aa r X i v : . [ m a t h . A T ] O c t Hamburger Beiträge zur Mathematik Nr. 858ZMP–HH/20-14
Principal ∞ -Bundles and Smooth String Group Models Severin Bunk
Abstract
We provide a general, homotopy-theoretic definition of string group models within an ∞ -categoryof smooth spaces, and we present new smooth models for the string group. Here, a smooth spaceis a presheaf of ∞ -groupoids on the category of cartesian spaces. The key to our definition andconstruction of smooth string group models is a version of the singular complex functor, whichassigns to a smooth space an underlying ordinary space. We provide new characterisations ofprincipal ∞ -bundles and group extensions in ∞ -topoi, building on work of Nikolaus, Schreiber,and Stevenson. These insights allow us to transfer the definition of string group extensions fromthe ∞ -category of spaces to the ∞ -category of smooth spaces. Finally, we consider smooth higher-categorical group extensions that arise as obstructions to the existence of equivariant structureson gerbes. These extensions give rise to new smooth models for the string group, as recentlyconjectured in joint work with Müller and Szabo. Contents
1. Introduction and overview
2. Smooth spaces and ∞ -topoi ∞ -topoi 8
3. Principal ∞ -bundles and group extensions in ∞ -topoi ∞ -categories 153.3 Principal ∞ -bundles 19
4. Homotopy-theoretic smooth string group models
A. Actions and category objects References The most direct way to define the string group is via the Whitehead tower of O ( n ) , · · · −→ String( n ) −→ Spin( n ) −→ SO ( n ) −→ O ( n ) . (1.1)By this approach, String( n ) is defined as a 3-connected topological space with a continuous map String( n ) → Spin( n ) which induces an isomorphism on all homotopy groups except for in degree1hree. So far, this defines String( n ) only as a space, but in [Sto96] Stolz constructed String( n ) as atopological group and the map String( n ) → Spin( n ) as a morphism of topological groups. In fact,he presented a construction that produces, for any compact, connected, and simply connected Liegroup H , a morphism String( H ) → H of topological groups whose underlying continuous map is athree-connected covering. A covering of this type is also called a string group extension of H . In theseconventions, we write String( n ) := String(Spin( n )) .The string group is important in geometry and topology in several ways. Originally, Killing-back [Kil87] and Witten [Wit] investigated the two-dimensional supersymmetric σ -model on back-ground manifolds M and found that this is well-defined only if the free loop space LM admits a spinstructure. Witten, moreover, computed the index of a hypothetical Dirac operator on LM based onphysical arguments, leading to the definition of the Witten genus. By now, it has been understood thatthe Witten genus is related to the cohomology theory of topological modular forms (TMF). The stringgroup enters in this story, for example by defining orientations in TMF [AHR, DHH11], analogouslyto how the spin group underlies orientations in real K-theory.Since the free loop space LM is less tractable than the manifold M itself, it is an importantquestion whether the condition that LM admit a spin structure can be recast as a condition on themanifold M itself. This is indeed the case: spin structures on LM correspond to string structureson M [ST, ST04, Wal15]. Topologically, a string structure on M is a lift of the classifying map M → B O ( n ) of the tangent bundle T M → M to a map M → BString( n ) . That is, a string structureis a reduction of the structure group of T M to String( n ) . From a geometric perspective, the interestultimately is in identifying consequences and constructions that are facilitated by a string structureon a manifold. Concrete examples include the Höhn-Stolz conjecture [Höh, Sto96] that the Wittengenus is trivial for any Riemannian k -manifold with positive Ricci curvature which admits a stringstructure, or the long-standing goal to define a Dirac operator on the loop space LM .In order to study the differential geometric, rather than topological, implications of string struc-tures, it is paramount to have models for String( n ) not just as a topological group, but as a groupobject in some geometric category. For instance, given a Riemannian manifold M , the construction ofthe Dirac operator associated with a spin structure on M depends on the ability to glue the tangentbundle T M from smooth
Spin( n ) -valued functions. Technically, one also needs to find local framesfor T M in which the Levi-Civita connection of M is represented by 1-forms valued in the Lie algebra spin ( n ) rather than so ( n ) ; however, since the fibre of the map Spin( n ) → SO ( n ) is discrete, these Liealgebras happen to be canonically isomorphic (for more background on spin geometry and Dirac oper-ators, we recommend [LM89]). Analogously to how spin structures on LM stem from string structureson M , a hypothetical Dirac operator on LM may well stem from a geometric operator on M itself(e.g. via some transgression procedure), obtained from a further lift of the Levi-Civita connection tothe Lie algebra string ( n ) . However, for this to make sense, one must work with a smooth , rather thantopological, model for String( n ) .Classical results on cohomology readily imply that it is impossible to construct String( H ) as a finite-dimensional Lie group (for any compact, connected, simply connected Lie group H ). Thus, to findgeometric models for String( H ) , one needs to look beyond the category of smooth, finite-dimensionalmanifolds. Indeed, a number of models for String( H ) have been found in (higher) categories of smoothspaces that generalise the notion of a manifold in various ways [BSCS07, Hen08, SP11, Wal12, NSW13,FRS16]. 2n each of these constructions, an extension A −→ String( H ) −→ H of a compact, simple, simply connected Lie group H is constructed within the chosen ambient categoryof smooth spaces. It is then argued that on the underlying ordinary spaces (meaning topological spacesor simplicial sets) one obtains a string group extension in the sense of (1.1). However, so far there is nogeneral definition of String( H ) in a smooth context that formalises this procedure. Consequently, ingeometric models for String( H ) the extending group A currently has to be chosen ad hoc as an explicitdelooping of the Lie group U (1) in a rather strict sense. This obscures the homotopy-theoretic natureof String( H ) , since from a homotopical point of view, not A is fixed, but only its homotopy type.In [BMS], studying symmetries of gerbes, we came across extensions of Lie groups H not by adelooping of the Lie group U (1) , but by the delooping of the diffeological group U (1) H of smoothmaps from H to U (1) . However, if H is simply connected, then the smooth group U (1) H is homotopyequivalent to U (1) . Therefore, extensions of H by the delooping B( U (1) H ) potentially have the correcthomotopy type to produce smooth string group extensions of H . Nevertheless, we could not make thisrigorous due to the lack of a homotopy-theoretic notion of smooth string group extensions that doesnot fix the extending group, but only its homotopy type.Here, we provide such a general definition of smooth string group extensions, and we prove thatthe string group models proposed in [BMS] fit within this definition. Let M fd denote the categoryof manifolds and smooth maps, and let C art ⊂ M fd be the full subcategory on those manifolds thatare diffeomorphic to R n for any n ∈ N . As our ambient category of smooth spaces, we choose the ∞ -category H ∞ := F un( C art op , S ) of presheaves of spaces on C art . This provides a very general notionof smooth space: for instance, H ∞ contains the categories of manifolds, diffeological spaces, and Liegroupoids. We write M for the image of a manifold M under the fully faithful inclusion M fd ֒ → H ∞ .The ∞ -category H ∞ is even an ∞ -topos, and there exists an established theory of group objects in ∞ -topoi [Lur09]. Moreover, there exists a notion of principal ∞ -bundles and extensions of group objectsin ∞ -topoi due to [NSS15]. A large part of this paper is devoted to developing this theory further. Inparticular, we show that group actions in ∞ -topoi automatically form groupoid objects (Theorem 3.19)and that principal ∞ -bundles essentially consists of an effective epimorphism and a principal groupaction (Theorem 3.31); this is analogous to the definition of principal bundles of topological spaces asa locally trivial map and a principal group action. Then, we provide the following characterisation ofextensions of group objects: Theorem 1.2
Let H be an ∞ -topos. Given a group object b A in H , denote its underlying object in H by A . Let b A b ι −→ b G b p −→ b H be a sequence of morphisms of group objects in H . The following are equivalent:(1) b A b ι −→ b G b p −→ b H is an extension of group objects in H , i.e. the sequence B A → B G → B H is a fibresequence of pointed connected objects in H .(2) The sequence b A b ι −→ b G b p −→ b H is a fibre sequence of group objects in H .(3) The sequence A ι −→ G p −→ H is a fibre sequence in H .(4) The map p : G → H together with the action of A on G induced by ι define a principal A -bundleover H . Point (1) in Theorem 1.2 is the definition of extensions of group objects in ∞ -topoi from [NSS15].In order to give a general homotopy-theoretic definition of string group extensions within H ∞ , we needto associate an underlying space to an object in H ∞ . In [Bunb] we investigated (a model categorical3resentation of) a functor S e : H ∞ → S from H ∞ to the ∞ -category S of spaces. It evaluates asmooth space X ∈ H ∞ on the extended affine simplices ∆ ke ∈ C art and then takes the geometricrealisation of the resulting simplicial object in S —that is, S e is a smooth version of the singularcomplex functor. Here, we give further interpretation and context to this functor: the adjunction e c ⊣ Γ , where Γ : H ∞ → S is the global-section functor, fits into a triple adjunction Π ⊣ e c ⊣ Γ ⊣ codisc ,where codisc is fully faithful and where Π preserves finite products. That is, the ∞ -topos H ∞ iscohesive. Theorem 1.3
The functor S e : H ∞ → S is part of the cohesion of H ∞ : there is a canonical equivalence Π ≃ S e . This has already been indicated in [BEBdBP] and proven on the level of model categories ofsimplicial presheaves in [Bunb]; here we provide an ∞ -categorical proof based on findings from [Bunb].Let L : H → H ′ be a functor between ∞ -topoi which preserves finite products and geometric realisationsof simplicial objects. We show that L maps principal ∞ -bundles in H to principal ∞ -bundles in H ′ and group extensions in H to group extensions in H ′ . (This relies on Theorem 3.19.) In particular,the functor S e : H ∞ → S has these properties. In S , a string group extension of a compact, connected,simply connected Lie group H can be defined as usual: it is an extension A → String( H ) → H ofgroup objects in S such that String( H ) is 3-connected and such that the morphism String( H ) → H induces an isomorphism on all homotopy groups except for in degree three. Using that S e M ≃ M forany manifold M [Bunb] and that S e preserves principal ∞ -bundles and group extensions, we are nowable to transfer this definition to H ∞ : Definition 1.4
Let H be a compact, simple, and simply connected Lie group, and let b H denote theinduced group object in H ∞ . An extension b A −→ \ String( H ) −→ b H of group objects in H ∞ is called a smooth string group extension of H if its image under S e is a string group extension in S . Finally, we show that the string group models conjectured in [BMS] fit within Definition 1.4. Let M be a manifold endowed with a bundle gerbe G (a categorified hermitean line bundle). In [BMS], weaddressed the question of when an action of a Lie group H on M lifts to an equivariant structure on G . We found that the obstruction to such a lift is an extension HLB
M i −→ Sym( G ) p −→ H (1.5)of H by the smooth 2-group HLB M of hermitean line bundles on M . Each of the above objects canbe interpreted as a group object in H ∞ via the nerve functor, and so (1.5) provides an extension of H as a group object in H ∞ . The case relevant for string group extensions is M = H , where H isa compact, connected and simply connected Lie group, acting on itself via left multiplication. Since H is 2-connected, there is an objectwise equivalence HLB H ≃ B( U (1) H ) , and since H is 1-connected,there is a smooth homotopy equivalence U (1) H ≃ U (1) . Therefore, the extending group in (1.5) hasthe correct homotopy type for a string group extension. We prove: Theorem 1.6
Let H be a compact, simple, simply connected Lie group, and let N be the nerve functor.We consider the left-action of H on itself via left multiplication. Let G ∈ G rb( H ) be a gerbe on H whoseclass in H ( H ; Z ) ∼ = Z is a generator. The sequence \ N HLB H \ N Sym( G ) b H c Ni c Np is a smooth string group extension of H . B U (1) in this case. It is interesting that the connection does not changethe homotopy type of the extension. A similar observation has been made in [BMS], where a secondextension of H , equivalent to (1.5), was constructed with a connection on the gerbe G acting as crucialauxiliary data. Since that second extension is equivalent to the one in (1.5) [BMS], it gives rise toa second smooth string group extension of H . Finally, we expect that most (or possibly all) of theaforementioned smooth string group models fit within Definition 1.4. Organisation.
