid:0) X , ( τ , g ) (cid:1) (cid:16) X (cid:96) ( τ , g ) −−−! H aef (cid:0) V , ( G , φ ) (cid:1)(cid:17) (295) between the sets of equivalence classes of: ( i) integrably ( G , φ ) -structured V -folds (Def. 4.42), (ii) local diffeomorphisms into the (cid:0) V , ( G , φ ) (cid:1) -Haefliger stack, hence objects in its ´etale topos (Def. 3.32).Proof. We proceed as in the proof of Prop. 4.46, but lifting the diagram there from H to the iterated slice (cid:0) H / B Aut ( T e V ) (cid:1) / B G (285). 86 i) First consider an integrably G -structured V -fold (cid:0) X , ( τ , g ) (cid:1) . We describe the construction of a local diffeomor-phism into the Haefliger stack from this: Pick any (cid:0) V , ( τ V , g V ) (cid:1) -atlas (cid:0) V , ( τ V , g V ) (cid:1) (cid:111) (cid:111) met (cid:0) U , ( τ U , g U ) (cid:1) met (cid:47) (cid:47) (cid:47) (cid:47) (cid:0) X , ( τ X , g X ) (cid:1) (287). By Def. 4.36, this is equivalently a choice of equivalence between the pullbacks to U of the G -structures on V and on X . Regarded in the iterated slice (285), this equivalently means that we have a square in (cid:0) H / B Aut ( T e V ) (cid:1) / B G (285), as shown on the left of the following: (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) U × X U ´et (cid:15) (cid:15) (cid:79) (cid:79) ´et ´et (cid:15) (cid:15) (cid:47) (cid:47) At (cid:0) G Frames ( V ) (cid:1) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) U ´et (cid:15) (cid:15) (cid:15) (cid:15) ´et (cid:47) (cid:47) V τ V (cid:15) (cid:15) (cid:15) (cid:15) (cid:96) Frames ( V ) (cid:15) (cid:15) X τ X (cid:47) (cid:47) (cid:96) Frames ( X ) (cid:48) (cid:48) B G (cid:37) (cid:37) B Aut ( T e V ) g X (cid:56) (cid:64) g V (cid:114) (cid:122) ⇔ (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) (cid:79) (cid:79) (cid:15) (cid:15) U × X U ´et (cid:15) (cid:15) (cid:79) (cid:79) ´et ´et (cid:15) (cid:15) ´et (cid:47) (cid:47) At ´et (cid:0) G Frames ( V ) (cid:1) ´et (cid:15) (cid:15) (cid:79) (cid:79) ´et ´et (cid:15) (cid:15) U ´et (cid:15) (cid:15) (cid:15) (cid:15) ´et (cid:47) (cid:47) V ´et (cid:15) (cid:15) (cid:15) (cid:15) (cid:96) Frames ( V ) (cid:20) (cid:20) τ V (cid:25) (cid:25) X (cid:96) ( τ X , g X ) (cid:47) (cid:47) (cid:96) Frames ( X ) (cid:48) (cid:48) τ X (cid:45) (cid:45) H aef (cid:0) V , ( G , φ ) (cid:1) τ H (cid:40) (cid:40) (cid:96) Frames ( H ) (cid:45) (cid:45) B G B φ (cid:37) (cid:37) B Aut ( T e V ) g X (cid:49) (cid:57) g X (cid:1) (cid:9) (296)Now we proceed as follows: (a) Observing (with Prop. 2.53) that fiber products in the iterated slice are actually given by the plain fiber productsin H equipped with canonical morphisms to the slicing objects, we find that passing to nerves (Example 2.69) ofthe vertical morphisms on the left of (296) yields a morphism from the ´etale groupoid induced by the given V -coverof X to the Atiyah groupoid of G Frames ( X ) (Def. 2.90) – just as in (293), but now equipped with coherent mapsto B φ . (b) Therefore, we obtain the factorization through the (cid:0) V , ( G , φ ) (cid:1) -Haefliger groupoid (the ´etalification of the Atiyahgroupoid of the G -frame bundle shown on the top right of (296)) just as in (293), but now, in addition, coherentlyequipped with maps to B φ . (c) After this ´etalification we may identify these maps: Since those on V remain unchanged by ´etalification over V , these still give the canonical ( G , φ ) -structure ( τ V , g V ) , as shown on the far right of (296). But since now thevertical simplicial morphisms are all local diffeomorphisms, pullback along which preserves ( G , φ ) -structure (byLemma 4.40) and in particular preserves tangent- and frame bundles (by Prop. 3.31) it follows that all stages of the (cid:0) V , ( G , φ ) (cid:1) -Haefliger groupoid in the top right are now equipped with the classifying map of their frame bundles. (d) Since colimits in the slice are given by colimits in the underlying topos (by Example 2.52), the colimit overthe simplicial sub-diagram on the far right of (296) still yields the (cid:0) V , ( τ , g ) (cid:1) -Haefliger stack (292), as shown, nowequipped with canonical maps to B φ . (e) We claim that the induced map from the Haefliger stack to B Aut ( T e V ) , denoted (cid:96) Frames ( H ) in (296), isindeed the classifying map of the frame bundle of the Haefliger stack: (cid:96) Frames ( H ) (cid:39) (cid:96) Frames (cid:16) H aef (cid:0) V , ( G , φ ) (cid:1)(cid:1) . (297)This follows because:• by (c) above, the component maps of the colimiting map classify the frame bundles of the stages of thesimplicial nerve;• therefore, the colimiting map classifies the colimit of the frame bundles of the simplicial nerve, by Prop.2.56,• but the colimit of the tangent bundles of the ´etale cover is the tangent bundle of the corresponding ´etale stack,by Prop. 3.37. 87 f) In particular, this implies that the induced homotopy which fills the bottom right part of (296) (cid:96)
Frames (cid:0) H (cid:1) g H (cid:43) (cid:51) B φ ◦ τ H , (298)canonically given by the colimit construction in the iterated slice, constitutes a ( G , φ ) -structure on the (cid:0) V , ( G , φ )) (cid:1) -Haefliger stack. (g) In conclusion, the dashed morphism on the right of (296) exists and is a local diffeomorphism, as in the proofof Prop. 4.46; but, by construction in the iterated slice, it is now exhibited as a local isometry to the Haefliger stackequipped with the induced ( G , φ ) -structure (298). (ii) The converse construction is now immediate: Given a local diffeomorphism of the form shown dashed on theright of (296), pulling back the ´etale atlas of the Haefliger stack along it yields a V -atlas for X (just as in the proofof this converse step in Prop. 4.46) and pulling (via Lemma 4.40) the ( G , φ ) -structure (298) around the resultingCartesian square makes this a (cid:0) V , ( G , φ ) (cid:1) -atlas that exhibits X as equipped with an integrable ( G , φ ) -structure.This construction is clearly injective on equivalence classes, by ∞ -functoriality of the pullback construction (283)of ( G , φ ) -structures; and it is surjective on equivalence classes by item (i) above. Hence this is a bijection onequivalence classes, as claimed. (cid:3) Tangential structures.
Closely akin to G -structures (Def. 4.36) are tangential structures (Def. 4.48 below) wherenot the structure group itself is lifted, but only its shape: Definition 4.48 (Tangential structure) . Let H be an elastic ∞ -topos (Def. 3.21), V ∈ Groups ( H ) (Prop. 2.74), ( G , φ ) ∈ Groups ( H ) / S Aut ( T e V ) (Def. 4.35) and X ∈ V Folds ( H ) (Def. 4.14). (i) We say that a tangential ( G , τ ) -structure on X is a lift ( τ , g ) through B φ of the composite of the frame bundleclassifying map (268) with the shape-unit (48): B G B φ (cid:15) (cid:15) V -fold X (cid:96) Frames ( X ) (cid:47) (cid:47) tangentialstructure τ (cid:50) (cid:50) B Aut ( T e V ) η S (cid:47) (cid:47) B S Aut ( T e V ) shape ofstructure groupof frame bundle (cid:56) (cid:64) g (299) (ii) We write Tangential ( G , φ ) Structures X ( H ) : = H / B S Aut ( T e V ) (cid:0) η S ◦ (cid:96) Frames ( X ) , B φ (cid:1) (300)for the ∞ -groupoid of ( G , φ ) -tangential structures on the V -fold X . Example 4.49 (Tangential structures on smooth manifolds) . Let H = JetsOfSmoothGroupoids ∞ (Example 3.24) G ∈ LieGroups (cid:44) −! Groups (cid:0) H (cid:1) (see (161)) and X ∈ SmoothManifolds (cid:44) −! H regarded as an R n -fold accordingto Example 4.17. In this case, the structure group of X (Def. 4.24) is the ordinary general linear group GL R ( n ) (Example 4.25). Hence here tangential structure in the general sense of Def. 4.48 is tangential structure in thetraditional sense of differential topology (popularized under this name in [GMTW06, 5], originally introduced as“ ( B , f ) -structure” [La63][St68, II], review in [Ko96, 1.4]). Example 4.50 (Cohesive refinement of tangential structure) . Every ( G , φ ) -structure (Def. 4.36) induces tangential ( S G , S φ ) -structure (Def. 4.48) by composition with the naturality square of η S on B φ : B G B φ (cid:15) (cid:15) η S B G (cid:47) (cid:47) B S G B S φ (cid:15) (cid:15) V -fold X (cid:96) Frames ( X ) (cid:47) (cid:47) ( G , φ ) -structure τ (cid:54) (cid:54) B Aut ( T e V ) η S B Aut ( TeV ) (cid:47) (cid:47) B S Aut ( T e V ) shape ofstructure groupof frame bundle (cid:72) (cid:80) g (301)Conversely, realizing a tangent structure as obtained from a G -structure this way means to find a geometric (differ-ential) refinement. 88 xample 4.51 (Orientation structure) . Let H = JetsOfSmoothGroupoids ∞ (Example 3.24) and X ∈ H an R n -fold (Def. 4.14) hence an ordinary manifold (Example 4.17) or, more generally, an ordinary ´etale Lie groupoid(Example 4.18). With the general linear and the (special) orthogonal group regarded as smooth groups via (161)SO ( n ) i SO (cid:47) (cid:47) O ( n ) i O (cid:47) (cid:47) GL ( n ) ∈ Groups ( SmoothManifolds ) (cid:47) (cid:47) Groups ( H ) (302)we have: (i) an O ( n ) -structure (Def. 4.36) on X is equivalently a Riemannian structure (Example 4.44); (ii) but a tangential S O ( n ) -structure (Def. 299) is equivalently no structure , since S O ( n ) S i O (cid:39) (cid:47) (cid:47) S GL ( n ) is anequivalence of underlying shapes (since O ( n ) is the maximal compact subgroup of GL ( n ) ), (iii) while a tangential S SO ( n ) -structure (Def. 299) is an orientation of X . (iv) A differential refinement, in the sense of Example 4.50, of such an orientation structure is an oriented Rie-mannian structure (via its induced volume form).