In Section 2 we investigate the functor S e : H ∞ → S . Further, we recall some basicnotions and facts about ∞ -topoi and prove Theorem 1.3.Section 3 is devoted to the theory of group objects, group extensions, and principal ∞ -bundles in ∞ -topoi. We recall the definitions of these notions from [NSS15] and provide new characterisations ofprincipal ∞ -bundles and group extensions. In particular, we prove Theorem 1.2.In Section 4, we use the results obtained thus far to transfer the definition of string group extensionsin S to the ∞ -topos H ∞ . After recalling from [BMS] the smooth 2-group extensions which obstruct theexistence of equivariant structures on gerbes, we show that these extensions give rise to new smoothmodels for string group, thus proving Theorem 1.6.Finally, in Appendix A we prove Theorem 3.19: we show that group actions in an ∞ -topos giverise to groupoid objects. Notation.
We usually make no notational distinction between ordinary categories and ∞ -categories;the nerve functor will be used implicitly where necessary.We write ∆ for the simplex category, and S et ∆ for the category of simplicial sets. In a simplicialcategory C , we denote the simplicially enriched hom-functor by C ( − , − ) : C op × C → S et ∆ .We write |−| = colim C ∆ op for the colimit of simplicial objects in an ∞ -category C . Moreover, wealso refer to | X | (if it exists) as the geometric realisation of a simplicial object X in C .Usually, we denote ∞ -categories by letters C , D , . . . , but for ∞ -topoi we use bold-face letters H . Inparticular, the ∞ -topos of spaces is denoted by S . We write D ( − , − ) : D op × D → S for the mappingspaces in an ∞ -category D .We model ∞ -categories by quasi-categories. Given an ∞ -category C and a simplicial set K ∈ S et ∆ ,we write F un( K, C ) = S et ∆ ( K, C ) = C K for the ∞ -category of functors from K to C .We let ∆ + denote the augmented simplex category, i.e. the category ∆ with an initial object adjoined.We usually do not distinguish notationally between augmented simplicial objects X ∈ F un( ∆ op+ , C ) inan ∞ -category C and their underlying simplicial objects. If we wish to make this distinction explicitfor clarity, we will denote the latter by the restriction X | ∆ op .If M is a simplicial model category, then M ◦ is the full simplicial subcategory on the cofibrant-fibrant objects of M . Recall from [Lur09] that the coherent nerve N ( M ◦ ) is an ∞ -category.If C is a (small) ∞ -category, we write P ( C ) = F un( C op , S ) for the ∞ -category of presheaves ofspaces on C . Acknowledgements.
The author would like to thank Lukas Müller, Birgit Richter, ChristophSchweigert, Walker Stern, and Konrad Waldorf for helpful discussions. The author acknowledgespartial support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) underGermany’s Excellence Strategy—EXC 2121 “Quantum Universe”—390833306.5
Smooth spaces and ∞ -topoi In this section we recall and develop some background on the ∞ -categories most relevant in this paper.Most importantly, we consider a presheaf ∞ -category H ∞ , whose objects can be interpreted as ageneral notion of smooth spaces. We study an ∞ -categorical version S e : H ∞ → S of a Quillen functorconsidered in [Bunb], which provides a version of the singular complex functor for smooth spaces.Subsequently, we briefly recall the definition of an ∞ -topos and of cohesion of ∞ -topoi, and we showthat S e is part of the cohesion of H ∞ . We let C art denote the (small) category whose objects are submanifolds of R ∞ that are diffeomorphicto R n for any n ∈ N , and whose morphisms are the smooth maps between these manifolds. We let H ∞ := P ( C art) = F un( C art op , S ) denote the ∞ -category of presheaves of spaces on C art . The ∞ -category H ∞ is presented by severalmodel categories of simplicial presheaves on C art —for example, there is a canonical equivalence [Lur09] H ∞ ≃ N (cid:0) ( H i ∞ ) ◦ (cid:1) , where H i ∞ is the category of simplicial presheaves on C art , endowed with the injective model structure.Let I := { c × R → c | c ∈ C art } denote the set of morphisms in C art of the form c × c R , where c R : R → ∗ is the map that collapses the real line to the point. We can localise both H i ∞ and H ∞ at this set of morphisms (or rather at its image under the Yoneda embedding), and there is still acanonical equivalence between the localisations [Lur09], N (cid:0) ( L I H i ∞ ) ◦ (cid:1) ≃ L I H ∞ . The simplicial model categories H i ∞ and L I H i ∞ were the subject of [Bunb]. On the level of theirunderlying ∞ -categories, one of the main results of that paper can be phrased as follows. For k ∈ N ,we let ∆ ke := { t ∈ R k +1 | P ki =0 t i = 1 } denote the extended (affine) k -simplex. This is a k -dimensionalaffine subspace of R k +1 , and hence forms a cartesian space. The face and degeneracy maps of thestandard topological simplices | ∆ k | extend to the extended affine simplices, turning them into a functor ∆ e : ∆ → C art , [ k ] ∆ ke . We let S e : H ∞ → S denote the composition of functors S e : H ∞ F un( ∆ op , S ) S . ∆ ∗ e colim (2.1)We refer to this functor as the smooth singular complex functor ; viewing the ∞ -category H ∞ as an ∞ -category of smooth spaces, S e thus assigns an underlying ordinary space to a smooth space. Theorem 2.2 [Bunb]
There exist adjunctions of ∞ -categories H ∞ L I H ∞ S S
Loc ⊥ S e ⊣ ι S Ie R e L Ie ⊣ R Ie ⊣ here S Ie is the restriction of S e to L I H ∞ ⊂ H ∞ . Furthermore, the following statements hold true:(1) The functor S e : H ∞ → S preserves and reflects I -local equivalences.(2) The morphism ι is fully faithful, i.e. Loc is a reflective localisation.(3) The three right-hand vertical functors are equivalences of ∞ -categories.(4) The diagram obtained by omitting the morphism L Ie is (weakly) commutative.Proof. The first claim follows readily from Proposition 3.6, Corollary 3.12, and Corollary 3.37 of [Bunb].(Note that model categorical presentations of H ∞ , L I H ∞ , and S are used in [Bunb], and the functorsin the statement are presented by Quillen functors.)Further, claim (1) follows readily from [Bunb, Cor. 3.15]. Claim (2) follows from general proper-ties of ∞ -categories underlying simplicial model categories and their Bousfield localisations [Lur09].Claim (3) is the version on the underlying ∞ -categories of Theorems 3.14 and 3.40 of [Bunb]. Claim (4)holds true because the diagram of the right-adjoints clearly commutes ( ι is an inclusion, and R e simplyfactors through L I H ∞ ⊂ H ∞ [Bunb]). Remark 2.3
There is a fully faithful embedding M fd ֒ → H ∞ from the category of manifolds into H ∞ : it sends a manifold M to the presheaf M of discrete spaces that maps a cartesian space c to theset M fd( c, M ) of smooth maps from c to M . By [Bunb, Thm. 5.1] there is a canonical equivalence ofspaces M ≃ S e M for any M ∈ M fd , which is natural in M . ⊳ Proposition 2.4
The localisation functor
Loc : H ∞ → L I H ∞ preserves finite products. The class W I of I -local equivalences in H ∞ is closed under finite products.Proof. By [Bunb, Prop. 2.13], the localisation L I H ∞ agrees with the localisation L W H ∞ of H ∞ at allcollapse morphisms c → ∗ , for c ∈ C art . The class W is stable under finite products in H ∞ , since C art has finite products. Therefore, the first claim follows from [Cis19, Cor. 7.1.16]. The second claim thenfollows since a morphism in H ∞ is in W I precisely if its image under Loc is an equivalence [Lur09,Prop. 5.5.4.15].
Proposition 2.5
For
X, Y ∈ H ∞ , let Y X ∈ H ∞ denote their internal hom object in H ∞ . Thelocalisation functor Loc : H ∞ → L I H ∞ is given (up to equivalence) by Loc ≃ colim ∆ op H ∞ (cid:0) ( − ) ∆ e (cid:1) . Proof.
By Theorem 2.2(4), there is a canonical equivalence S Ie ◦ Loc ≃ S e . Combining this withTheorem 2.2(3), we obtain canonical equivalences Loc ≃ L Ie ◦ S Ie ◦ Loc ≃ L Ie ◦ S e . Consider the adjunction ˜ c : S ⇄ H ∞ : ev ∗ , where ˜ c assigns to a space K the constant presheaf withvalue K , and where ev ∗ evaluates a presheaf on the final object ∗ ∈ C art . These functors induce anequivalence ˜ c : S ⇄ L I H ∞ : ev ∗ [Bunb, Thm. 2.17], and there is a canonical equivalence ev ∗ ≃ S Ie of functors L I H ∞ → S by [Bunb, Prop. 2.7, Cor. 3.15]. By adjointness, we also obtain a canonicalequivalence ˜ c ≃ L Ie . Consequently, there is a canonical equivalence Loc ≃ ˜ c ◦ S e .