Example 4.52 (Higher Spin structure [SSS09][SSS12]) . Let H = JetsOfSmoothGroupoids ∞ (Example 3.24) and X ∈ H an R n -fold (Def. 4.14) hence an ordinary manifold (Example 4.17) or, more generally, an ordinary ´etaleLie groupoid (Example 4.18). The sequence of groups (302) in Example 4.51 is, under shape, the beginning of the Whitehead tower of S O ( n ) (cid:39) S GL ( n ) . The tangential structures (Def. 4.48, Example 4.49) corresponding to thestages in this tower are the Spin structure and its higher analogues: (cid:15) (cid:15) B S Fivebrane ( n ) (cid:15) (cid:15) B S String ( n ) (cid:15) (cid:15) B S Spin ( n ) (cid:15) (cid:15) B S SO ( n ) (cid:15) (cid:15) X (cid:96) Frames ( X ) (cid:44) (cid:44) Riemannianstructure (cid:47) (cid:47)
Orientationstructure (cid:48) (cid:48)
Spinstructure (cid:50) (cid:50)
Stringstructure (cid:51) (cid:51)
Fivebranestructure (cid:53) (cid:53) B O ( n ) η S B O ( n ) (cid:47) (cid:47) (cid:15) (cid:15) B S O ( n ) (cid:39) (cid:15) (cid:15) B GL ( n ) η S B GL ( n ) (cid:47) (cid:47) B S GL ( n ) (303) Flat V -folds.Definition 4.53 (Flat V -folds) . Let H be an elastic ∞ -topos (Def. 3.21), V ∈ Groups ( H ) (Prop. 2.74) and X ∈ V Folds ( H ) (Def. 4.14). We say that X is flat if the classifying map (268) of its frame bundle (Prop. 4.26) factorsthrough the (cid:91) -counit (49), hence if it carries ( G , φ ) -structure (Def. 4.36) for ( G , φ ) = ( (cid:91) Aut ( T e V ) , ε (cid:91) Aut ( T e V ) ) : (cid:91) B Aut ( T e V ) ε (cid:91) B Aut ( TeV ) (cid:15) (cid:15) X (cid:96) Frames ( X ) (cid:47) (cid:47) τ (cid:53) (cid:53) B Aut ( T e V ) (cid:55) (cid:63) (304)By the universal property of ε (cid:91) and since (cid:91) commutes with B , this means equivalently that X carries G -structurefor any discrete group G (cid:39) (cid:91)(cid:91) G . 89 roposition 4.54 (Flat frame bundles are V -folds) . Let H be an elastic ∞ -topos (Def. 3.21), V ∈ Groups ( H ) (Prop.2.74) and X ∈ V Folds ( H ) (Def. 4.14). If X is flat (Def. 4.53), then (i) its flat frame bundle ( (cid:91) Aut ( T e V )) Frames ( X ) (278) is itself a V -fold (Def. 4.14) and (ii) the bundle morphism is a local diffeomorphism (Def. 3.26): ( (cid:91) Aut ( T e V )) Frames ( X ) ´et (cid:47) (cid:47) X .Proof.
First consider (ii) : We need to show that the left square in the following pasting diagram is Cartesian: ( (cid:91) Aut ( T e V )) Frames ( X ) η ℑ ( (cid:91) Aut ( TeV )) Frames ( X ) (cid:47) (cid:47) p (cid:15) (cid:15) ℑ (cid:0) ( (cid:91) Aut ( T e V )) Frames ( X ) (cid:1) (cid:47) (cid:47) ℑ p (cid:15) (cid:15) (pb) ℑ ∗ (cid:15) (cid:15) X η ℑ X (cid:47) (cid:47) ℑ X ℑ τ (cid:47) (cid:47) ℑ (cid:91) Aut ( T e V ) Here the right square is Cartesian, by definition (278) and since ℑ , being a right adjoint, preserves Cartesian squares(by Prop. 2.26). Hence, by the pasting law (Prop. 2.23) it is sufficient to show that the total rectangle is Cartesian.But, by the naturality of η ℑ , the total rectangle is equivalent to that of the following pasting diagram: ( (cid:91) Aut ( T e V )) Frames ( X ) (cid:15) (cid:15) (cid:47) (cid:47) (pb) ∗ (cid:15) (cid:15) η ℑ ∗ (cid:47) (cid:47) (pb) ℑ ∗ (cid:15) (cid:15) X τ (cid:47) (cid:47) (cid:91) B Aut ( T e V ) η ℑ (cid:91) B Aut ( TeV ) (cid:47) (cid:47) ℑ (cid:91) B Aut ( T e V ) Here the left square is Cartesian by the definition (278), while the right square is Cartesian since its two horizontalmorphisms are equivalences, by elasticity. Hence the total rectangle is Cartesian by the pasting law (Prop. 2.23).Regarding (i) : We need to exhibit a V -atlas (252) for the flat frame bundle. So let V (cid:111) (cid:111) ´et U ´et (cid:47) (cid:47) (cid:47) (cid:47) X be a V -atlasfor X , and consider the following pullback diagram: U × X ( (cid:91) Aut ( T e V )) Frames ( X ) ´et (cid:15) (cid:15) (cid:15) (cid:15) ´et (cid:47) (cid:47) (cid:47) (cid:47) (pb) ( (cid:91) Aut ( T e V )) Frames ( X ) ´et (cid:15) (cid:15) (cid:15) (cid:15) U ´et (cid:15) (cid:15) ´et (cid:47) (cid:47) (cid:47) (cid:47) XV Observe that all four morphisms in the square are effective epimorphisms (Def. 2.63) and local diffeomorphisms(Def. 3.26): The bottom one by definition, the right one by (ii) and hence the other two since both classes ofmorphisms are closed under pullback (Lemma 2.65 and Lemma 3.27). Finally, since the class of local diffeomor-phisms is also closed under composition (Lemma 3.27), the total vertical morphisms is a local diffeomorphism,and hence the total outer diagram is a V -atlas of the flat frame bundle. (cid:3) Proposition 4.55 ( (cid:91) G -frame bundles are V -folds) . Let H be an elastic ∞ -topos (Def. 3.21), V ∈ Groups ( H ) (Prop.2.74) X ∈ V Folds ( H ) (Def. 4.14), ( G , φ ) ∈ Groups ( H ) / Aut ( T e V ) (Prop. 2.74, Def. 2.86, Example 3.30) with G (cid:39) (cid:91) Gdiscrete, and ( τ , g ) ∈ ( G , φ ) Structures X ( H ) . Then the corresponding G-frame bundle (278) is itself a V -fold:G (cid:39) (cid:91) G ⇔ G Frames ( X ) ∈ V Folds ( H ) . Proof.
The proof proceeds verbatim as that for Prop. 4.54, just with the structure group restricted along (cid:91) G ! (cid:91) Aut ( T e V ) . (cid:3) In summary, we have found the general abstract version of the local model spaces of orbifolds:
Proposition 4.56 (Local orbifold model spaces) . Let H be an elastic ∞ -topos (Def. 3.21), G , V ∈ Groups ( H ) (Prop. 2.74), with G (cid:39) (cid:91) G discrete, and ( V , ρ ) ∈ G Actions ( H ) (Prop. 2.79) a restriction (Prop. 2.85) of the action ( V , ρ Aut ) by group-automorphisms (Prop. 2.102). Then the homotopy quotient (108) V (cid:12) G ∈ H of V regarded with its canonical framing (Prop. 4.32) i) is a flat V -fold (Def. 4.53); (ii) with G-structure (Def. 4.36) (iii) whose G-frame bundle (278) is G-equivariantly (Def. 2.83) equivalent to V itself:G Frames (cid:0) V (cid:12) G (cid:1) (cid:39) V . Proof.
First observe that V (cid:12) G is a V -fold, by Prop. 4.21 applied to Example 4.16. That this is flat (i) is implied by (ii) , since G is assumed to be discrete. For (ii) and (iii) observe that the canonical framing on V is G -equivariant,by Prop. 4.33, so that Prop. 4.34 implies G -structure on V (cid:12) G classified by the action morphism ρ itself. But thismeans that its homotopy fiber, hence the corresponding G -frame bundle (Def. 278) is V itself, by (107) (and inaccord with Prop. 4.55). (cid:3) Example 4.57 (Ordinary orbifold singularities) . Let H : = JetsOfSmoothGroupoids ∞ (Example 3.24) and V : =( R n , +) as in Example 4.17. Then a group automorphism of V is a linear isomorphism, hence Aut Grp ( R n , +) (cid:39) GL ( n ) . Therefore, in this case the assumptions of Prop. 4.56 hold precisely for V a linear representation of thediscrete group G , and thus we recover the traditional local orbifold models V (cid:12) G from [Sa56] (in their incarnationas ´etale groupoids). Orbi- V -folds. Finally, we promote V -folds to orbifolds proper, in that we promote the ∞ -category of ´etale stacksto a proper ∞ -category of higher orbifolds: Definition 4.58 (Orbi- V -folds) . Let H be a singular-elastic ∞ -topos (Def. 3.51) and V ∈ Groups ( H ⊂ ) . We say thatan orbi-V -fold is an object X ∈ H whose purely smooth aspect (208) is a V -fold (Def. 4.14). (i) We write V Orbifolds ( H ) ⊂ H for the full sub- ∞ -category on orbi- V -folds: X ∈ V Orbifolds ( H ) ⇔ ⊂ X ∈ V Folds ( H ) . This means, equivalently, that the orbi- V -folds in H are the orbi-singularizations (208) of the V -folds in H ⊂ : V Folds ( H ⊂ ) (cid:111) (cid:111) SmthOrbSnglr (cid:39) (cid:47) (cid:47) V Orbifolds ( H ) Smth ( X ) (cid:111) (cid:111) (cid:31) : = X : = X (cid:31) (cid:47) (cid:47) OrbSnglr ( X ) (305) (ii) Similarly, given, in addition, ( G , φ ) ∈ Groups ( H ) / Aut ( T e V ) (Def. 4.35), we write ( G , φ ) Structured V Orbifolds ( H ) ⊂ H for the full sub- ∞ -category on ( G , φ ) -structured orbi- V -folds (Def. 4.41): ( G , φ ) Structured V Folds ( H ⊂ ) (cid:111) (cid:111) SmthOrbSnglr (cid:39) (cid:47) (cid:47) ( G , φ ) Structured V Orbifolds ( H )( Smth ( X ) , ( τ , g )) (cid:111) (cid:111) (cid:31) : = ( X , ( τ , g )) : = ( X , ( τ , g )) (cid:31) (cid:47) (cid:47) ( OrbSnglr ( X ) , ( τ , g )) (306) Remark 4.59 (Coefficients for orbifold cohomology) . The point of Def. 4.58 is that, by regarding a V -fold inthe elastic ∞ -topos H ⊂ equivalently as an orbi- V -fold in the larger singular-elastic ∞ -topos H , a larger class ofcoefficients for intrinsic cohomology theories (22) becomes available, notably coefficients of the form S ≺ ( A (cid:12) G ) (see Lemma 4.7 below). This is what gives rise, in §5, to proper orbifold cohomology (Def. 5.15 below) in contrastto the coarser cohomology of underlying ´etale groupoids (Def. 5.11 below).91 emark 4.60 (The proper ∞ -category of higher orbifolds) . While (306) is an equivalence of abstract ∞ -categories, (i) it is not an equivalence of sub- ∞ -categories of the ambient singular-elastic ∞ -topos H : ∞ -category ofof ´etale groupoids V Folds ( H ⊂ ) (cid:115)(cid:19) Smth (cid:37) (cid:37) (cid:54)(cid:39) proper ∞ -categoryof orbifolds V Orbifolds ( H ) (cid:107)(cid:75) OrbSnglr (cid:120) (cid:120) ∈ ( Categories ∞ ) / H H(ii)
To bring out this distinction, also in view of Remark 4.59, we call V Orbifolds ( H ) (Def. 4.58) the proper ∞ -category of orbifolds , in contrast to the ∞ -category V Folds ( H ⊂ ) (253) of ´etale ∞ -groupoids. (iii) It is a happy coincidence that proper is also the technical adjective chosen in [DHLPS19] for equivarianthomotopy theories presented by ∞ -presheaves over categories of orbits with compact – hence finite if discrete –isotropy groups: In this terminology the singular-cohesive ∞ -topos H is, according to Def. 3.48, indeed a proper global equivariant homotopy theory. Example 4.61 ( V -folds) . Let H be a singular-elastic ∞ -topos (Def. 3.51) and V ∈ Groups ( H ) (Prop. 2.74) anygroup object, not necessarily smooth. Then a V -fold according to Def. 4.14 is, in particular, an orbi- ⊂ V -foldaccording to Def. 4.58, hence a V -fold for V : = ⊂ V the purely smooth aspect of V : V Folds ( H ) (cid:31) (cid:127) (cid:47) (cid:47) ( ⊂ V ) Orbifolds ( H ) . But, in general, being a V -fold is a much stronger condition than being an ( ⊂ V ) -orbifold, even (and in particular)if V is already smooth: For a V -fold X not only the full underlying ⊂ X is required to be locally equivalent to ⊂ V , but moreover, for each K ∈ Groups fin , the geometric K -fixed locus of ⊂ X is required to be locally equivalentto the geometric K -fixed locus of ⊂ V . Example 4.62 (Subcategories of smooth and of flat orbifolds) . Let H be an elastic ∞ -topos (Def. 3.21), V ∈ Groups ( H ) (Prop. 2.74) and ( G , φ ) ∈ Groups ( H ) / Aut ( T e V ) (Prop. 2.74, Def. 2.86, Example 3.30). We have fullyfaithful inclusions into the ∞ -category of ( G , φ ) -structured orbi- V -folds (Def. 4.58) ( G , φ ) Structured V Orbifolds ( H )( G , φ ) Structured V Folds ( H ) (cid:39) (cid:7) i ⊂ s m o o t h o r b i f o l d s (cid:52) (cid:52) ( (cid:91) G , φ ◦ ε (cid:91) ) Structured V Orbifolds ( H ) (cid:57) (cid:89) i (cid:91) fl a t o r b i f o l d s (cid:107) (cid:107) (307)of (i) smooth ( G , φ ) -structured V -folds, via Lemma 3.65; (ii) flat ( (cid:91) G , φ ◦ ε (cid:91) ) -structured V -folds (Def. 4.53). 92 Orbifold cohomology
With an internal ∞ -topos-theoretic characterization of orbifolds in hand (from §4), we immediately obtain aninduced notion of (differential, geometric, ´etale) orbifold cohomology , given by the intrinsic cohomology (22) ofthe ambient singular-cohesive ∞ -topos. Here we discuss how this new intrinsic notion of orbifold cohomology- subsumes proper equivariant cohomology theory (§5.1)- and unifies it with tangentially twisted cohomology (§5.2). Proper equivariant cohomology.Definition 5.1 (Borel equivariant cohomology) . Let H ⊂ be a cohesive ∞ -topos (Def. 3.1) G ∈ Groups ( H ⊂ ) (Prop.2.74) and ( X , τ ) , ( A , ρ ) ∈ G Actions ( H ⊂ ) (Prop. 2.79). Then the Borel equivariant cohomology of X with coeffi-cients in A is the intrinsic cohomology (22) in the slice H / B G (Prop. 2.46) of the homotopy quotient (108) of X with coefficients in the shape (128) of the homotopy quotient of A : Borel equivariantcohomology H Borel ( X , A ) : = π H / B G (cid:0) ( X (cid:12) G ) , S ( A (cid:12) G ) (cid:1) = ( X (cid:12) G ) cocycle c (cid:47) (cid:47) τ (cid:35) (cid:35) A (cid:12) G ρ (cid:124) (cid:124) B G (cid:116) (cid:124) (308) Definition 5.2 (Proper equivariant cohomology) . Let H be a singular-cohesive ∞ -topos (Def. 3.48) G ∈ Groups ( H (cid:91) ) (Prop. 2.74) a discrete ∞ -group, and ( X , τ ) , ( A , ρ ) ∈ G Actions ( H ) (Prop. 2.79). Then we say that the proper equiv-ariant cohomology of X with coefficients in A is the intrinsic cohomology (22) in the slice H / ≺ B G (Prop. 2.46) ofthe orbi-singularization (209) of the homotopy quotient (108) of X with coefficients in the shape (128) of theorb-singularization of the homotopy quotient of A : proper equivariantcohomology H G ( X , A ) : = π H / ≺ B G (cid:0) ≺ ( X (cid:12) G ) , S ≺ ( A (cid:12) G ) (cid:1) = ≺ ( X (cid:12) G ) cocycle c (cid:47) (cid:47) ≺ ( τ ) (cid:37) (cid:37) S ≺ ( A (cid:12) G ) (cid:0) η S ≺ B G (cid:1) − ◦ S ≺ ( ρ ) (cid:121) (cid:121) ≺ B G (cid:115) (cid:123) (309) Recovering traditional G -equivariant cohomology. We discuss how in the case of a finite group G , traditional G -equivariant cohomology (see §B) is a special case of proper equivariant cohomology (Def. 5.2). We take thekey observation from [Re14] (Prop. 5.6 below). Definition 5.3 ( G -equivariant cohesive ∞ -topos) . Let H ⊂ be a cohesive ∞ -topos (Def. 3.1) and G ∈ Groups fin afinite group (215). We write G H ⊂ : = Sheaves ∞ (cid:0) G Orbits , H ⊂ (cid:1) = Func ∞ (cid:0) G Orbits op , H ⊂ (cid:1) (310)for the ∞ -topos of H ⊂ -valued ∞ -sheaves on the G -orbit category (Def. B.8), to be called the corresponding G-equivariant cohesive ∞ -topos . Remark 5.4 (Proper equivariant cohomology theory in singular ∞ -toposes) . In the case H ⊂ (cid:39) Groupoids ∞ (35),Def. 5.3 reduces to the ∞ -category G Groupoids ∞ (Def. B.4) of traditional G -equivariant homotopy theory (recalledin §B). The intrinsic cohomology (22) of the ∞ -topos G Groupoids ∞ – or of its tangent ∞ -topos T (cid:0) G Groupoids ∞ (cid:1) (Example 2.51) in the twisted abelian case (Remark 2.96) – is proper equivariant cohomology (following termi-nology in [DHLPS19]), including G -Bredon cohomology [Br67a][Br67b] (review in [Blu17, §1.4][tD79, §7]), G -equivariant K-theory [Se68][AS69] (which is proper equivariant by [AS04, A3.2][FHT07, A.5][DL98]), G -equivariant Cohomotopy theory [Se71][tD79, §8][SS19][BSS19], etc.Hence, by Remark 3.20, to the extent that the objects of the cohesive ∞ -topos H ⊂ in Def. 5.3 are ∞ -groupoidsequipped with further geometric or differential-geometric structure, the intrinsic cohomology theory (22) in G H ⊂ (310) is an enhancement of plain G -equivariant cohomology to a flavor of proper G-equivariant differential coho-mology theory (by Remark 3.20). 