7e observe that there exists a canonical equivalence S e = colim ∆ op S (cid:0) ∆ ∗ e ( − ) (cid:1) ≃ ev ∗ ◦ colim ∆ op H ∞ (cid:0) ( − ) ∆ e (cid:1) . By [Bunb, Prop. 6.2], we have that colim H ∞ ∆ op (( − ) ∆ e ) is a functor H ∞ → L I H ∞ ; that is, it takes valuesin I -local objects. It follows that there are canonical equivalences Loc ≃ ˜ c ◦ S e ≃ ˜ c ◦ ev ∗ ◦ colim ∆ op H ∞ (cid:0) ( − ) ∆ e (cid:1) ≃ colim ∆ op H ∞ (cid:0) ( − ) ∆ e (cid:1) . This completes the proof. ∞ -topoi In this section, we briefly recall some background on ∞ -topoi. Most of the material in this sectioncan be found in [Lur09, Sch, NSS15]. For n ∈ N and a subset S ⊂ [ n ] , let ∆ S ⊂ ∆ n be the full ∞ -subcategory on the vertices that lie in S . There is a canonical isomorphism ∆ S ∼ = ∆ | S | as simplicialsets, where | S | is the cardinality of S . The simplicial set ∆ S can equivalently be seen as the image ofan inclusion ∆ | S | ֒ → ∆ n that sends the i -th vertex of ∆ | S | to the vertex of ∆ n which corresponds to the i -th element of S (with the order induced from the inclusion S ⊂ [ n ] ). Given a simplicial object in an ∞ -category C , i.e. b X ∈ F un( ∆ op , C ) , we set b X ( S ) := b X (∆ | S | ) . This comes with a canonical morphism b X n → b X ( S ) , induced by the inclusion S ⊂ [ n ] . Definition 2.6
Let C be an ∞ -category. A groupoid object in C is a simplicial object b X ∈ F un( ∆ op , C ) such that, for every n ∈ N and every partition [ n ] = S ∪ S ′ (as finite sets) with S ∩ S ′ ∼ = {∗} consistingof a single element, the diagram b X n b X ( S ′ ) b X ( S ) b X is a pullback diagram in C . We denote the full subcategory of F un( ∆ op , C ) on the groupoid objects by G pd( C ) ⊂ F un( ∆ op , C ) . Let ∆ + denote the simplex category with an initial object [ − adjoined. For n ∈ N , let ∆ + , ≤ n ⊂ ∆ + bethe full subcategory on the objects [ − , . . . , [ n ] . In particular, ∆ op+ , ≤ is the category with two objectsand one non-trivial morphism [0] → [ − . Therefore, any morphism p : P → X in an ∞ -category C defines an object { p } ∈ F un( ∆ op+ , ≤ , C ) . Definition 2.7
Given a morphism p : P → X in an ∞ -category C , its Čech nerve ˇ Cp (if it exists) isthe augmented simplicial object obtained as the right Kan extension ∆ op+ , ≤ C ∆ op+ { p } ı ˇ Cp That is, ˇ Cp = Ran ı { p } , where ı is the inclusion ∆ op+ , ≤ ֒ → ∆ op+ . Proposition 2.8 [Lur09, Prop. 6.1.2.11]
Let C be an ∞ -category, and let b X : ∆ op+ → C be an augmentedsimplicial object. The following are equivalent:(1) b X is a right Kan extension of b X | ∆ op+ , ≤ .(2) The underlying simplicial object b X | ∆ op is a groupoid object in C and the diagram b X | ∆ op+ , ≤ = b X b X b X b X − d d is a pullback square in C . In this situation, it follows that, for every n ≥ , the spine decomposition [ n ] = [1] ⊔ [0] · · · ⊔ [0] [1] induces a canonical equivalence ( ˇ Cp ) n ≃ P × X · · · × X P | {z } n +1 factors . Definition 2.9
Let C be an ∞ -category, and let p : P → X be a morphism in C . Then, p is an effectiveepimorphism if ˇ Cp ∈ F un( ∆ op+ , C ) ∼ = F un(( ∆ op ) ⊲ , C ) is a colimiting cocone in C . That is, the morphism p : P → X is an effective epimorphism precisely if the induced morphism | ˇ Cp | → X is an equivalence. Let b X : ∆ op+ → C be an augmented simplicial object in an ∞ -category C . We denote the morphism b X → b X − by p . Suppose that its Čech nerve ˇ Cp exists. Observe that { p } = ı ∗ b X as objects in F un(∆ , C ) ∼ = F un( ∆ op+ , ≤ , C ) . By the adjointness property of the right Kan extension, there is acanonical equivalence F un( ∆ op+ , ≤ , C )( ı ∗ b X, { p } ) ≃ F un( ∆ op+ , C )( b X, ˇ Cp ) . The identity ı ∗ b X = { p } thus induces a canonical morphism η : b X −→ ˇ Cp . (2.10)We define ∞ -topoi in terms of the Giraud-Lurie-Rezk axioms [Lur09, Def. 6.1.0.4, Thm. 6.1.0.6]: Definition 2.11 An ∞ -topos is an ∞ -category H satisfying the following axioms:(1) H is presentable. In particular, H has all limits and colimits. We denote its initial object by ∅ ∈ H and its final object by ∗ ∈ H .(2) Colimits in H are universal: for any diagram D : K → H , any cocone D : K ⊲ → H under D withapex Y ∈ H , and any morphism f : X → Y in H , the induced morphism colim K H ( D × c Y c X ) −→ (cid:0) colim K H D (cid:1) × Y X is an equivalence (on the left-hand side, c X, c Y : K → H are the constant diagrams with values X and Y , respectively, and the pullback is formed in F un( K, H ) ).(3) Coproducts in H are disjoint: for every pair of objects X, Y ∈ H , the pushout diagram ∅ XY X ⊔ Y is also a pullback diagram.
4) Groupoids in H are effective: given any groupoid object b X ∈ G pd( H ) , let p : b X → | b X | denotethe canonical morphism which is part of the colimiting cocone. Then, the comparison morphism η : b X → ˇ Cp constructed in (2.10) is an equivalence of simplicial objects in H . In particular, p isan effective epimorphism. Example 2.12
We list some examples of ∞ -topoi; we will mostly be using the first two cases.(1) The ∞ -category of spaces S is an ∞ -topos.(2) Any ∞ -category P ( C ) of presheaves of spaces on a (small) ∞ -category C is an ∞ -topos.(3) Any accessible, left-exact, reflective localisation of an ∞ -category P ( C ) of presheaves on a small ∞ -category C is an ∞ -topos; in fact, every ∞ -topos is equivalent to an ∞ -topos of this form [Lur09,Thm. 6.1.0.6, Prop. 6.1.5.3]. ⊳ An important notion of morphism between ∞ -topoi is that of a geometric morphism, which is moreadapted to the additional structure on ∞ -topoi than a mere functor of ∞ -categories: Definition 2.13
Let X , Y be ∞ -topoi. A geometric morphism of ∞ -topoi from X to Y is a functor F ∗ : X → Y admitting a left exact left adjoint F ∗ : Y → X . One can show that the ∞ -category S of spaces is final in the ∞ -category of ∞ -topoi and geometricmorphisms [Lur09, Prop. 6.3.4.1]. That is, for every ∞ -topos H there exists an essentially uniquegeometric morphism H → S . We will denote the corresponding adjunction by e c : S ⇄ H : Γ and referto Γ as the global-section functor . Example 2.14
Consider a Grothendieck ∞ -site, i.e. a small ∞ -category C with a Grothendieck(pre)topology. Suppose C additionally has a final object. If H is the ∞ -category of sheaves of spaceson C , then the global section functor Γ of H agrees with the evaluation of sheaves at the final object of C . In particular, this applies to H ∞ , the ∞ -topos of presheaves of spaces on C art from Section 2.1. ⊳ Definition 2.15 An ∞ -topos H is called cohesive if the adjunction e c : H ⇄ S : Γ can be extendedto a triple adjunction Π ⊣ c ⊣ Γ ⊣ codisc , in which the left adjoint Π preserves finite products and inwhich the right adjoint codisc is fully faithful. Cohesive ∞ -topoi have been studied extensively in [Sch] and related works. Theorem 2.16
The ∞ -topos H ∞ is cohesive, i.e. there exists a triple adjunction Π ⊣ e c ⊣ Γ ⊣ codisc as in Definition 2.15, and there is a canonical equivalence Π ≃ S e . Remark 2.17
The fact that H ∞ is cohesive is not new, see [Sch]. The second statement has beenindicated in [BEBdBP] and has been worked out in detail in a model categorical presentation in [Bunb].Here, we give an ∞ -categorical proof of this fact for completeness. ⊳ Proof.
The ∞ -topos H ∞ = P ( C art) admits a right-adjoint to its global-section functor Γ by abstractarguments: evaluation of a presheaf at any object preserves colimits, and since both H ∞ and S arepresentable, Γ must admit a further right adjoint. It is well-known that this can actually be extendedinto a triple adjunction which establishes that H ∞ is cohesive [Sch].For the second part of the statement, we show that S e is left-adjoint to the functor e c . Recall fromSection 2.1 that, in this situation, e c simply sends a space K ∈ S to the constant presheaf on C art with value K . Further, recall from the proof of Proposition 2.4 (and [Bunb, Prop. 2.13]) that the10 -local objects in H ∞ are precisely the essentially constant presheaves, i.e. those X ∈ H ∞ for whichthe canonical morphism X ( ∗ ) → X ( c ) is an equivalence for every c ∈ C art . Equivalently, X is I -localif and only if the canonical morphism e c Γ X → X is an equivalence in H ∞ . Further, by Theorem 2.2the right adjoint R e to S e factors through the localisation L I H ∞ ⊂ H ∞ ; this is precisely the full ∞ -subcategory of H ∞ on the I -local objects.Consider the two adjunctions S e : H ∞ ⇄ S : R e and e c : S ⇄ H ∞ : Γ . They induce an adjunction S e ◦ e c : S S : Γ ◦ R e . ⊥ By the definition (2.1) of S e , for any space K ∈ S we have a canonical natural equivalence S e ◦ e c ( K ) = colim ∆ op S (cid:0)e c ( K )(∆ e ) (cid:1) ≃ K , because left-hand side is the colimit of a constant diagram over an indexing category whose nerve iscontractible in the Kan-Quillen model structure on S et ∆ (see Lemma A.7, Example A.8). In otherwords, there is a canonical natural equivalence S e ◦ e c ≃ S . Consequently, there is also a canonicalequivalence on the right adjoints, Γ ◦ R e ≃ S . We obtain natural equivalences e c ≃ e c ◦ Γ ◦ R e ≃ R e . In the second equivalence we have used that R e takes values in L I H ∞ ⊂ H ∞ and that on objects in L I H ∞ the morphism e c ◦ Γ → H ∞ is an equivalence. From the equivalence R e ≃ e c and the adjunction S e ⊣ R e we infer that S e is a further left adjoint to e c . Hence, it is equivalent to the functor Π .Theorem 2.16 shows that the smooth singular complex functor S e : H ∞ → S has a deep homotopicalmeaning for assigning homotopy types to objects in H ∞ and for studying these homotopy types. Italso provides an additional, refined, perspective on the good homotopical properties of the functor S e that were found and studied in [Bunb]. Finally, note that it also follows that there is a naturalequivalence S e ( F ) = colim ∆ op S (cid:0) F (∆ e ) (cid:1) ≃ colim C art op S ( F ) . That is, S e computes the colimit of C art op -shaped diagrams of spaces. ∞ -bundles and group extensions in ∞ -topoi In this section, starting from the theory introduced in [NSS15], we develop characterisations of principal ∞ -bundles and of extensions of group objects in ∞ -topoi. These characterisations are interestingalready in their own right, but in Section 4 they will also allow us to transfer the definition of stringgroup extensions from S to H ∞ and to construct explicit smooth models for the string group. Here, we recall the definitions of group objects and their extensions in ∞ -topoi [NSS15]. We investigatehow to compute limits of group and groupoid objects in ∞ -topoi, and how group objects and theirclassifying objects behave under functors between ∞ -topoi that preserve finite products and geometricrealisations. 11et H be an ∞ -topos, and let G pd( H ) be the ∞ -category of groupoid objects in H . Further, let EEpi( H ) ⊂ F un(∆ , H ) denote the full ∞ -subcategory on the effective epimorphisms in H . Recallthat by Definition 2.11(4) there is a canonical equivalence G pd( H ) ≃ EEpi( H ) , (3.1)given by forming colimits and Čech nerves, respectively. Lemma 3.2
In an ∞ -topos H , effective epimorphisms are stable under pullback and under pushout.Proof. The fact that effective epimorphisms are stable under pullback is [Lur09, Prop. 6.2.3.15]. Thestability under pushouts follows from the facts that the effective epimorphisms in H are precisely the -connective morphisms [Lur09, Def. 6.5.1.10], and that n -connected morphisms in an ∞ -topos arestable under pushout for any n ≤ − [Lur09, Prop. 6.5.1.17]. (The n -connected morphisms are evenstable under colimits, since they form the left class of an orthogonal factorisation system on H .) Definition 3.3
Let C be an ∞ -category. Let G rp( C ) ⊂ G pd( C ) denote the full ∞ -subcategory on thosegroupoid objects where X is a final object of C . We call G rp( C ) the ∞ -category of group objects in C . Proposition 3.4
For any ∞ -topos H , there are reflective localisations F un( ∆ op , H ) G pd( H ) G rp( H ) . ⊥ ⊥ Proof.