93 roposition 5.5 (Cohesive Elmendorf theorem) . Consider a cohesive ∞ -topos H ⊂ (Def. 3.1) with an ∞ -site Charts of charts (Def. 3.9). Then for G ∈ Groups fin a finite group, we have an equivalence of ∞ -categoriesG H ⊂ (cid:39) Sheaves ∞ (cid:0) Charts , G Groupoids ∞ (cid:1) , (311) where G Groupoids ∞ is the ∞ -category of D-topological G-spaces (Def. B.4).Proof. Consider the following sequence of ∞ -functors G H ⊂ : = Sheaves ∞ (cid:0) G Orbits , H ⊂ (cid:1) = Sheaves ∞ (cid:0) G Orbits , Sheaves ∞ ( Charts ) (cid:1) (cid:39) ! Sheaves ∞ (cid:0) G Orbits × Charts (cid:1) (cid:39) ! Sheaves ∞ (cid:0) Charts , Sheaves ∞ ( G Orbits ) (cid:1) (cid:39) ! Sheaves ∞ (cid:0) Charts , G Groupoids (cid:1) . That the first and second of these ∞ -functors are equivalences follows by the product/hom-adjunction for ∞ -functors. With that, the last equivalence follows, objectwise, by Elmendorf’s theorem (Prop. B.10). (cid:3) Proposition 5.6 (G-equivariant homotopy theory embeds into G -singular cohesion) . Let H be a singular-cohesive ∞ -topos (Def. 3.48) over Groupoids ∞ (35) and let G ∈ Grp fin be a finite group (215) . (i) Then there is a full sub- ∞ -category inclusionG H ⊂ (cid:31) (cid:127) ∆ G (cid:39) (cid:47) (cid:47) H / ≺ G (312) of the G-equivariant non-singular cohesive ∞ -topos (Def. 5.3) into the slice of H (Prop. 2.46) over the genericG-orbi singularity (213) . (ii) This is such that, when pre-composed with the cohesive Elmendorf equivalence (Prop. 5.5), a cohesive sheaf(on
Charts ) of G
Groupoids (353) presented (356) by D-topological G-spaces X U (Def. B.1) is sent to the presheafon Singularities that is given as follows:
Sheaves ∞ (cid:0) Charts , G Groupoids ∞ (cid:1) (cid:39) (cid:47) (cid:47) G H ⊂ ∑ ≺ G ∆ G (cid:47) (cid:47) Sheaves ∞ (cid:0) Charts × Singularities (cid:1)(cid:0) U Shp G Top (cid:0) X U (cid:1)(cid:1) (cid:31) (cid:47) (cid:47) (cid:18)(cid:0) U , ≺ K (cid:1) Shp
Top (cid:18)(cid:18) (cid:70) φ ∈ Groups ( K , G ) X φ ( K ) U (cid:19) × G EG (cid:19)(cid:19) (313) where on the right we have the topological shape (36) of the Borel construction by the residual G-action on thefixed point subspaces X φ ( K ) U ⊂ X U (349) .Proof. For H ⊂ (cid:39) Groupoids ∞ this is [Re14, Prop. 3.5.1]; our expression Shp Top (cid:0) X φ ( K ) U × G EG (cid:1) is, up to conventionof notation, the expression for B Fun ( H , G (cid:121) X U ) that is spelled out in [Re14, p. 7][Lu19, 3.2.17] (using that our G is discrete). The generalization here follows immediately by applying this equivalence objectwise in the ∞ -siteCharts. (cid:3) The following is our key class of examples:
Example 5.7 (Cohesive shape of G -orbi-singular space is G -homotopy type) . In the cohesive ∞ -topos H ⊂ : = SmoothGroupoids ∞ (Example 3.18) consider a 0-truncated object X ∈ H ⊂ , equipped with a G -action (Def. 107)of a discrete group G , and with corresponding Cohesive G -orbispace (Prop. 4.6) X : = OrbSnglr ( X (cid:12) G ) in H : = SingularSmoothGroupoids ∞ (Example 3.56), which is either of: (i) a smooth G -orbifold: X ∈ SmoothManifolds (cid:31) (cid:127) (cid:47) (cid:47)
DiffeologicalSpaces (cid:31) (cid:127) (cid:47) (cid:47) H ⊂ (Example 4.10) (ii) a topological G -orbi space: X ∈ TopologicalSpaces
Cdfflg (cid:47) (cid:47)
DTopologicalSpaces (cid:31) (cid:127) (cid:47) (cid:47) H ⊂ (Example 4.11)94hen the cohesive shape (208) of the G -orbi-singular space X ∈ H is equivalent, under the identification of Prop.5.6, to the G -topological shape (356) of the underlying topological G -space of X : (i) By Prop. 4.12, comparing (250) with (313) we have: G SmoothManifolds form G -topological shape Shp G Top ( Dtplg ( − )) (cid:15) (cid:15) form Fr´echet-smooth orbifold OrbSnglr (( − ) (cid:12) G ) (cid:47) (cid:47) SingularSmoothGroupoids ∞ / ≺ G Shp formcohesive shape (cid:15) (cid:15) G Groupoids ∞ (cid:31) (cid:127) include G -equivariant homotopy theory ∆ G (cid:47) (cid:47) SingularGroupoids ∞ / ≺ G (314) (ii) By Prop. 4.13, comparing (251) with (313) we have: G TopologicalSpaces form G -topological shape Shp G Top (cid:15) (cid:15) form topological G -orbi space OrbSnglr ( Cdfflg ( − ) (cid:12) G ) (cid:47) (cid:47) SingularSmoothGroupoids ∞ / ≺ G Shp formcohesive shape (cid:15) (cid:15) G Groupoids ∞ (cid:31) (cid:127) include G -equivariant homotopy theory ∆ G (cid:47) (cid:47) SingularGroupoids ∞ / ≺ G (315) Lemma 5.8 ( ∆ G commutes with Disc) . The construction ∆ G from Prop. 5.6 commutes with embedding of discretecohesive structure (207) : Sheaves ∞ (cid:0) Singularities , Groupoids ∞ (cid:1) / ≺ G Disc (cid:45) (cid:45) G Groupoids
Disc (cid:45) (cid:45) ∆ G (cid:49) (cid:49) Sheaves ∞ (cid:0) Singularities , H ⊂ (cid:1) / ≺ G G H ⊂ ∆ G (cid:49) (cid:49) Theorem 5.9 (Cohomology of good orbispaces is proper equivariant cohomology) . Consider the singular-cohesive ∞ -topos H : = SingularCohesiveGroupoids ∞ (Example 3.56) and let G ∈ Groups fin be a discrete group (215) . Thenthe intrinsic cohomology (22) (i) of a G-orbi-singular space X ∈ H / ≺ G (Def. 4.4) which is either (a) a topological G-orbi-space (Example 4.11) with universal covering space (Def. 4.5) X G top ∈ G TopologicalSpaces(346) ; (b) a Fr´echet-smooth G-orbifold (Example 4.10) with universal covering space (Def. 4.5) X ∈ Fr´echetManifoldsand underlying G-topological space X G top : = Dtplg ( X ) (32) ; (ii) with coefficients in a cohesively discrete G- ∞ -groupoid A (353) (hence the G-topological shape (356) of sometopological G-space A G top ) regarded as a geometrically discrete orbi-singular ∞ -groupoid A via (312) :G TopologicalSpaces
Shp G Top (cid:47) (cid:47) G Groupoids ∞ Disc (cid:47) (cid:47) G H ⊂ ∆ G (cid:47) (cid:47) H / ≺ G A top (cid:47) (cid:47) A (cid:31) (cid:47) (cid:47) A equals the proper G-equivariant cohomology (Def. B.6) of X G top with coefficients in A: H / ≺ G (cid:0) X , A (cid:1) (cid:39) G Groupoids ∞ (cid:0) Shp G Top ( X G top ) , A (cid:1) hence: π n H / ≺ G (cid:0) X , A (cid:1) (cid:39) H − nG ( X G top , A ) intrinsicequivariant differential cohomologyin ∞ -topos ofsingular smooth ∞ -groupoids proper G -equivariant cohomology roof. (i) By Example 4.11 the topological G -orbi space X is given by X (cid:39) OrbSnglr (cid:0)
Cdfflg ( X ) (cid:12) G (cid:1) . With this, we compute as follows: H / ≺ G (cid:0) X , A (cid:1) = H / ≺ G (cid:16) OrbSnglr (cid:0)
Cdfflg ( X top ) (cid:12) G (cid:1) , ∆ G Disc ( A ) (cid:17) (cid:39) H / ≺ G (cid:16) OrbSnglr (cid:0)
Cdfflg ( X top ) (cid:12) G (cid:1) , Disc ( ∆ G A ) (cid:17) (cid:39) ( Groupoids ∞ ) / ≺ G (cid:16) Shp (cid:0)
OrbSnglr ( Cdfflg ( X top ) (cid:12) G ) (cid:1) , ∆ G A (cid:17) (cid:39) ( Groupoids ∞ ) / ≺ G (cid:0) ∆ G X , ∆ G A (cid:1) (cid:39) G Groupoids (cid:0)
Shp G Top ( X top ) , A (cid:1) . (316)Here the first step, after unwinding the definitions, is Lemma 5.8. The second step is the Shp (cid:97) Disc f c -adjunction(207). The third step is Prop. 4.13. The last step is Prop. 5.6 (ii)
By Example 4.10 the Fr´echet-smooth G -orbifold X is given by X (cid:39) OrbSnglr (cid:0) X (cid:12) G (cid:1) . With this, we compute just as in (316) only that now in the third step we use Prop. 4.12. (cid:3)
Example 5.10 (Orientifold cohomology) . Take the singular elastic ∞ -topos H = SingularJetsOfSmoothGroupoids ∞ (Example 3.56) and V = ( R n , +) ∈ H ⊂ (259). Then a X ⊂ ∈ V Folds ( H ⊂ ) (Def. 4.14) is an ordinary n -dimensionalorbifold or, more generally, an n -dimensional ´etale ∞ -stack (by Example 4.18) with structure group (Def. 4.24)the ordinary general linear group Aut ( T e V ) (cid:39) GL ( n ) (by Example 4.25). Hence, the composition of the delooping(101) of the ordinary determinant group homomorphism GL ( n ) det −! Z with the classifying map (cid:96) Frames ( X ⊂ ) (268) of the frame bundle of X (Def. 4.26) realizes X ⊂ as an object in the slice ∞ -topos (Prop. 2.46) over B Z .Consequently, it realizes its orbi-singualrization X : = ≺ X ⊂ ∈ H (3.52) as an object in the slice over ≺ Z (202): X ⊂ B det ◦ (cid:96) Frames ( X ⊂ ) (cid:15) (cid:15) B Z ∈ (cid:0) H ⊂ (cid:1) / B Z ⇔ X ≺ (cid:0) B det ◦ (cid:96) Frames ( X ⊂ ) (cid:1) (cid:15) (cid:15) ≺ Z ∈ (cid:0) H ⊂ (cid:1)(cid:14) ≺ Z . (317)This is the incarnation of the orbifold as an orbi-orientifold [DFM11][FSS15, 4.4][SS19]. In particular, if thecovering space (Def. 4.5) X : = fib (cid:0) B det ◦ (cid:96) Frames ( X ⊂ ) (cid:1) happens to be an R n -fold (Example 4.17), we have just a plain orientifold (without further orbifolding) and then theintrinsic cohomology (22) of X regarded in the slice over ≺ Z (317) is, by Theorem 5.9 the proper Z -equivariantcohomology of X , such as, for instance, Real K-theory [At66] (see [Mas11] for the perspective in proper equivariantcohomology) or Z -Equivariant Cohomotopy [tD79, 8.4][SS19]. We introduce general ´etale cohomology of ´etale ∞ -stacks (Def. 5.11), which is sensitive to geometric G -structureand to tangential structure (Def. 5.13). Promoting this to the proper incarnation of orbifolds (Remark 4.60),we finally obtain tangentially twisted proper orbifold cohomology (Def. 5.15) which we prove unifies tangen-tially twisted topological cohomology away from orbifold singularities with proper equivariant cohomology at thesingularities (Theorem 5.16). As a fundamental class of examples, we construct J-twisted proper orbifold Coho-motopy theories (Def. 5.28) and observe, as an application, that these subsume the relevant cohomology theoriesfor non-perturbative string theory, according to “Hypothesis H” (Remark 5.30).96 ohomology of V -´etale ∞ -stacks.Definition 5.11 ( ´Etale cohomology) . Let H be an elastic ∞ -topos (Def. 3.21), V ∈ Groups ( H ) (Prop. 2.74), ( G , φ ) ∈ Groups ( H ) / Aut ( TeV ) (Def. 4.35), and X ∈ Integrably ( G , φ ) Structured V Folds ( H ) (Def. 4.42). The ´etalecohomology of (cid:0) X , ( τ , g ) (cid:1) is its intrinsic cohomology (22) when regarded (via Prop. 4.46)Integrably ( G , φ ) Structured V Folds ( H ) (cid:47) (cid:47) (cid:16)(cid:0) H / B Aut ( T e V ) (cid:1) / ( B G , B φ ) (cid:17)(cid:14) (cid:16) H aef (cid:0) V , ( G , φ ) (cid:1) , ( τ H , g H ) (cid:17) (cid:0) X , ( τ , g ) (cid:1) (cid:18)(cid:0) X , ( τ , g ) (cid:1) (cid:96) ( τ , g ) −−−! met H aef (cid:0) V , ( G , φ ) (cid:1)(cid:19) in the iterated slice of (285) over the (cid:0) V , ( G , φ ) (cid:1) -Haefliger stack (Def. 4.45) equipped with its canonical ( G , φ ) -structure ( τ H , g H ) (Prop. 4.46), hence is G-structure-twisted cohomology (Remark 2.94): ´etale cohomology H ( τ , g ) (cid:0) X , A (cid:1) : = (cid:16)(cid:0) H / B Aut ( T e V ) (cid:1) / ( B G , B φ ) (cid:17)(cid:14) (cid:16) H aef (cid:0) V , ( G , φ ) (cid:1) , ( τ H , g H ) (cid:17) (cid:16)(cid:0) X , ( τ , g ) (cid:1) , (cid:0) A , p (cid:1)(cid:17) = X cocycle c (cid:47) (cid:47) (cid:96) ( τ , g ) (cid:36) (cid:36) A p (cid:122) (cid:122) H aef ( V , ( G , φ )) (cid:113) (cid:121) (318) Remark 5.12 ( ´Etale cohomology is geometric) . As the notation in Def. 5.11 indicates, ´etale cohomology is a“geometric cohomology theory” in that it does depend (in general) on the G -structure g on the V -fold X , (forinstance its complex- or symplectic- or Riemannian- or Lorentzian structure structure, by Example 4.44).Next we consider cohomology theories that are not sensitive to the metric part g of a G -structure ( τ , g ) , but justto its tangential structure τ . Definition 5.13 (Tangentially twisted cohomology) . Let H be an elastic ∞ -topos (Def. 3.21), V ∈ Groups ( H ) (Prop. 2.74), ( G , φ ) ∈ Groups ( H ) / Aut ( TeV ) (Def. 4.35), ( A , ρ ) ∈ G Actions ( H ) and X ∈ ( G , φ ) Structured V Folds ( H ) (286). Then, for A ∈ H / B S G , the tangentially twisted cohomology of V with coefficients in A is (see Remark 2.94) tangentially twistedcohomology H S τ (cid:0) X , A (cid:1) : = H / S Aut ( T e V ) (cid:0) ( X , η S ◦ τ ) , ( A (cid:12) G , ρ ) (cid:1) = X cocycle c (cid:47) (cid:47) η S ◦ τ (cid:32) (cid:32) ( S A ) (cid:12) ( S G ) S ρ (cid:121) (cid:121) B S G (cid:118) (cid:126) (319) Remark 5.14 (Need for G -Structure vs. tangential structure) .(i) The notion of tangentially twisted cohomology in Definition 5.13 make sense more generally for V -foldsequipped only with tangential structure (Def. 4.48) instead of full G -structure (Def. 4.36) (hence only with areduction of the shape of their structure group, instead of the actual structure group (Def. 4.24)) and it only need A to be equipped with a S G -action. (ii) We state the definition in the more restrictive form above just in order to bring out the following promotion ofthis notion to its proper orbifold version (Remark 4.60), in Def. 5.15 below. The process of orbi-singularizationis in fact sensitive to the full G -structure, and not just to its tangential shape. More precisely, it is sensitive to the geometric fixed point spaces of the G -structure and not just its homotopy fixed point spaces (as per Remark 3.68Example 3.71). 97 angentially twisted proper orbifold cohomology. We now promote tangentially twisted cohomology of V -folds(Def. 5.13) to a proper orbifold cohomology theory in the sense of Def. 4.58. Definition 5.15 (Tangentially twisted proper orbifold cohomology) . Let ◦ H be a singular-elastic ∞ -topos (Def. 3.51). ◦ V ∈ Groups ( H ⊂ ) (Prop. 2.74). ◦ ( G , φ ) ∈ Groups ( H ⊂ ) / Aut ( T e V ) (Prop. 2.74, Def. 2.86, Example 3.30). ◦ X ⊂ ∈ V Folds ( H ⊂ ) (Def. 4.14). ◦ ( τ , g ) ∈ ( G , φ ) Structures X ⊂ ( H ⊂ ) (Def. 4.36). ◦ ( A , ρ ) ∈ G Actions ( H ⊂ ) .and set A : = ≺ ( A (cid:12) G ) and X = ≺ X ⊂ .The tangentially twisted proper orbifold cohomology of X with coefficients in S A is (see Remark 2.94) H S ≺ τ (cid:0) X , A (cid:1) : = π H / S ≺ B G (cid:16) ( X , η S ◦ ≺ ( τ )) , ( S A , S ≺ ρ ) (cid:17) = X cocycle c (cid:47) (cid:47) η S ◦ ≺ ( τ ) (cid:34) (cid:34) S ≺ (cid:0) A (cid:12) G (cid:1) S ≺ ( ρ ) (cid:121) (cid:121) S ≺ B G (cid:14) ∼ Theorem 5.16 (Tangentially twisted orbifold cohomology at and away from singularities) . Consider the tangen-tially twisted orbifold cohomology of Def. 5.15 restriction to (1) smooth and (2) flat orbifolds, according toExample 4.62. Then (see the first diagram on p. 10): (i)
The tangentially twisted orbifold cohomology of flat orbifolds for 0-truncated coefficients A is naturally equiv-alent to the proper equivariant cohomology (Def. 5.2) of their (cid:91)
G-frame bundle (278) : tangentially twistedorbifold cohomology H S ≺ τ (cid:0) i (cid:91) X flatorbifold , A (cid:1) (cid:39) properequivariant cohomology H (cid:91) G (cid:0) ( (cid:91) G ) Frames ( X ⊂ ) (cid:91) G -frame bundle , A (cid:1) . (ii) The tangentially twisted orbifold cohomology of smooth (non-orbi-singular) orbifolds is equivalently the tan-gentially twisted cohomology (Def. 5.11) of the underlying V -folds: tangentially twistedorbifold cohomology H S ≺ τ (cid:0) i ⊂ X smoothorbifold , A (cid:1) (cid:39) tangentially twisted V -fold cohomology H S τ (cid:0) X ⊂ V -fold , A (cid:1) . Proof.
The case (i) means that the classifying map of the G -structure in question factors as follows, where we useProp. 4.37 to identify the leftmost morphism ρ as exhibiting the action (107) of (cid:91) G on ( (cid:91) G ) Frames ( X ) : ( (cid:91) G ) Frames ( X ) (cid:12) (cid:91) G ρ (cid:47) (cid:47) (cid:96) τ (cid:50) (cid:50) (cid:96) Frames ( X ) (cid:52) (cid:52) B (cid:91) G ε (cid:91) B G (cid:47) (cid:47) B G (cid:47) (cid:47) B Aut ( T e V ) . Now we observe (a) with Def. 3.48 that S acts objectwise over Singularities, (b) with Prop. 2.39 that the pullback of presheaves over Singularities is computed objectwise, (c) and with Lemma 4.7 that ≺ ( A (cid:12) G ) is objectwise over Singularities a homotopy quotient by G ,so that Lemma 3.6 applies objectwise over Singularities to give the pullback square shown on the right here:98 ≺ (cid:0) A (cid:12) (cid:91) G (cid:1) (pb) (cid:15) (cid:15) (cid:47) (cid:47) S ≺ (cid:0) A (cid:12) G (cid:1) (cid:15) (cid:15) ≺ (cid:0) ( (cid:91) G ) Frames ( X ) (cid:12) ( (cid:91) G ) (cid:1) (cid:51) (cid:51) ≺ ρ (cid:47) (cid:47) η S ◦ ≺ τ (cid:51) (cid:51) ≺ B (cid:91) G (cid:47) (cid:47) S ≺ B G By the universal property of the pullback, this means that every cocycle factors naturally as shown by the dashedmorphism. But by Def. 5.2 this dashed morphism is equivalently a cocycle in proper equivariant cohomology, asclaimed.The case (ii) means (using Lemma 3.65) that the orbi-singular space X is in fact smooth X (cid:39) ⊂ X (cid:39) NnOrbSnglr (cid:0) X ⊂ (cid:1) . Therefore, we have the following natural equivalences of spaces of dashed morphisms: S ≺ (cid:0) A (cid:12) G (cid:1) (cid:15) (cid:15) ⊂ X (cid:39) X (cid:58) (cid:58) (cid:47) (cid:47) S ≺ B G ⇔ ⊂ S ≺ (cid:0) A (cid:12) G (cid:1) (cid:15) (cid:15) ⊂ X (cid:57) (cid:57) (cid:47) (cid:47) ⊂ S ≺ B G ⇔ S (cid:0) A (cid:12) G (cid:1) (cid:15) (cid:15) X ⊂ (cid:59) (cid:59) (cid:47) (cid:47) S B G ∈ ∈ H (cid:14) S ≺ B G (cid:16) X , S ≺ (cid:0) A (cid:12) G (cid:1)(cid:17) (cid:39) H (cid:14) S B G (cid:16) X ⊂ , S (cid:0) A (cid:12) G (cid:1)(cid:17) (320)Here the first equivalence is by the adjunction NnOrbSnglr (cid:97) Smth and the fully faithfulness of NnOrbSnglr (207).The second step uses ⊂ ◦ S (cid:39) S ◦ ⊂ (Lemma 3.67) and ⊂ ◦ ≺ (cid:39) ⊂ (Remark 3.53) But on the right of (320) we seethe tangentially twisted cohomology of X ⊂ , as claimed. (cid:3) J-Twisted orbifold Cohomotopy theory.
We discuss now the example of tangentially twisted proper orbifoldcohomology (Def. 5.15) where the coefficients are (shapes of) spheres, specifically of
Tate V -spheres (Def. 5.19),In this case the tangential twist is the
J-homomorphism (Def. 5.24) whence we speak of
J-twisted Cohomotopytheory (Def. 5.28).