First, the right adjoints in the above sequence of adjunctions are fully faithful by definition.The first morphism has a left adjoint by [Lur09, Prop. 6.1.2.9]. For the second left adjoint, we use theequivalence (3.1) : this equivalence induces a commutative square G pd( H ) G rp( H )EEpi( H ) EEpi ∗ ( H ) ≃ ≃ where EEpi ∗ ( H ) ⊂ EEpi( H ) is the full ∞ -subcategory on those effective epimorphisms f : X → X − where X is a final object. A left adjoint to the bottom morphism is given by the functor that sendsan effective epimorphism f : X → X − to the morphism g : ∗ → X − ⊔ X ∗ induced by the pushout.Since f is an effective epimorphism, Lemma 3.2 implies that so is g .For a group object b G ∈ G rp( H ) in an ∞ -topos H , we set G := b G ∈ H and B G := colim ∆ op H b G = | b G | ∈ H . Note that in an ∞ -topos H , the map X → colim H ∆ op b X is an effective epimorphism for any groupoidobject b X ∈ G pd( H ) . Hence, given a group object b G in H , the morphism ∗ ≃ G → B G is aneffective epimorphism. Moreover, the functor B is part of an equivalence [Lur09, Lemma 7.2.2.11] (seealso [NSS15, Thm. 2.19]) H ∗ / ≥ G rp( H ) , ⊥ ΩB Note that there is a shift in counting between [Lur09] and the nLab: A morphism f in H is n -connect ive in theconventions of [Lur09] if and only if it is ( n − -connect ed in the conventions used on the nLab. This proof goes back to a mathoverflow answer by Jacob Lurie, see https://mathoverflow.net/questions/140639/is-the-category-of-group-objects-in-an-infty-1-topos-reflective-as-a-subcat/140742 . H ∗ / ≥ is the ∞ -category of pointed, connected objects in H .Unravelling the definition, we obtain that a group object in an ∞ -category C with a final object ∗ ∈ C is a equivalently simplicial object b G in C such that b G ≃ ∗ and, for any [ n ] ∈ ∆ and any partition [ n ] = S ∪ S ′ as finite sets with S ∩ S ′ ∼ = {∗} consisting of a single element, the diagram b G n b G ( S ) b G ( S ′ ) b G ≃ ∗ is a pullback diagram in C . That is, there is a canonical equivalence b G n ≃ −→ b G ( S ) × b G ( S ′ ) . In particular,iterating this for the spine partition [ n ] = [1] ⊔ [0] · · · ⊔ [0] [1] , we obtain a canonical equivalence b G n ≃ −→ G n . Proposition 3.5
Let
L : H → H ′ be a functor of ∞ -topoi.(1) If L preserves finite products, then it preserves group objects.(2) If L additionally preserves geometric realisations, then, for any group object b G in H , there is acanonical equivalence B(L G ) ≃ L(B G ) . Proof.
The first part of the Proposition is known [Lur09]; we include its proof only for completeness.Any functor F : C → D between ∞ -categories preserves simplicial objects, i.e. it induces a functor F un( ∆ op , C ) −→ F un( ∆ op , D ) . Suppose that b A ∈ F un( ∆ op , H ) is a group object in H . Since L preservesfinite products, it preserves final objects, so that (L b A ) ≃ ∗ is final in H ′ . For n = 0 and any partition [ n ] = S ∪ S ′ with S ∩ S ′ ∼ = {∗} , we obtain a commutative diagram (L b A ) n = L( b A n ) L (cid:0) b A ( S ) × b A ( S ′ ) (cid:1) L b A ( S ) × L b A ( S ′ ) ≃ ≃ The top morphism is an equivalence since b A is a group object in H and the vertical morphism is anequivalence since L preserves products. This proves claim (1). Using that B A = colim H ∆ op b A = | b A | , thesecond part is now immediate. Remark 3.6
We will prove a number of statements about functors as in Proposition 3.5(2), i.e. func-tors between ∞ -topoi which preserve geometric realisations and finite products. An important classof such functors arises is given by the additional left-adjoints of cohesive ∞ -topoi—see Definition 2.15.In particular, the functor S e : H ∞ → S from Section 2.1 is of this type by Theorem 2.16. ⊳ Lemma 3.7
Let H be an ∞ -topos.(1) A morphism b X → b Y in G pd( H ) is an equivalence if and only if X i → Y i is an equivalence in H for i = 0 , .(2) A morphism b G → b H in G rp( H ) is an equivalence if and only if G → H is an equivalence in H . roof. Proposition 3.4 implies that an equivalence of groupoid objects b X ≃ −→ b Y in H is the sameas an objectwise equivalence of the underlying simplicial objects in H : b X and b Y are local objects in F un( ∆ op , H ) with respect to the localisation G pd( H ) ⊂ F un( ∆ op , H ) , so that the local equivalencesbetween them are precisely the original, i.e. the levelwise, equivalences. In particular, this implies the‘only if’ part of claim (1).Conversely, if we are given a morphism b X ≃ −→ b Y of groupoid objects in H such that b X i → b Y i is anequivalence for i = 0 , , then it follows that b X ≃ −→ b Y is a levelwise equivalence of simplicial objects;this is because there are canonical equivalences b X n ≃ b X × b X · · · × b X b X , natural in b X ∈ G pd( H ) . Itthen follows that the morphism b X → b Y is also an equivalence in G pd( H ) .The same line of argument shows the second claim. Lemma 3.8
Let H be an ∞ -topos, and let K ∈ S et ∆ be a simplicial set.(1) A diagram b X : K ⊳ → G pd( H ) of groupoid objects in H is a limit diagram if and only if thecomposition ι b X : K ⊳ → G pd( H ) ֒ → F un( ∆ op , H ) is a limit diagram.(2) A diagram b X : K ⊳ → G pd( H ) of groupoid objects in H is a limit diagram if and only if the induceddiagrams b X i : K ⊳ → H are limit diagrams for i = 0 , .(3) A diagram b G : K ⊳ → G rp( H ) of group objects in H is a limit diagram if and only if the thecomposition b G : K ⊳ → G rp( H ) ֒ → G pd( H ) is a limit diagram.(4) A diagram b G : K ⊳ → G rp( H ) of group objects in H is a limit diagram if and only if the induceddiagram b G = G : K ⊳ → H is a limit diagram.Proof. The ‘only if’ direction of claims (1) and (2) is readily seen as follows: we first note that sincethe inclusion G pd( H ) ⊂ F un( ∆ op , H ) is a right adjoint, we have that if b X : K ⊳ → G pd( H ) is a limitdiagram, then so is b X : K ⊳ → F un( ∆ op , H ) . Further, since limits in presheaf (or diagram) categoriesare computed pointwise, this is equivalent to the functor b X i : K ⊳ → H being a limit diagram in H forevery [ i ] ∈ ∆ .For the converse direction in claim (1), we first show that limits of diagrams in G pd( H ) can becomputed in F un( ∆ op , H ) . More precisely, a functor b X : K ⊳ → G pd( H ) is a limit diagram wheneverits composition with the inclusion ι : G pd( H ) ֒ → F un( ∆ op , H ) is so, i.e. the inclusion reflects limits.Equivalently, the ∞ -subcategory G pd( H ) ֒ → F un( ∆ op , H ) is closed under limits in F un( ∆ op , H ) . Thisis seen as follows: consider a functor b X : K ⊳ → G pd( H ) and a decomposition [ n ] = S ∪ S ′ with S ∩ S ′ = {∗} . This induces an equivalence b X n ≃ −→ b X ( S ) × b X b X ( S ′ ) in F un( K ⊳ , H ) . Setting b Y := lim F un( ∆ op , H ) K ( ι b X ) and using that limits commute with limits, we have b Y n ≃ (cid:0) lim K F un( ∆ op , H ) ( ι b X ) (cid:1) n (3.9) ≃ lim K H ( ι b X n ) ≃ lim K H (cid:0) ι b X ( S ) × ι b X ι b X ( S ′ ) (cid:1) ≃ (cid:0) lim K H ι b X ( S ) (cid:1) × (lim K H ι b X ) (cid:0) lim K H ι b X ( S ′ ) (cid:1) ≃ b Y ( S ) × b Y b Y ( S ′ ) , b Y ∈ F un( ∆ op , H ) is local with respect to the localisation G pd( H ) ֒ → F un( ∆ op , H ) ,i.e. that b Y ∈ G pd( H ) . Since the inclusion ι : G pd( H ) ֒ → F un( ∆ op , H ) is fully faithful, b Y is also a limitof the diagram b X : K → G pd( H ) . Consequently, if ι b X : K ⊳ → F un( ∆ op , H ) is a limit diagram, then b X : K ⊳ → G pd( H ) is a limit diagram.For the converse direction in claim (2), suppose that b X : K ⊳ → G pd( H ) is a diagram such thatthe functors b X i : K ⊳ → H are limit diagrams for i = 0 , . By part (1) it suffices to show that thecomposition ι b X : K ⊳ → F un( ∆ op , H ) is a limit diagram; that is, it suffices to show that b X i : K ⊳ → H is a limit diagram for every [ i ] ∈ ∆ .Since b X : K ⊳ → F un( ∆ op , H ) is valued in groupoid objects, and since limits in H commute [Lur09,Lemma 5.5.2.3], it follows from (3.9) that for every [ n ] ∈ ∆ the diagram b X n : K ⊳ → H is equivalent toa limit diagram b X × b X · · · × b X b X : K ⊳ → H , and is hence a limit diagram itself.The proof of claim (3) proceeds along the exact same line as the proof of part (2): the key insightis the fact that if b G : K ⊳ → G rp( H ) is a diagram such that the composition b G : K ⊳ → G pd( H ) is alimit diagram, then lim G pd( H ) K ( b G ) is still local with respect to the localisation G rp( H ) ⊂ G pd( H ) .Claim (4) is then the combination of claims (2) and (3).Having established several properties of the ∞ -category of group objects in H , we now defineextensions of group objects: Definition 3.10 [NSS15, Def. 4.26]
Let b A and b H be group objects in an ∞ -topos H . An extension ofgroup objects of b H by b A is a sequence b A → b G → b H in the ∞ -category G rp( H ) such that the sequence B A → B G → B H is a fibre sequence in H ∗ / ≥ . Remark 3.11
This definition of a group extension has advantages from a theoretical perspective.Nevertheless, it appears that there should be a simpler definition that more directly generalises exten-sions of groups in S et to the ∞ -categorical setting. For group objects in S et , a group extension is asequence A → G → H of group homomorphisms such that A is the fibre of the morphism G → H atthe identity element of H . We will prove in Theorem 3.49 that one can indeed define group extensionsin ∞ -topoi along these principles. ⊳ ∞ -categories We now investigate actions of group objects in ∞ -topoi. For a simplicial set K ∈ S et ∆ , we let obj( K ) be the set K of vertices of K , seen as a discrete simplicial set. Let J := ∆ [ f − ] be the localisationof ∆ at its non-trivial edge (see e.g. [Cis19, Sec. 3.3]). Lemma 3.12
Let C be an ∞ -category, and let K be a simplicial set.(1) The inclusion ι : obj( K ) ֒ → K induces a morphism ι ∗ : C K = F un( K, C ) −→ F un (cid:0) obj( K ) , C (cid:1) = C obj( K ) of simplicial sets, which is a fibration between fibrant objects in the Joyal model structure.(2) Consider either of the inclusions ∆ { i } ֒ → J , where i = 0 , . The induced morphism F un( J, C K ) −→ C K × C obj( K ) F un (cid:0) J, C obj( K ) (cid:1) is a trivial Kan fibration.
3) Let g : K → C and g ′ : obj( K ) → C be functors. For any equivalence η : ι ∗ g ≃ −→ g ′ , consider thespace of pairs (ˆ g ′ , ˆ η ) , where ˆ g ′ is a lift of g ′ to a functor ˆ g ′ : K → C , and where ˆ η is an equivalence g ≃ −→ ˆ g ′ such that ι ∗ ˆ η = η . This space is a contractible Kan complex.Proof. Part (1) follows since obj( K ) ֒ → K is a cofibration in the Joyal model category S et ∆ J , C is afibrant object in S et ∆ J , and S et ∆ J is a (closed) symmetric monoidal model category.For part (2), we apply [Cis19, Cor. 3.6.4] to the categorical anodyne extension ∆ { i } ֒ → J = ∆ [ f − ] and the Joyal fibration (i.e. isofibration) from part (1).Part (3) is obtained by taking the fibre of the morphism from part (2), which is a contractible Kancomplex since it is the fibre of a trivial Kan fibration. This fibre is equivalently described as the spaceof lifts in the commutative diagram ∆ { i } C K J C obj( K ) g ι ∗ η which is precisely the space of pairs (ˆ g ′ , ˆ η ) of lifts ˆ g ′ : K → C of g ′ and equivalences ˆ η : g ≃ −→ ˆ g ′ suchthat ι ∗ ˆ η = η . Example 3.13
Let b G be a group object in an ∞ -category C with a final object. This is, in particular,a simplicial object b G : ∆ op → C (we suppress the canonical inclusion functors G rp( C ) ֒ → G pd( C ) ֒ → F un( ∆ op , C ) ). Consider the functor [0] ⋆ ( − ) : ∆ −→ ∆ , [ n ] [0] ⋆ [ n ] ∼ = [ n + 1] , where ⋆ denotes the join of categories (and where we view partially ordered sets as categories). Theinduced pullback functor Dec := (cid:0) [0] ⋆ ( − ) (cid:1) ∗ : F un( ∆ op , C ) −→ F un( ∆ op , C ) is also called the decalage functor; see [Ste12] for more background. For any n ≥ , the partition [ n ] = { , } ⊔ { } { , . . . , n } induces an equivalence γ n : (Dec b G ) n = b G n +1 ≃ G × b G n . (3.14)We can phrase this as an equivalence of functors Dec b G ≃ G × b G : obj( ∆ op ) → C . From Lemma 3.12we obtain that there exists an essentially unique way to lift these data to a functor ∆ op → C , whichwe denote by G//G , and an equivalence γ : Dec b G ≃ −→ G//G in F un( ∆ op , C ) , whose components areexactly the equivalences γ n from (3.14). One can now check that G//G is the simplicial object in C that describes the right action of b G on itself via the group multiplication in b G . ⊳ Definition 3.15
Let C be an ∞ -category with pullbacks and a final object, let b G be a group object in C , and let P ∈ C be an object in C . An action of b G on P is a simplicial object P //G ∈ F un( ∆ op , C ) such that(1) for each n ∈ N , we have ( P //G ) n = P × G n − ,(2) the morphism d : P × G → P agrees with the canonical projection onto P , the morphism s : P → P × G agrees with the morphism P × ( ∗ → G ) , and(3) the collapse morphism P → ∗ induces a morphism P //G → b G in F un( ∆ op , C ) . P //G , we set a := d : P × G → P . It follows by the pasting law forpullbacks that there are canonical equivalences of morphisms between d : P × G n → P × G n − and a × G n − : P × G n → P × G n − , and similarly between d n : P × G n → P × G n − and the projectiononto the first n factors. Remark 3.16
Definition 3.15 is taken from [NSS15, Def. 3.1] almost verbatim, but it differs fromthat source in that we do not require group actions to be groupoid objects. Instead, we show inTheorem 3.19 that this is a consequence of the axioms in Definition 3.15. A second (minor) differenceis that we also fix the level-zero degeneracy map s : P → P × G . ⊳ Example 3.17
For any group object b G ∈ G rp( C ) there is a canonical trivial action ∗ //G on the finalobject ∗ ∈ C , coming from the canonical equivalence ∗ × b G ≃ b G of simplicial objects. That is, there isa canonical equivalence b G ≃ ∗ //G in F un( ∆ op , C ) . ⊳ Example 3.18
We can now give a precise meaning to the last sentence of Example 3.13: the object
G//G ∈ F un( ∆ op , C ) is an action of G on itself via right multiplication. ⊳ Given an action of a group object b G on an object P in C , we would like to think of the simplicialobject P //G as the action groupoid associated with this action. This is indeed justified:
Theorem 3.19
Let C be an ∞ -category with finite limits, let b G ∈ G rp( C ) be a group object in C , andlet P //G ∈ F un( ∆ op , C ) be an action of b G on an object P ∈ C . Then, P //G is a groupoid object in C . Remark 3.20
Theorem 3.19 is important for us since will need to show that functors
L : H → H ′ between ∞ -topoi that preserve finite products and geometric realisations map group actions to groupactions (see Theorem 3.32). In [NSS15], group actions are defined to be groupoid objects, but functors L : H → H ′ as above do not preserve groupoid objects in general. However, Theorem 3.19 showsthat, as in the classical case of (set-theoretic) group actions, actions of group objects in ∞ -topoi are automatically a groupoid objects. ⊳ We prove Theorem 3.19 in Appendix A. For the remainder of this section, let H be an ∞ -topos. Definition 3.21
Let b G ∈ G rp( H ) be a group object. A G -action over an object X ∈ H is an augmentedsimplicial object b X ∈ F un( ∆ op+ , H ) whose underlying simplicial object is a G -action P //G on someobject P ∈ H , and whose augmenting object is X , i.e. b X − = X . Writing p : P → X for the morphism b X | ∆ + , ≤ , we also denote a G -action over X by P //G p −→ X ∈ F un( ∆ op+ , H ) . A morphism of G -actions over X ∈ H , ( P //G → X ) f −→ ( Q//G → X ) , is a morphism f in F un( ∆ op+ , H ) as above such that(1) f − = 1 X is the identity on X , and(2) the collapse morphisms P → ∗ and Q → ∗ induce a (weakly) commutative diagram P //G Q//G ∗ //G f | ∆ op of simplicial objects in H . he ∞ -category of G -actions over X ∈ H is the full ∞ -subcategory of F un( ∆ op , H ) / ( X × ( ∗ //G )) on thoseobjects whose underlying simplicial object is a G -action. Observe that an ordinary G -action is equivalent to a G -action over the final object ∗ ∈ H . Example 3.22
For a group object b G ∈ G rp( H ) and an action P //G of b G on an object P ∈ H ,let q : ( ∆ op ) ⊲ → H be a colimiting cocone of the simplicial diagram P //G in H . Observing that ( ∆ op ) ⊲ ∼ = ( ∆ ⊳ ) op ∼ = ( ∆ + ) op , this defines an augmented simplicial object in H , which we denote as q : P //G −→ colim ∆ op H ( P //G ) = | P //G | . Therefore, the data
P //G → |
P //G | form a G -action over | P //G | . In particular, the canonical morphism ∗ //G → B G is of this form. ⊳ Another example of a morphism of this type is the collapse morphism
G//G → ∗ , as we show now:
Proposition 3.23 If b G is a group object in H , then the canonical morphism | G//G | ≃ −→ ∗ is an equivalence.Proof. Since H is presentable, there exists a combinatorial simplicial model category M and an equiv-alence of ∞ -categories H ≃ N ( M ◦ ) [Lur09, Prop. A.3.7.6]. Under this equivalence, colimits in H overdiagrams indexed by ordinary categories correspond to homotopy colimits in M [Lur09, Cor. 4.2.4.8].It now suffices to observe that any simplicial object in M obtained as the decalage of another simplicialobject has an augmentation and extra degeneracies [Rie14, Ste12].Any morphism b A → b G of group objects induces an action of b A on G by the following construction: Proposition 3.24
Let b f : b A → b G be a morphism in G rp( H ) . Define a simplicial object G//A as thepullback
G//A G//G ∗ //A ∗ //G (3.25) in F un( ∆ op , H ) . Then, G//A is an action of b A on G .Proof. We check the axioms in Definition 3.15: axiom (1) follows from the pasting law for pullbacksand the diagram ( G//A ) n ( G//G ) n G ( ∗ //A ) n ( ∗ //G ) n ∗ in which the right-hand square is a pullback for any n ∈ N by construction of G//G .Axiom (2) is readily seen from applying the maps d and s to the diagram (3.25), for n = 0 , .Axiom (3) follows since the morphism G//A −→ ∗ //A induced by the above diagram agrees with themorphism obtained by collapsing the first factor G .18 .3 Principal ∞ -bundles In this subsection, we characterise principal ∞ -bundles and group extensions in ∞ -topoi. Throughoutthis section, let H be an ∞ -topos and let b G ∈ G rp( H ) be a group object in H . Definition 3.26 [NSS15, Def. 3.4] A G -principal ∞ -bundle on X ∈ H is a G -action P //G → X over X such that the augmented simplicial object P //G → X is a colimiting cocone for the simplicialdiagram P //G ∈ F un( ∆ op , H ) . In other words, the augmenting map p : P → X induces an equivalence colim H ∆ op ( P //G ) ≃ −→ X in H .A morphism of G -principal ∞ -bundles on X , denoted ( P //G → X ) −→ ( Q//G → X ) , is a mor-phism of the underlying G -actions over X . The ∞ -category B un G ( X ) of G -principal ∞ -bundles over X is the full ∞ -subcategory of F un( ∆ op , H ) / ( X × ( ∗ //G )) (cf. Definition 3.21) on the G -principal ∞ -bundleson X . Example 3.27
Let b G be a group object in H . For any G -action P //G in H , the morphism P //G →| P //G | is a principal G -bundle in H over | P //G | . As concrete examples of this type, we have alreadyseen that G//G turns G into a principal G -bundle over ∗ ∈ H (Proposition (3.23)), and that ∗ //G turns ∗ into a principal G -bundle over B G (by the definition of B G ). ⊳ We now provide an alternative characterisation of principal ∞ -bundles in an ∞ -topoi. Let b G ∈ G rp( H ) be a group object in H , and let p : P //G → X be a G -action over an object X ∈ H . Let ı : ∆ op+ , ≤ ֒ → ∆ op+ be the inclusion. The identity provides a canonical equivalence η : { p } = ı ∗ ( P //G → X ) ≃ −→ ı ∗ ( ˇ Cp ) = { p } in F un( ∆ op+ , ≤ , H ) ≃ F un(∆ , H ) . Since right Kan extension is a right adjoint, there is an equivalence F un( ∆ op+ , ≤ , H ) (cid:0) ı ∗ ( P //G → X ) , { p } (cid:1) ≃ F un( ∆ op+ , H ) (cid:0) ( P //G → X ) , ˇ Cp (cid:1) of mapping spaces (compare also (2.10)). We denote the image of η under this equivalence by α : ( P //G → X ) −→ ˇ Cp .