Definition 5.17 (Complement of neutral element) . Let H be an ∞ -topos (Def. 2.30) and V ∈ Groups ( H ) (Prop.2.74). Let ( V , ρ Aut
Grp ) ∈ Aut
Grp ( V ) Actions ( H ) denote the group-automorphism action on V (Prop. 2.102). (i) Consider those subobjects (Def. 2.61) of the homotopy quotient V (cid:12) Aut
Grp (125) whose pullback along themorphism ∗ (cid:12) Aut
Grp ( V ) e (cid:12) Aut
Grp ( V ) (cid:47) (cid:47) V (cid:12) Aut
Grp ( V ) , which exhibits the neutral element as a fixed point of the group-automorphism action (Prop. 2.102), is empty.These are the subobjects forming the poset in the top left of the following Cartesian square (of ∞ -categories):SubObjects e / (cid:0) V (cid:12) Aut
Grp ( V ) (cid:1) (pb) (cid:47) (cid:47) (cid:127) (cid:95) (cid:15) (cid:15) ∗ (cid:127) (cid:95) ∅ (cid:15) (cid:15) SubObjects (cid:0) V (cid:12) Aut
Grp ( V ) (cid:1) ( e (cid:12) Aut
Grp ( V )) ∗ (cid:47) (cid:47) SubObjects ( ∗ ) (321) (ii) Consider next the union of these subobjects, hence the colimit over the left vertical functor in (321), which wedenote as follows: (cid:0) V \ { e } (cid:1) (cid:12) Aut
Grp ( V ) : = lim −! (cid:16) SubObjects e / (cid:0) V (cid:12) Aut
Grp ( V ) (cid:1) (cid:44) −! SubObjects (cid:0) V (cid:12) Aut
Grp ( V ) (cid:1)(cid:17) . (322) (iii) We call the homotopy fiber V \ { e } of the canonical morphism from this object (322) to B Aut
Grp ( G ) the complement of the neutral element of V \ { e } fib (cid:0) ρ Aut
Grp \ { e } (cid:1) (cid:47) (cid:47) ( V \ { e } ) (cid:12) Aut
Grp ( V ) ρ Aut
Grp \ { e } (cid:120) (cid:120) (cid:127) (cid:95) (cid:15) (cid:15) V (cid:12) Aut
Grp ( V ) ρ Aut
Grp (cid:15) (cid:15) B Aut
Grp ( V ) (323) (iv) We regard the complement of the neutral element as equipped with the Aut
Grp ( V ) -action which is exhibitedby the homotopy fiber sequence (323) (by Prop. 323): (cid:0) V \ { e } , ρ Aut
Grp \ { E } (cid:1) ) ∈ Aut
Grp ( V ) Actions ( H ) . Proposition 5.18 (Basic properties of complement of neutral element) . Let H be an ∞ -topos (Def. 2.30) andV ∈ Groups ( H ) (Prop. 2.74). Then the complement V \ { e } of the neutral element (Def. 5.17) (i) is a subobject (Def. 2.61) of V V \ { e } (cid:31) (cid:127) (cid:47) (cid:47) V (324) (ii) which is disjoint from the neutral element: ∅ (cid:47) (cid:47) (cid:15) (cid:15) (pb) V \ { e } (cid:127) (cid:95) (cid:15) (cid:15) ∗ e (cid:47) (cid:47) VProof.
For (i) we use the pasting law (Prop. 2.23) and the homotopy fiber characterization of the group-automorphismaction (126) to decompose (323) as the pasting of two Cartesian squares, as follows: V \ { e } (cid:47) (cid:47) (cid:127) (cid:95) (cid:15) (cid:15) (pb) ( V \ { e } ) (cid:12) Aut
Grp ( V ) ρ AutGrp \{ e } (cid:121) (cid:121) (cid:127) (cid:95) (cid:15) (cid:15) V (cid:47) (cid:47) (cid:15) (cid:15) (pb) V (cid:12) Aut
Grp ( V ) ρ AutGrp (cid:15) (cid:15) ∗ (cid:47) (cid:47) B Aut
Grp ( V ) Since monomorphisms are preserved by pullback (by Prop. 2.66), this shows the first claim from the construction(322).For (ii) we paste to the middle horizontal morphism in this diagram the square (122) which exhibits the neutralelement as a fixed point of the group-automorphims action (Prop. 2.102) and then we pull back the right verticalmorphism along the boundary of that square, as shown in the following: ∅ (cid:15) (cid:15) (cid:36) (cid:36) (cid:47) (cid:47) ∅ (cid:39) lim −! i ∅ (cid:15) (cid:15) (cid:42) (cid:42) V \ { e } (cid:47) (cid:47) (cid:127) (cid:95) (cid:15) (cid:15) ( V \ { e } ) (cid:12) Aut
Grp ( V ) : = lim −! i U i (cid:127) (cid:95) (cid:15) (cid:15) ∗ e (cid:39) (cid:39) (cid:47) (cid:47) ∗ (cid:12) Aut
Grp ( V ) e (cid:12) Aut
Grp ( V ) (cid:43) (cid:43) V (cid:47) (cid:47) V (cid:12) Aut
Grp ( V ) Here the right square is Cartesian since colimits in an ∞ -topos are preserved by pullback (54) and using thedefinition (321), as indicated in the top right. Similarly the rear square is Cartesian, since pullback preserves theinitial object (this being the empty colimit, Example 2.33). With this, and since the front square is Cartesian by (i) ,the pasting law (Prop. 2.23) implies that also the left square is Cartesian, which was to be shown. (cid:3) efinition 5.19 (Tate V -sphere) . Let H be an ∞ -topos (Def. 2.30) and V ∈ Groups ( H ) (Prop. 2.74). Then we saythat the Tate V -sphere is the homotopy cofiber S V : = V / ( V \ { e } ) of the inclusion (324) of the complement of the neutral element into V (Def. 5.17), hence the object in thishomotopy pushout square: V \ { e } (cid:31) (cid:127) (cid:47) (cid:47) (cid:15) (cid:15) (po) V (cid:15) (cid:15) ∗ (cid:47) (cid:47) S V (325) Example 5.20 (Tate sphere in unstable motivic homotopy theory) . For H : = Sheaves ∞ (cid:0) Schemes
Nis (cid:1) and V : = A the Tate V -sphere of Def. 5.19 is the Tate sphere in the traditional sense of (unstable) motivic homotopy theory,see [VRO07, 2.22]. Example 5.21 (Tate spheres with shape of ordinary spheres) . Let H = JetsOfSmoothGroupoids ∞ (Def. 3.24) and V : = ( R n , +) as in Example 4.17. Then Aut Grp ( R n , +) = GL ( n ) (as in Example 4.25) and the complement of theneutral element (Def. 5.17) is the ordinary complement R n \ { } , whose shape is that or the ordinary n − S (cid:0) R n \ { } (cid:1) (cid:39) S S n − . (326)Hence the Tate R n -sphere (Def. 5.19) is the homotopy pushout shown on the left here: R n \ { e } (cid:15) (cid:15) (cid:31) (cid:127) (cid:47) (cid:47) (pb) R n (cid:15) (cid:15) ∗ (cid:47) (cid:47) S ( R n ) S S S n − (cid:15) (cid:15) (cid:31) (cid:127) (cid:47) (cid:47) (pb) ∗ (cid:15) (cid:15) ∗ (cid:47) (cid:47) S S ( R n ) Since the shape modality (128) is left adjoint it preserves homotopy pushouts (Prop. 2.26), so that the shape of theTate R n -sphere is that of the ordinary n -sphere: S S R n (cid:39) S S n . (327)In contrast, the Tate R n -sphere itself is the “germ of a smooth sphere”. Proposition 5.22 (Canonical action on Tate V -sphere) . Let H be an ∞ -topos (Def. 2.30) and V ∈ Groups ( H ) (Prop.2.74). The Tate V -sphere (Def. 5.19) inherits a canonical action (Prop. 2.79) of the group-automorphism group Aut
Grp ( V ) (Def. 2.101), associated (via Prop. 2.87) to a group homomorphism Aut
Grp ( V ) (cid:47) (cid:47) Aut ( S V ) (328) whose homotopy quotient (108) is given by the following homotopy pushout ( V \ { e } ) (cid:12) Aut
Grp ( V ) (cid:31) (cid:127) (cid:47) (cid:47) (cid:15) (cid:15) (po) V (cid:12) Aut
Grp ( V ) (cid:15) (cid:15) ∗ (cid:12) Aut
Grp ( V ) (cid:47) (cid:47) S V (cid:12) Aut
Grp ( V ) (329) of the defining morphisms in (323) .Proof. Since the forgetful ∞ -functor H / B Aut
Grp ( V ) −! H preserbes colimits (Example 2.52), the diagram (329)extends to a diagram over B Aut
Grp ( V ) . Pulling this back along the point inclusion (103) and using that colimitsin an ∞ -topos are preserved by pullback (54), we find that the homotopy fiber of S V (cid:12) Aut
Grp ( V ) ! B Aut
Grp ( V ) isgiven by the defining homotopy pushout (329) of the Tate V -sphere. (cid:3) efinition 5.23 (Linear group) . Let H be an elastic ∞ -topos (Def. 3.21) and V ∈ Groups ( H ) (Prop. 2.74). (i) We say that V is a linear group if it is equipped with an equivalenceAut ( T e V ) (cid:39) exp (cid:47) (cid:47) Aut
Grp ( V ) ∈ Groups ( H ) (330)between (a) the plain automorphism group of the local neighborhood of the neutral element (Def. 4.24) and (b) the group-automorphism group of V (Def. 2.101) (ii) We write LinearGroups ( H ) ∈ Categories ∞ for the ∞ -category of linear groups in H . Definition 5.24 (Tate J-homomorphism) . Let H be an elastic ∞ -topos (Def. 3.21) and V ∈ LinearGroups ( H ) (Prop.5.23). (i) The
Tate J-homomorphism is the composite J V : Aut ( T e V ) (cid:39) exp (cid:47) (cid:47) Aut
Grp ( V ) (cid:47) (cid:47) Aut ( S V ) (331)of (a) the defining equivalence (330) with (b) the homomorphism (328) which reflects the canonical Aut Grp ( V ) -action on the Tate V -sphere (Def. 5.22). (ii) The corresponding Aut ( T e V ) -actions on S V and on S ( S V ) , by restriction along (331) and (333) of the canonicalautomorphism actions (Prop. 2.87), we denote, respectively, by ( S V , ρ J ) ∈ Aut ( T e V ) Actions ( H ) . (332) (iii) The actual
J-homomorphism is the shape of the further composite with the homomorphism Aut ( η S S V ) fromProp. 131: J V : S Aut ( T e V ) (cid:39) S exp (cid:47) (cid:47) S J V (cid:53) (cid:53) S Aut
Grp ( V ) (cid:47) (cid:47) S Aut ( S V ) S Aut (cid:0) η S Aut ( S V ) (cid:1) (cid:47) (cid:47) S Aut (cid:0) S S V (cid:1) . (333) Example 5.25 (Ordinary J-homomorphism) . Let H = SingularJetsOfSmoothGroupoids ∞ (Example 3.56) and V : =( R n , +) as in Example 4.17. This is a linear group in the sense of Def. 5.23, with Aut ( T R n ) (cid:39) GL ( n ) (Example4.25). Via Example 5.21 the induced action on the shape of the Tate R n -sphere (Def. 5.24) is the classical J-homomorphism (going back to [Wh42], reviewed in [Rav86, p. 4]): J : S O ( n ) (cid:39) S GL ( n ) (cid:47) (cid:47) Aut (cid:0) S S n (cid:1) (334)being the image under topological shape (Def. 36) of the defining action of GL ( n ) on R n and hence on its one-pointcompactification S n . Definition 5.26 (Representation spheres) . Let H be a singular-elastic ∞ -topos (Def. 3.51), V Groups ( H ⊂ ) (Prop.2.74), and ( G , φ ) ∈ Groups ( H ⊂ ) / Aut ( T e V ) (Prop. 2.74, Def. 2.86, Example 3.30). Then we say that the represen-tation sphere S V φ of the G -action φ on V (via Prop. 2.87) is the shape (Def. 3.1) of the orbi-singularization (Def.3.52) of the homotopy quotient (108) of the Tate V -sphere (Def. 5.19) by the restricted action (Prop. 2.85) along φ of the action ρ J (332) induced by the J-homomorphism (Def. 5.24): S V φ : = S ≺ (cid:0) S V (cid:12) φ G (cid:1) ∈ H / ≺ G . Example 5.27 (Ordinary representation spheres) . Let H = SingularJetsOfSmoothGroupoids ∞ (Example 3.55) and V : = ( R n , +) as in Example 4.17, whence Aut ( T e V ) (cid:39) GL ( n ) (Example 4.25). For G (cid:31) (cid:127) φ (cid:47) (cid:47) GL ( n ) ⊂ Aut ( T e V ) a finite subgroup, hence a linear G -representation, we have that the representation sphere S R n φ according to Def.5.26 is the ordinary representation sphere, as an object in G -equivariant homotopy theory.102 efinition 5.28 ( J -twisted proper orbifold Cohomotopy theory) . Let H be a singular-elastic ∞ -topos (Def. 3.51) V ∈ Groups ( H ) (Prop. 2.74), W ∈ LinearGroups ( H ) (Def. 5.23) and φ : Aut ( T e W ) (cid:47) (cid:47) Aut ( T E V ) . Then