Observe that, by construction, the restriction of α along ı is η . We will not distinguish notationallybetween α as defined here and its restriction along the inclusion ∆ op ⊂ ∆ op+ (since α − = 1 X ). Definition 3.28 A G -action P //G −→ X over X ∈ H is called principal if the canonical morphism α : P //G −→ ˇ Cp is an equivalence in F un( ∆ op , H ) . This is an ∞ -categorical version of the principality condition for a group action. It is, in fact,equivalent to the usual principality condition—that the morphism P × G → P × X P is an equivalence—in the following sense (in particular, this implies the converse to [NSS15, Prop. 3.7]): Lemma 3.29
Let
P //G → X be a G -action over X ∈ H . The following are equivalent:(1) The G -action is principal.(2) The diagram P × G PP X d =pr P a = d pp (3.30) is a pullback diagram in H . roof. (1) implies (2) since the action P //G p −→ X is principal precisely if it is equivalent, as anaugmented simplicial object in H , to the Čech nerve ˇ Cp = Ran ι { p } . Thus, the implication followsfrom Proposition 2.8.Conversely, (2) also implies (1): we know from Theorem 3.19 that P //G is a groupoid object. If weadditionally have that (3.30) is a pullback diagram, then we can again apply Proposition 2.8 to obtainthe claim.We can use Lemma 3.29 to give a characterisation of principal ∞ -bundles which can be understoodas encoding directly the classical criteria for principal bundles: a locally trivial map p : P → X and aprincipal G -action over X . Proposition 3.31
Let
P //G p −→ X be a G -action over an object X ∈ H . The following are equivalent:(1) P //G p −→ X is a principal ∞ -bundle (in the sense of Definition 3.26).(2) The morphism p is an effective epimorphism and the action P //G is principal.Proof.
First, observe that since
P //G is a groupoid object in H , and since by assumption the canonicalmorphism | P //G | → X is an equivalence, it follows from Definition 2.11(4) that the canonical morphism α : P //G → ˇ Cp is an equivalence in F un( ∆ op , H ) . In particular, p is an effective epimorphism. Further,it has been shown in [NSS15, Prop. 3.7] that if P //G → X is a principal bundle, then the action P //G satisfies condition (2) of Lemma 3.29, and so the action is principal.To see the other direction, consider the commutative diagram | P //G | | ˇ Cp | X | α | In this case, both the top and the right-hand morphisms in diagram are equivalences. It thus followsthat also the left-hand morphism is an equivalence, which amounts to the fact that
P //G → X is aprincipal G -bundle in the sense of Definition 3.26. Theorem 3.32
Let
L : H → H ′ be a functor between ∞ -topoi which preserves geometric realisationsand finite products. Suppose b G is any group object in H .(1) L maps G -actions P //G p −→ X over X ∈ H to L G -actions L P // L G L p −→ L X over L X ∈ H ′ .(2) If the action P //G → X is a principal G -bundle, then the action L P // L G L p −→ L X is a principal L G -bundle.Proof. Since L preserves finite products, the first claim follows readily from Definition 3.15.For the second claim, recall that P //G → X is a principal G -bundle precisely if the map | P //G | → X is an equivalence. Applying the functor L to this morphism, we obtain an equivalence L | P //G | ≃ −→ L X .Since L preserves geometric realisations, and using claim (1), we obtain canonical equivalences | L P // L G | ≃ −→ L | P //G | ≃ −→ L X , which establishes the action L P // L G L p −→ L X as a principal L G -bundle over X . Remark 3.33
The proof of Theorem 3.32 would fail if it were not automatic that group actions aregroupoid objects (Theorem 3.19), since L does not preserve groupoid objects in general. ⊳ roposition 3.34 Let b G be a group object in H , and let P //G → Y be a G -principal ∞ -bundle in H .For any morphism f : X → Y in H , there is a canonical G -action over X on the pullback Q := X × Y P that makes Q//G → X into a G -principal ∞ -bundle on X .Proof. Let c : H −→ F un( ∆ op , H ) be the constant-diagram functor. Consider the pullback diagram c X × c Y ( P //G ) P //G c X c Y ˆ ff ∗ p p c f (3.35)in F un( ∆ op , H ) (or, equivalently, in G pd( H ) ). For any [ n ] ∈ ∆ there exists a canonical equivalence (cid:0) c X × c Y ( P //G ) (cid:1) n ≃ X × Y ( P × G n − ) ≃ ( X × Y P ) × G n − . We use Lemma 3.12 to obtain from these equivalences a canonical pair (up to contractible choices) ofan object ( X × Y P ) //G ∈ F un( ∆ op , H ) , with (( X × Y P ) //G ) n = ( X × Y P ) × G n for all n ∈ N , togetherwith an equivalence ( X × Y P ) //G ≃ −→ c X × c Y ( P //G ) (3.36)of simplicial objects in H . By a slight abuse of notation, we also denote the composition ( X × Y P ) //G ≃ −→ c X × c Y ( P //G ) −→ c X by f ∗ p . It follows by construction that ( X × Y P ) //G f ∗ p −→ X is a G -action over X . We are hence leftto show that it is a principal ∞ -bundle.To that end, we will show that the morphism | ( X × Y P ) //G | −→ X is an equivalence (compare Definition 3.26). Diagram (3.35) is a diagram of the form ∆ × ∆ −→ F un( ∆ op , H ) . Composing with the functor colim H ∆ op = |−| : F un( ∆ op , H ) −→ H we obtain a diagram | ( X × Y P ) //G | | P //G | X Y | ˆ f | f ∗ p p ≃ f (3.37)in H . The right-hand morphism is an equivalence since P //G → Y is assumed to be a principal ∞ -bundle. Using the equivalence (3.36), diagram (3.37) is equivalent to the diagram | c X × c Y ( P //G ) | | P //G | X Y | ˆ f | f ∗ p p ≃ f (3.38)21y the universality of colimits in H , we have a canonical equivalence | c X × c Y ( P //G ) | ≃ X × Y | P //G | . This establishes that the morphism f ∗ p in diagram (3.38) is the pullback of an equivalence in H , andhence that f ∗ p is an equivalence itself.One can now even show that every G -principal ∞ -bundle arises as a pullback of the bundle ( ∗ //G ) → B G . This insight is not new, but has been observed in [NSS15, Prop. 3.13, Thm. 3.17] already. However,in Section 4.3 it will be important to have a good understanding of the classifying map of a principal ∞ -bundle, and so we include a brief treatment of these maps. We start with two short technicallemmas, before constructing for each G -principal ∞ -bundle in H its classifying map. Lemma 3.39
Let b G be a group object in H , and let f : ∗ → B G be the base point of B G . The pullbackof the canonical bundle ( ∗ //G ) → B G along f agrees with the bundle G//G → ∗ .Proof.
Consider the commutative square
G//G ∗ //G ∗ c B G in F un( ∆ op , H ) . By the canonical equivalence G ≃ ΩB G , this diagram is level-wise a pullback, i.e. itis a pullback diagram in F un( ∆ op , H ) . That proves the claim by Proposition 3.34. Definition 3.40 A G -principal ∞ -bundle P //G → X is trivial if it is equivalent in B un G ( X ) to the trivial G -principal ∞ -bundle X × ( G//G ) → X , i.e. if there is an equivalence of simplicial objects in H /X between P //G and X × ( G//G ) that commutes with the canonical morphisms to ∗ //G . Lemma 3.41 [NSS15, Prop. 3.12]
For every G -principal ∞ -bundle P //G → X in H , there exists aneffective epimorphism U → X such that the pullback bundle U × X ( P //G ) is trivial.Proof. We give an alternative proof to [NSS15]. Given a G -principal ∞ -bundle P //G → X in H , con-sider the effective epimorphism P → X and the pullback bundle P × X ( P //G ) . We have a commutativediagram P × ( G//G ) P × X ( P //G ) P //G c P c X pr a × ψ in F un( ∆ op , H ) , where a × acts on P with the first copy of G and as the identity on the remainingcopies of G . The induced morphism ψ is a morphism of G -principal ∞ -bundles (since the triangles inthe diagram commute and since a × is a morphism of G -actions). It is thus equivalent to a morphism ψ ′ : ˇ C ( P × G → P ) −→ ˇ C (cid:0) ( P × X ( P //G )) −→ P (cid:1)
22f Čech nerves over P . The level-zero component of ψ ′ is precisely the equivalence P × G → P × X P which establishes that P //G → X is principal (cf. Proposition 3.29). Since ψ ′ is the image of ψ under the right Kan extension Ran ι (compare Definition 2.7), it follows that ψ ′ , and hence ψ , is anequivalence. Proposition 3.42
For every G -principal ∞ -bundle P //G → X in H , the diagram P //G ∗ //G c X c B G p | p | (3.43) is a pullback diagram in F un( ∆ op , H ) : there is an equivalence ( P //G → X ) ≃ × B G ( ∗ //G ) of G -principal ∞ -bundles over X . In particular, every G -principal ∞ -bundle is a pullback of the bundle ∗ //G → B G . This is a refinement of [NSS15, Prop. 3.13] to a statement on the level of simplicial objects, ratherthan only on their zeroth level.
Proof.