J-twistedproper orbifold Cohomotopy is the tangentially twisted proper orbifold cohomotopy (Def. 5.15) with coefficients ( A , ρ ) : = ( S V , ρ J ) the Tate W -sphere (Def. 5.19) with its Tate J-homomorphism action (Def. 5.24): J-twistedorbifold Cohomotopy π S ≺ τ ( − ) : = tangentially twisted orbifold cohomology H S ≺ τ (cid:0) − , ( S V , ρ J ) Tate V -sphere with J -homomorphism action (cid:1) . Hence for a structured orbifold (Def. 4.58) (cid:0) X , ( τ , g ) (cid:1) ∈ (cid:0) Aut ( T e W ) , φ (cid:1) Structured V Orbifolds ( H ) , we have: J-twistedorbifold Cohomotopy π S ≺ τ ( X ) = orbifold X cocycle c (cid:47) (cid:47) η S ◦ ≺ ( τ ) tangentialtwist (cid:38) (cid:38) S ≺ (cid:0) orbi-singularizedTate W -sphere S W (cid:12) Aut ( T e W ) (cid:1) S ≺ (cid:0) ρ J (cid:1) twisting viaorbi-singularizedJ-homomorphism (cid:117) (cid:117) S ≺ B Aut ( T e W ) (cid:113) (cid:121) (cid:14) ∼ Example 5.29 ( J -Twisted proper orbifold Cohomotopy of ordinary orbifolds) . Let H = SingularJetsOfSmoothGroupoids ∞ (Example 3.55) and V : = ( R n , +) , W : = ( R p , +) as in Example 4.17,with p ≤ n , and φ : ( R p , +) (cid:44) ! ( R n , +) be the canonical inclusion. Then the corresponding J-twisted properorbifold Cohomotopy theory π S ≺ τ (Def. 5.28) is defined on ordinary n -dimensional orbifolds (by Example 4.18)with GL ( p ) -structure (by Example 4.25) and it unifies the following two special cases (by Theorem 5.16, see thesecond diagram on p. 10)): (i) On smooth orbifolds, i.e., on ordinary manifolds (Example 4.17) it reduces to non-abelian cohomology withcoefficients the shape of the ordinary p -sphere (by Example 5.21) and tangentially twisted via the traditional J-homomorphism (by Example 334). This is the J-twisted Cohomotopy theory considered in [FSS19b][FSS19c][BSS19]. (ii)
On flat orbifolds, such as the vicinity of ordinary orbifold singularities R p (cid:12) G for finite subgroups G φ (cid:44) ! GL ( p ) (by Example 4.57), hence for linear G -representations φ , it reduces to proper equivariant cohomology in RO-degree φ and with coefficients the representation sphere S R n φ (by Example 5.27). This is the tangentially RO-gradedequivariant Cohomotopy theory considered in [SS19][BSS19].By way of conclusion and outlook, we highlight the following: Remark 5.30 (Orbifold cohomology in non-perturbative string theory and
Hypothesis H ) . Traditional discus-sion of orbifold cohomology has been strongly motivated by its application to perturbative string theory (e.g.[AMR02][ARZ06][ALR07][BU09][DFM11]). However, perturbative string theory is famously in need of a non-perturbative completion (“M-theory”, see [HSS18, 2][FSS19a] for review and pointers) whose mathematical for-mulation has remained an open problem. Therefore, it is to be expected that the historically rich interaction betweenorbifold cohomology theory and string theory is just the tip of an iceberg, whose full scope is a cohomology theoryof M-theoretic orbifolds.Elsewhere we have put forward a precise hypothesis as to what this mathematical theory should be. This
Hy-pothesis H says that: (i) far from singularities, M-theory is controlled by twisted Cohomotopy theory [FSS19b][FSS19c][BSS19][FSS20a]; (ii) at singularities, M-theory is controlled by RO-graded equivariant Cohomotopy theory [HSS18][SS19][BSS19].(See these references for various consistency checks of this hypothesis.)The impact of Theorem 5.16, in its specialization to Example 5.29, is to show that these two cases are indeedtwo aspects of a single unified cohomology theory: J-twisted proper orbifold Cohomotopy theory.103
Model category presentations
We recall some basics of model categories (e.g. [GJ99, 2]) of simplicial presheaves ([Ja87][Ja96][Ja15]) as pre-sentations of ∞ -toposes ([Lu09a, A.2, A.3]). Model categories of simplicial presheaves.Definition A.1 (Model category of simplicial presheaves) . Let C be a site. We write (i) sPSh ( C ) loc ∈ HomotopicalCategories (335)for the category of simplicial presheaves on C , regarded as a homotopical category with weak equivalences thelocal weak homotopy equivalences of simplicial sets. (ii) sPSh ( C ) inj / proj , loc ∈ ModelCategories (336)for the same category regarded as either the corresponding injective or projective model category. (iii) sPSh (cid:96) (cid:47) (cid:47) L lwhe sPSh loc = : H (337)for the corresponding simplicial localization. Lemma A.2 (Cofibrancy in projective model structure [Du01, Cor. 9.4]) . Let C be a site. For a simplicial presheafX • ∈ sPSh ( C ) proj , loc in the projective model structure (336) to be cofibrant it is sufficient that X • is degreewise (i) a coproduct of representables, such that (ii) the degenerate cells split off as a direct summand. Lemma A.3 (Simplicial presheaf represents its own hocolim [DHI04, 2.1][Sc13, 2.3.21]) . Let C be a site andX • ∈ sPSh ( C ) a simplicial presheaf (Def. A.1). Then its image under simplicial localization (337) is equivalentlythe simplicial homotopy colimit over the images of its component presheaves: (cid:96) ( X • ) (cid:39) lim −! ( (cid:96) X ) • ∈ H . Topological mapping stacksExample A.4 (Model category presentation of smooth ∞ -groupoids) . Let C = CartesianSpaces (Def. 2.5). Thenthe simplicial localization (337) of sPSh ( C ) loc (336) is SmoothGroupoids ∞ (Example 3.18): L lwhe sPSh ( CartesianSpaces ) loc (cid:39) SmoothGroupoids ∞ . Lemma A.5 (Mapping stack from delooping of discrete group to topological stack) . In SmoothGroupoids ∞ (Ex-ample 3.18) consider (i) a finite group embedded via (215) G ∈ Groups
Disc (cid:47) (cid:47)
Groups (cid:0)
SmoothGroupoids ∞ (cid:1) , (338) (ii) a topological groupoid, embedded via (148)TopologicalGroupoids Cdfflg (cid:47) (cid:47)
SmoothGroupoids ∞ X top X ⊂ (339) Then the mapping stack (56) formed in
SmoothGroupoids ∞ is the degreewise image under Cdfflg (32) of thetopological groupoid representing the mapping stack of topological groupoids (which exists by [No10] since G isfinite, hence compact):
Maps (cid:0) B G , X ⊂ (cid:1) (cid:39) Cdfflg
Maps (cid:0) B G , X top (cid:1) . (340)104 roof. Since (by Example 3.18)SmoothGroupoids ∞ (cid:39) Sheaves ∞ ( CartesianSpaces ) (cid:111) (cid:111) L (cid:31) (cid:127) ⊥ (cid:47) (cid:47) PreSheaves ∞ ( CartesianSpaces ) it is sufficient to show that we have an equivalence of ∞ -presheaves of the form R n (cid:31) (cid:47) (cid:47) PreSheaves ∞ (cid:0) CartesianSpaces (cid:1) ( R n × B G , X ⊂ ) (cid:39) PreSheaves ∞ (cid:0) CartesianSpaces (cid:1) ( R n , Cdfflg
Maps ( B G , X top )) (341)By Example A.4, we may model this in the global projective model structure on simplicial presheaves overCartesianSpaces:sPSh ( CartesianSpaces ) (cid:96) (cid:47) (cid:47) L lwhe sPSh ( CartesianSpaces ) proj (cid:39) PreSheaves ∞ ( CartesianSpaces ) (342)with, by Lemma A.3, the following models: (a) A model under (cid:96) (342) of the Cartesian product R n × B G with the delooping B G (cid:39) lim −! G × • (104), is given bythe simplicial presheaf R n × G × • ∈ sPSh ( CartesianSpaces ) proj . (343) (b) A model under (cid:96) (342) for the image (339) of a topological groupoid X top is given by its nerve regarded as asimplicial presheaf, componentwise via (147) N • ( X top ) ∈ sPSh ( CartesianSpaces ) proj . (344)Moreover:• The object (343) is projectively cofibrant, by Lemma A.2, as is its Cartesian product with an k simplex ∆ [ k ] .• The object (344) is projectively fibrant (objectwise a Kan complex) by the groupoid property of X top .Therefore to get (341) it is, in turn, sufficient to exhibit for R n ∈ CartesianSpaces a natural isomorphism of simpli-cial sets of the form (cid:90) [ k ] ∈ ∆ PSh (cid:0) R n × (cid:0) G × k × ∆ ( k , • ) (cid:1) , Cdfflg ( N k ( X top )) (cid:1) (cid:39) PSh (cid:18) R n , Cdfflg (cid:18) (cid:90) [ k ] ∈ ∆ N k ( X top ) ( G × k × ∆ ( k , • )) (cid:19)(cid:19) , (345)where the end (cid:82) [ k ] ∈ ∆ ( − ) expresses the limit that computes the morphism of simplicial sets as a subset of the productof the function spaces of components. We obtain this as the following composite of natural isomorphisms: (cid:90) [ k ] ∈ ∆ PSh (cid:16) R n × (cid:0) G × k × ∆ ( k , • ) (cid:1) , Cdfflg (cid:0) N k ( X top ) (cid:1)(cid:17) (cid:39) (cid:90) [ k ] ∈ ∆ PSh (cid:16) R n , (cid:0) Cdfflg ( N k ( X top )) (cid:1) ( G × k × ∆ ( k , • )) (cid:17) (cid:39) (cid:90) [ k ] ∈ ∆ PSh (cid:16) R n , Cdfflg (cid:16)(cid:0) N k ( X top ) (cid:1) ( G × k × ∆ ( k , • )) (cid:17)(cid:17) (cid:39) PSh (cid:18) R n , (cid:90) [ k ] ∈ ∆ Cdfflg (cid:16)(cid:0) N k ( X top ) (cid:1) ( G × k × ∆ ( k , • )) (cid:17)(cid:19) (cid:39) PSh (cid:18) R n , Cdfflg (cid:18) (cid:90) [ k ] ∈ ∆ (cid:0) N k ( X top ) (cid:1) ( G × k × ∆ ( k , • )) (cid:19)(cid:19) . Here the first step is the definition of function spaces ( − ) ( − ) , the second step uses that Cdfflg, being a right adjoint,preserves products (Prop. 2.26). The third step uses that the Hom-functor preserves limits (hence ends) in itssecond argument, while the fourth step uses that Cdfflg, being a right adjoint, preserves limits (hence ends), againby Prop. 2.26. (cid:3) Equivariant homotopy theory
For reference, we recall some basics of unstable equivariant homotopy theory (see [May96][Blu17]). We focushere on finite groups, for simplicity and since this is what we need in the main text (Remark 3.64), but all statementsin the following, notably Elmendorf’s theorem (Prop. B.10 below) generalize to compact Lie groups.