Consider the diagram
G//G ∗ //GP × ( G//G ) P //G ∗ c B G c P c X (3.44)in F un( ∆ op , H ) . Here, the front and back squares are pullbacks (by Lemmas 3.41 and 3.39), and thediagram is obtained as a morphism of pullback diagrams. We need to show that the right-hand face isa pullback square in F un( ∆ op , H ) .First, we show that the top square of (3.44) is a pullback. By Lemma 3.8 it suffices to check thislevel-wise: at simplicial level n = 0 , it is trivial. For n ∈ N , the square consists of the the image underthe functor ( − ) × G n − of the diagram P × G PG ∗ a pr (3.45)This a pullback diagram: there is a commutative diagram P × G P × G PG ∗ pr pr g a pr
23n which the dashed morphism is given by g = ( a × G ) ◦ (1 P × inv × G ) ◦ (1 P ◦ ∆ G ) , where ∆ G : G → G is the diagonal morphism, and where inv : G → G is the choice of an inverse in G : since the groupobject b G ∈ G rp( H ) is in particular a groupoid object, we have a diagram b G ≃ b G × ∗ b G × b G ≃ b G (Λ ) b G b G , ≃ d where we have used the characterisation of groupoid objects as certain category objects from Propo-sition A.5. Choosing an inverse for the right-facing morphism defines the morphism inv .Since g is an equivalence (because b G is a group object), diagram (3.45) is a pullback in H , andsince the span category { , } ← { } → { , } has contractible nerve, the pullback (3.45) is preservedby ( − ) × G n − (see Lemma A.9). We thus obtain that the top square in diagram (3.44) is a pullback.Next, we prove that the bottom square of (3.44) is a pullback. We define Y := ∗ × B G X ∈ H , andwe consider the diagram (omitting constant-diagram functors) Y × X ( P //G ) Y ∗ P //G X B G Both squares in this diagram are pullbacks in F un( ∆ op , H ) , so that the pasting law yields a canonicalequivalence of simplicial objects Y × X ( P //G ) ≃ ∗ × B G ( P //G ) . Observe that Y × X ( P //G ) → Y is a G -principal ∞ -bundle by Proposition 3.34, so that Y ≃ (cid:12)(cid:12) Y × X ( P //G ) (cid:12)(cid:12) ≃ (cid:12)(cid:12) ∗ × B G ( P //G ) (cid:12)(cid:12) . Now we use that the morphism
P //G → B G factors through ∗ //G (by Definitions 3.15 and 3.26) andthat ∗ × B G ( ∗ //G ) ≃ G//G (by Lemma 3.39). Applying the pasting law to the diagram ( P //G ) × ( ∗ //G ) ( G//G ) G//G ∗ P //G ∗ //G B G in F un( ∆ op , H ) , in which both squares are pullbacks, we obtain a canonical equivalence ∗ × B G ( P //G ) ≃ ( P //G ) × ( ∗ //G ) ( G//G ) . The right-hand side is precisely the pullback described by the top square in diagram (3.44), and ishence canonically equivalent to P × ( G//G ) in F un( ∆ op , H ) . Thus, it follows that Y ≃ (cid:12)(cid:12) ∗ × B G ( P //G ) (cid:12)(cid:12) ≃ (cid:12)(cid:12) P × ( G//G ) (cid:12)(cid:12) ≃ P .
The last equivalence can be seen either by combining Proposition 3.23 with the fact that |−| preservesfinite products (because ∆ op is sifted [Lur09]), or simply by recalling that P × ( G//G ) → P is a G -principal ∞ -bundle on P . This shows that the bottom square in (3.44) is a pullback.24inally, we prove that the right-hand square in (3.44) is a pullback as well. Consider the commu-tative diagram of solid arrows P //G X × B G ( ∗ //G ) ∗ //GX B G ϕ which induces an essentially unique morphism ϕ of simplicial objects in H . By the commutativity ofthe right-hand triangle in this diagram ϕ is even a morphism of G -actions, and by the commutativity ofthe left-hand triangle it is a morphism of G -actions over X . Since its source and target are G -principal ∞ -bundles, ϕ is equivalent to a morphism of Čech nerves ϕ ′ : ˇ C ( P → X ) −→ ˇ C (cid:0) ( X × B G ∗ ) → X (cid:1) . That is, ϕ ′ is the image under Ran ι (compare Definition 2.7) of the square P X × B G ∗ X X ϕ ′ This ϕ ′ is an equivalence precisely because the bottom square of (3.44) is a pullback. Consequently, themorphism ϕ is an equivalence in F un( ∆ op , H ) , and thus the right-hand face in (3.44) is a pullback. Corollary 3.46
Let
P //G → X be a G -principal ∞ -bundle in H . For any morphism x : ∗ → X , wehave a pullback diagram G//G P //G ∗ X x in F un( ∆ op , H ) . In particular, any fibre of P → X is canonically equivalent to G in H . Remark 3.47
In fact, for any group object b G in H and any object X ∈ H , there is an equivalence B un G ( X ) ≃ H ( X, B G ) between the ∞ -category of G -principal ∞ -bundles on X and the mapping space H ( X, B G ) [NSS15,Thm. 3.17]. This implies that every morphism of principal G -bundles on X is an equivalence. Proposi-tion 3.42 feeds into the proof of this equivalence by showing that the functor H ( X, B G ) → B un G ( X ) ,sending a morphism X → B G to the principal ∞ -bundle X × B G ( ∗ //G ) , is fully faithful. ⊳ In particular, under the equivalence of Remark 3.47, the morphism | p | : X → B G in diagram (3.43)is a classifying morphism for the bundle P //G → X . Proposition 3.48
Let
L : H → H ′ be a functor of ∞ -topoi that preserves finite products and geometricrealisations. If P //G → X is a G -principal ∞ -bundle in H , classified (up to canonical equivalence) bya morphism | p | : X → B G , then the L G -principal ∞ -bundle L P // L G −→ L X (compare Theorem 3.32)in H ′ is classified by the morphism | L p | ≃ L | p | . roof. Consider the commutative diagram L( P //G ) L( ∗ //G )L P // L G ∗ // L G L | P //G | LB G | L P // L G | BL G L pq ≃ ≃ L | p || q |≃ ≃ The morphism q is the canonical morphism induced from the collapse morphism L P → ∗ . The frontface of this diagram is a pullback in H ′ , witnessing | q | as the classifying morphism L X → BL G ofthe bundle L P // L G −→ L X . Since all diagonal morphisms are equivalences, the back face of thediagram is a pullback as well, showing that L | p | is a classifying morphism of the L G -principal ∞ -bundle L( P //G ) −→ L X , which is equivalent to the bundle L P // L G −→ L X . Finally, since thediagonal morphisms arise from the natural equivalences L ◦ |−| ≃ |−| ◦ L , it follows that | q | ≃ | L p | .We now state several alternative characterisations of group extensions in ∞ -topoi. These clarifythe relation between the original notion of an extension of group objects from Definition 3.10 and moredirect categorifications of several perspectives on group extensions in S et . The last of these alternativecharacterisations will be important in Section 4.3. Theorem 3.49
Let H be an ∞ -topos, and let b A b ι −→ b G b p −→ b H be a sequence of morphisms in G rp( H ) .The following are equivalent:(1) b A b ι −→ b G b p −→ b H is an extension of group objects in H (see Definition 3.10).(2) The sequence b A b ι −→ b G b p −→ b H is a fibre sequence in G rp( H ) .(3) The sequence A ι −→ G p −→ H is a fibre sequence in H .(4) The map p : G → H together with the action G//A of A on G induced by ι define a principal A -bundle over H .Proof. (1) ⇒ (3) : This implication has been proven in [NSS15] already. We import the proof forcompleteness: consider the diagram A G ∗∗ H B A ∗∗ B G B H ι ′ p ′ B ι B p (3.50)in H . Every square in diagram (3.50) is a pullback square (this assumes (1)). It thus follows thatthe sequence A ι ′ −→ G p ′ −→ H is a fibre sequence in H . By this construction, the morphisms ι ′ and p ′ coincide with the morphisms Ω ◦ B( ι ) and Ω ◦ B( p ) , respectively. The natural equivalence Ω ◦ B ≃ id H then yields that also A ι −→ G p −→ H is a fibre sequence in H . Observe that every vertical morphism indiagram (3.50) is an effective epimorphism since ∗ → B G is an effective epimorphism for every groupobject b G ∈ G rp( H ) and since effective epimorphisms are stable under pullback.26 ⇒ (2) : This follows from Proposition 3.8. (2) ⇒ (1) : This implication holds because the delooping functor B is a right adjoint. (4) ⇒ (3) : Consider the commutative diagram A G ∗∗ H B A The outer rectangle is a pullback diagram by construction of B A , and the right-hand square is apullback by Corollary 3.46. Thus, the implication follows by applying the pasting law for pullbacks. (1) ⇒ (4) : Recall that the morphism ∗ → B H is an effective epimorphism and that effectiveepimorphisms are stable under pullback. It then readily follows from the pasting diagram (3.50)that p : G → H is an effective epimorphism. By Proposition 3.34, the morphism H → B A inducesan A -principal ∞ -bundle over H as the pullback of ∗ //A → B A . We know from the pasting con-struction (3.50) that the level-zero object of the induced pullback principal ∞ -bundle is (canonically)equivalent to G . Further, we have that the map G → H induced from the pullback constructioncoincides with p : G → H under this equivalence (as in the proof of “ (1) ⇒ (3) ”).It thus remains to show that the action of A on G obtained via the upper central pullback squarein (3.50) and the pullback construction in Proposition 3.34 coincides with the action G//A inducedfrom the morphism b ι : b A → b G in G rp( H ) (cf. Proposition 3.24).To that end, we consider the following diagram in F un( ∆ op , H ) : G//G ∗ //G ( H × B A ∗ ) //A ∗ //A ∗ c B G c H c B A pr H g ι c B ι (3.51)The simplicial object in the front, upper-left corner is obtained from the pullback construction for A -principal ∞ -bundles (Proposition 3.34). Therefore, the front face of the cube in diagram (3.51)is a pullback square in F un( ∆ op , H ) . Further, the bottom square is a pullback by assuming (1) andby (3.50), and the back square is a pullback diagram because it is so objectwise (see also Lemma 3.39).Forgetting for a moment about the two upper left objects, the remaining diagram can be viewed as amorphism of cospans in H . This induces an essentially unique morphism of the left and right pullbacksquares, and this is how we define the upper-left diagonal morphism, labelled g , and the remainingcoherence data of the cube (3.51).Applying the pasting law to the front and bottom square, we deduce that the diagonal square ( H × B A ∗ ) //A ∗ //A ∗ c B G
27s a pullback diagram in F un( ∆ op , H ) . Its vertical morphisms are the diagonals of the left and rightfaces of the cube in diagram (3.51). This, in turn, yields a commutative diagram ( H × B A ∗ ) //A G//G ∗∗ //A ∗ //G c B G in which the outer square and the right-hand square are pullbacks. (These squares are precisely the topand the back squares of diagram (3.51).) The pasting law for pullbacks thus implies that the left-handsquare is a pullback diagram in F un( ∆ op , H ) as well. By construction of the action G//A induced fromthe morphism b ι : b A → b G in G rp( H ) (Proposition 3.24) as a pullback in F un( ∆ op , H ) , we thus obtain acanonical equivalence ( H × B A ∗ ) //A ≃ G//A in G pd( H ) . That is, the two actions of A on G agree, as desired. Since the action ( H × B A ∗ ) //A isprincipal by Proposition 3.34, it follows that the action G//A is principal as well. This proves theclaim.
Corollary 3.52
Suppose b A b ι −→ b G b p −→ b H is an extension of group objects in H . Then, there is a canonicalequivalence in H , | G//A | ≃
H .
This is the ∞ -categorical analogue of the canonical isomorphism G/A ∼ = H for ordinary (set-theoretic) group extensions A → G → H . Corollary 3.53
Let
L : H → H ′ be a functor between ∞ -topoi that preserves geometric realisationsand finite products. Suppose b A b ι −→ b G b p −→ b H is an extension of group objects in H . Then, the sequence L b A L b ι −→ L b G L b p −→ L b H is an extension of group objects in H ′ .Proof. This statement now follows from combining Theorem 3.32 and Theorem 3.49.