Definition B.1 (Topological G -spaces) . Let G ∈ Groups fin be a finite group. (i) We write G DTopologicalSpaces (cid:31) (cid:127) (cid:47) (cid:47) G TopologicalSpaces ∈ Categories (346)for the categories whose objects G (cid:121) X : = (cid:0) X , G × X ρ −! X (cid:1) (347)are topological spaces X (as in Def. 2.2) or specifically D-topological spaces (as in Def. 2.2), respectively, equippedwith continuous left G -actions ρ , and whose morphisms are the G -equivariant continuous functions: G TopologicalSpaces (cid:0) G (cid:121) X , G (cid:121) X (cid:1) : = X continuous f (cid:47) (cid:47) X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G × X f (cid:15) (cid:15) ρ (cid:47) (cid:47) X f (cid:15) (cid:15) G × X ρ (cid:47) (cid:47) X . (348) (ii) For G (cid:121) X a (D-)topological G -space and H ι (cid:44) ! G a subgroup, we write X H : = (cid:110) x ∈ X | ∀ h ∈ H ⊂ G ρ ( h , x ) = x (cid:111) (349)for the topological subspace of H -fixed points (which, if X is D-topological, is itself again D-topological, by Prop.2.4). (iii) For G (cid:121) X and G (cid:121) X two (D-)topological G -spaces, the mapping space (28) between their underlying (D-)topological spaces canonically becomes a G -space via the conjugation action and the corresponding fixed pointspace (349) Maps ( X , X ) G (cid:31) (cid:127) (cid:47) (cid:47) Maps ( X , X ) (350)is the subspace on the G -equivariant functions (348). Example B.2 ( G -cells) . For G ∈ Grp fin , H ⊂ G a subgroup and n ∈ N we have the G -spaces (Def. B.1) (cid:0) G / H (cid:1) × D n , (cid:0) G / H (cid:1) × S n − ∈ G DTopologicalSpacesbeing the product spaces of the discrete orbit spaces with the standard topological unit disk and unit circle, re-spectively, the latter equipped with the trivial G -action. The boundary inclusions ∂ D n = S n − ι n (cid:44) ! D n induce G -equivariant maps ι n , H : (cid:0) G / H (cid:1) × S n − (cid:31) (cid:127) ( id , ι n ) (cid:47) (cid:47) (cid:0) G / H (cid:1) × D n (351)for all n ∈ N , H ⊂ G . Definition B.3 ( G -CW-complexes) .(i) A G-CW-complex X is a D-topological G -space (Def. B.1) which is equipped with the realization as a colimit X (cid:39) lim −! n X n ∈ G DTopologicalSpacesover a sequence X − −! X −! X −! X −! · · · ∈ G DTopologicalSpaceswhere X − = ∅ and where each X n ! X n − is given by a set of attachments of G -cells along (351), hence by apushout of the form: 106 H ⊂ Gi ∈ In G / H × S n − (cid:127) (cid:95) ( ι n , H ) n , H (cid:15) (cid:15) (cid:47) (cid:47) (po) X n − (cid:127) (cid:95) (cid:15) (cid:15) ∏ H ⊂ Gi ∈ In G / H × D n (cid:47) (cid:47) X n (ii) We write G Sets (cid:31) (cid:127) (cid:47) (cid:47) G CWComplexes (cid:31) (cid:127) (cid:47) (cid:47) G DTopologicalSpaces (352)for the full subcategories on those D-topological G -spaces which admit the structure of G -CW-complexes. Definition B.4 (Homotopy theory of D-topological G -spaces) . The homotopy theory of topological G-spaces isthe ∞ -category G Groupoids ∞ ∈ Categories ∞ (353)which has the same objects as G CWComplexes (Def. B.3), and with ∞ -groupoids the topological shapes (Def. 36)of the mapping spaces (350) of G -equivariant maps: G Groupoids ∞ (cid:0) G (cid:121) X , G (cid:121) X (cid:1) : = Shp
Top (cid:16)
Maps (cid:0) X , X (cid:1) G (cid:17) . (354) Definition B.5 (Shape of G -topological spaces) .(i) We write Shp G Top : G CWComplexes (cid:47) (cid:47) G Groupoids ∞ (355)for the canonical ∞ -functor (topologically enriched functor) from the 1-category of G -CW-complexes (Def. B.3)to the ∞ -category of G - ∞ -groupoids (Def. B.4), which is the identity on objects and which on Hom-spaces is thecontinuous map given by the identity fuction from the discrete set of G -equivariant maps (348) to the topologicalspace of G -equivariant maps (354). (ii) For any choice of G -CW-approximation functor G TopologicalSpaces ( − ) cof (cid:47) (cid:47) G CWComplexwe get the corresponding shape functor on all of G TopologicalSpaces (Def. B.1) and hence on G DTopologicalspaces,which we denote by the same symbol:Shp G Top : G TopologicalSpaces ( − ) cof (cid:47) (cid:47) G CWComplexes
Shp G top (cid:47) (cid:47) G Groupoids ∞ . (356) Definition B.6 (Proper G -equivariant generalized cohomology of topological G -spaces) . For G ∈ Groups fin , wesay that the proper G-equivariant cohomology of a topological G -space (Def. B.1) X ∈ G TopologicalSpaces withcoefficients in a (pointed) G - ∞ -groupoid (Def. B.4), A ∈ G Groupoids ∞ , is H − nG ( X , A ) : = π n (cid:16) G Groupoids (cid:0)
Shp G Top ( X ) , A (cid:1)(cid:17) , where on the right we have the n th homotopy group (at the given basepoint) of the hom- ∞ -groupoid (354) from the G -topological shape of X (356) to A . Elmendorf’s theorem.Definition B.7 (Orbit of action of a finite group) . Let G be a finite group. If G (cid:121) S is a set equipped with an actionby G , then an orbit of G in S is a subset of points { g ( s ) | g ∈ G } ⊂ S obtained from any single point s ∈ S by actingon it with all elements of G . Definition B.8 (Orbit category of a finite group) . The category of G-orbits or orbit category of GG Orbits (cid:44) −! G Sets ∈ Categoriesis the category whose objects correspond to subgroup inclusions H ι (cid:44) −! G and whose morphisms are G -equivariantfunctions, hence morphisms of G -sets (352), between the corresponding coset spaces G / H −! G / H .107 xample B.9 (Systems of fixed point spaces) . Consider a topological space equipped with a G -action G (cid:121) X ∈ G DTopologicalSpaces (Def. B.1) and H ⊂ G a subgroup. Then a G -equivariant function G / H f −! X from thecorresponding G -orbit (Def. B.8) is determined by its image f (cid:0) [ e ] (cid:1) ∈ X of the class of the neutral element, and thatimage has to be fixed by the action of H ⊂ G of X . Therefore, the corresponding G -equivariant mapping spaces(350) Maps (cid:0) G / H , X (cid:1) G (cid:39) X H : = (cid:26) x ∈ X | ∀ h ∈ H ⊂ G ( h ( x ) = x ) (cid:27) ⊂ X are the topological subspaces of H -fixed points inside X (349). By functoriality of the mapping space construction,these fixed point spaces are exhibited as arranging into a topological presheaf on the G -orbit category (Def. B.8): X ( − ) : G Orbits op Maps ( − , X ) G (cid:47) (cid:47) TopologicalSpaces
Proposition B.10 (Elmendorf’s theorem [El83], see [Blu17, Thm. 1.3.6 and 1.3.8]) . Let G be a finite group. Thefunctor which sends a G-space G (cid:121)
X (Def. B.1) to its system of H-fixed point spaces (Example B.9) constitutes anequivalence of ∞ -categories G Groupoids ∞ (cid:39) (cid:47) (cid:47) Sheaves ∞ (cid:0) G Orbits (cid:1) G (cid:121) X (cid:31) (cid:47) (cid:47) X ( − ) = Maps (cid:0) − , X (cid:1) G (357) Acknowledgements.
The authors would like to thank Vincent Braunack-Mayer, David Carchedi, Dmitri Pavlov,Charles Rezk, and David Roberts for useful discussions.Hisham Sati,
Mathematics, Division of Science, New York University Abu Dhabi, UAE.
Urs Schreiber,
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