In this section, we present a definition of string group extensions within the ∞ -category H ∞ of smoothspaces. This relies on the singular complex functor S e : H ∞ → S for smooth spaces from Section 2and the theory of group extensions in ∞ -topoi from Section 3. We begin by recalling the definition ofa string group extension in the ∞ -category S of spaces. Then, we use our results thus far to transferthis definition to H ∞ along the functor S e , leading to a homotopy-theoretic definition of smooth stringgroup extensions (Definition 4.2).After recalling some background on bundle gerbes in Section 4.2, we provide new smooth modelsfor the string group in Section 4.3, building on recent constructions of smooth 2-group extensionsin [BMS]. (Already in that paper, evidence was given that these smooth 2-group extensions can modelthe string group; here we provide a full formal framework and proof for that conjecture.)28 .1 The definition of smooth string groups The definition of a string group via the Whitehead tower (see Section 1) is originally purely homotopy-theoretic. In particular, in a string group extension A → String( H ) → H the extending group A is notfixed, but only its homotopy type. So far, to our knowledge there does not exist a definition of stringgroup extensions in a smooth context that contains this flexibility—the extending group A is usuallychosen ad hoc to be some smooth version of B U (1) . Here, we provide a smooth version of the originalhomotopy-theoretic definition (see Definition 4.2). In particular, only the underlying homotopy typeof the extending smooth group A is fixed in this definition.Recall that any compact, simple, and simply connected Lie group H is also 2-connected and satisfies H ( H ; Z ) ∼ = Z . Any Lie group H defines a group object b H in the ∞ -topos of spaces S . We start byreformulating the definition of a string group extension of topological groups within the ∞ -category ofspaces: Definition 4.1
Let H be a compact, simple, and simply connected Lie group, and denote by b H ∈ G rp( S ) its associated group object in S . A string group extension of H is an extension of group objects b A b ι −−→ \ String( H ) b p −−→ b H in S such that(1) A is an Eilenberg-MacLane space K ( Z , , and(2) under the isomorphism π S ( H, B A ) ∼ = π S (cid:0) H, K ( Z , (cid:1) ∼ = H ( H ; Z ) ∼ = Z , the classifying morphism H → B A of the A -principal ∞ -bundle String( H ) //A → H (compareRemark 3.47 and Theorem 3.49(4)) represents a generator of Z . Given condition (1), condition (2) is equivalent to saying that the map G → H of spaces induces anisomorphism π i ( G ) → π i ( H ) for i = 3 and that π ( G ) ∼ = 0 . This is a consequence of the Hurewicz The-orem, the Universal Coefficient Theorem, and the long exact sequence of homotopy groups associatedto a (homotopy) fibre sequence of spaces. That is, G is a 3-connected approximation to H .Recall the ∞ -topos H ∞ = P ( C art) from Section 2.1. There we also introduced the localisation L I H ∞ of H ∞ at the set I = { c × R → c | c ∈ C art } and the smooth singular complex functor S e : H ∞ → S . Further, recall the fully faithful embedding ( − ) : M fd ֒ → H ∞ , with M ( c ) = M fd( c, M ) ;under this embedding, any Lie group H gives rise to a group object b H in H ∞ . We can now use ourresults from Section 3 to transfer the definition of a string group extension to the ∞ -topos H ∞ : Definition 4.2
Let H be a compact, simple, and simply connected Lie group. A smooth string groupextension of H is an extension b A b ι −−→ \ String( H ) b p −−→ b H of group objects in H ∞ such that its imageunder S e is a string group extension in S . Note that by Theorem 3.53 the functor S e maps group extensions in H ∞ to group extensions in S .Further, even though S e induces an equivalence between S and the localisation L I H ∞ rather than thefull ∞ -category H ∞ , we do not need to demand that A , String( H ) and H are local objects, because S e sends all I -local equivalences in H ∞ to equivalences in S (Theorem 2.2(1)).Definition 4.2 is a generalisation as well as a weakening of the following approach to smooth stringgroup extensions (see, for instance, [FRS16]): there, one works in the localisation L τ H ∞ of H ∞ at thedifferentiably good open coverings { c a → c } a ∈ Λ of cartesian spaces c ∈ C art , and one defines a string29roup extension of H via the pullback BString( H ) ∗ B H B U(1) p (4.3)Here, p denotes the fractional first Pontryagin class, which is a generator of H (B H ; Z ) ∼ = Z . However,this definition of String( H ) is considerably stricter than the original perception of String( H ) as a 3-connected covering of H by another group object (Definition 4.1). For instance, the definition of astring group extension based on (4.3) enforces that String( H ) → H is a B U (1) -principal ∞ -bundle.(Note that if H is an ∞ -topos and b A ∈ G rp( H ) is a group object whose multiplication lifts to an E -algebra structure, then B A is canonically the level-zero object of a group object in H [NSS15].)However, from the purely homotopy-theoretic point of view, not the actual fibre of this map shouldbe fixed, but the homotopy type of its underlying space (which must be a K ( Z ; 2) ). Definition 4.2emphasises this latter, homotopy-theoretic aspect of string group extensions.More concretely, for smooth string group extensions b A b ι −→ \ String( H ) b p −−→ b H in the sense of Def-inition 4.2 it is enough if there is an I -local equivalence A ≃ U (1) in H ∞ . Therefore, this setup isconsiderably more general than working with the pullback (4.3); in particular, two different smoothstring group extensions of a Lie group H need not be equivalent in H ∞ , but only in L I H ∞ . Remark 4.4
It will be interesting to see a Lie-algebra version of Definition 4.2. The ∞ -groups b A ∈ G rp( H ∞ ) that we allow to appear in string group extensions can have much larger Lie algebrasthan those which appear in the stricter definition via (4.3). This is true, in particular, for the smoothstring group extension we present in Section 4.3 below. There might hence be a Lie-algebra version of I -local equivalences of group objects in H ∞ . ⊳ We now work towards establishing a new string group model that fits Definition 4.2, but does notfit into the pullback (4.3). (The reason will be that the extension is not by B U (1) , but by a muchlarger, but also homotopically correct, ∞ -group.) Before we can present our smooth string group extension, we need to recall some background on bundlegerbes. We will not give full definitions or details here; for these, we refer the reader to [Wal07, Bun17,BS17, BSS18]. Bundle gerbes provide an explicit, geometric model for categorified line bundles. Wepoint out that there also exists a notion of connection on a bundle gerbe, but here we will only beworking with bundle gerbes without connection. (This is the main technical cause for the distinctionbetween our smooth string group model and that in [FRS16].) Further, bundle gerbes are a verygeometric, 2-categorical model for higher line bundles, so that our smooth string group model willarise as an explicitly defined smooth 2-group, making it particularly tangible.To any manifold M , we can assign a symmetric monoidal 2-groupoid ( G rb( M ) , ⊗ ) of bundle gerbeson M . Given a bundle gerbe G ∈ G rb( M ) , the monoidal groupoid G rb( M )( G , G ) of automorphisms of G is canonically equivalent to the symmetric monoidal groupoid (HLB( M ) , ⊗ ) of hermitean line bundleson M with the usual tensor product (which we also denote by ⊗ ). Note that (HLB( M ) , ⊗ ) is even a ; that is, it is a symmetric monoidal groupoid in which every object has an inverse with respect30o the monoidal product. Every smooth map f : N → M of manifolds induces a monoidal 2-functor f ∗ : G rb( M ) −→ G rb( N ) . Isomorphism classes of gerbes are in canonical bijection with the third integer cohomology of M : thereis an isomorphism of abelian groups π (cid:0) G rb( M ) , ⊗ (cid:1) ∼ = H ( M ; Z ) . (4.5)The class associated to a gerbe G under this isomorphism is called the Dixmier-Douady class of G .We let H ≤ denote the following 2-category: its objects are functors π : C → C art that areGrothendieck fibrations in groupoids (that is, π is a Grothendieck fibration and all its fibres aregroupoids). Its morphisms ( π : C → C art) −→ ( π ′ : C ′ → C art) are functors F : C → C ′ such that π ′ ◦ F = π , and its 2-morphisms are natural transformations η : F → F ′ such that π ′ η is the iden-tity natural transformation C art → C art . Note that the 2-category H ≤ is canonically equivalent tothe 2-category of pseudo-functors C art op → G pd from C art op to the 2-category of groupoids via theGrothendieck construction. We make the following definitions; for more background, see [BMS, SP11]. Definition 4.6 [BMS]
The is the 2-category of group objects in the2-category H ≤ . Example 4.7
Let H be a Lie group. We associate to it the following category, denoted by R H : itsobjects are pairs ( c, h ) of a cartesian space c ∈ C art and a smooth map h : c → H . A morphism ( c, h ) → ( c ′ , h ′ ) is a smooth map f : c → c ′ such that h ′ ◦ f = h , and the category R H comes witha canonical projection functor R H → C art . The product on H -valued maps turns R H into a smooth2-group in the sense of Definition 4.6. Note that R H is simply the Grothendieck construction of thepresheaf of sets H on C art . ⊳ Example 4.8
Let M be a manifold, and define a category HLB M as follows: its objects are pairs ( c, L ) of a cartesian space c ∈ C art and a hermitean line bundle L ∈ HLB( c × M ) . A morphism ( c, L ) → ( c ′ , L ′ ) is a pair ( f, ψ ) of a smooth map f : c → c ′ and an isomorphism ψ : L → ( f × M ) ∗ L ′ of hermitean line bundles over c . This category comes with a projection functor HLB M → C art . Thetensor product of hermitean line bundles turns HLB M into a smooth 2-group. ⊳ Let M be a manifold, and let G ∈ G rb( M ) be a gerbe on M . Further, let H be a connectedLie group acting smoothly on M from the left; we denote the action by Φ : H × M → M . Giventhese data, we define a category Sym( G ) as follows: an object in Sym( G ) is a triple ( c, h, A ) , where c ∈ C art is a cartesian space and where h : c → H is a smooth map. These give rise to a smooth map Φ h : c × M → c × M , defined as the composition Φ h : c × M ∆ × M −−−−→ c × c × M c × h × M −−−−−−→ c × H × M c × Φ −−−→ c × M , where ∆ : c → c × c is the diagonal map. Then, A is a 1-isomorphism A : pr ∗ M G −→ Φ ∗ h G of gerbes on the manifold c × M . A morphism ( c, h, A ) → ( c ′ , h ′ , A ′ ) is a pair ( f, ψ ) , where f is asmooth map f : c → c ′ such that h ′ ◦ f = h , and where ψ is a 2-isomorphism ψ : A −→ ( f × M ) ∗ A ′ .(Here we have implicitly used that there is a canonical 1-isomorphism ( f × M ) ∗ Φ ∗ h ′ G ∼ = Φ ∗ h G .) Observethat there is a projection functor p : Sym( G ) → R H , acting as ( c, h, A ) ( c, h ) and ( f, ψ ) f .31 emark 4.9 In this set-up, the following statements hold true:(1) The connectedness of H ensures that the functor p is surjective on objects. It is an essentiallysurjective Grothendieck fibration in groupoids [BMS, Thm. 5.27].(2) The equivalence G rb( N )( G ′ , G ′ ) ≃ (HLB( N ) , ⊗ ) for any gerbe G ′ on any manifold N implies thatthe diagram HLB M Sym( G ) ∗ R H e H is a pullback in H ≤ , where e H is the functor that sends c ∈ C art to the constant map c → H with value the unit element of H . Since p is a Grothendieck fibration in groupoids, this pullbackis even a homotopy pullback [BMS, App. A.1]. ⊳ Theorem 4.10 [BMS, Thms. 5.23, 5.27]
Let
Φ : H × M → M be a smooth action of a connected Liegroup H on a manifold M . Let G ∈ G rb( M ) be a bundle gerbe on M .(1) Sym( G ) is a smooth 2-group.(2) The functor p fits into a sequence HLB
M i −→ Sym( G ) p −→ R H (4.11) of smooth 2-groups. Further, p is a Grothendieck fibration in groupoids and surjective on objects. The nerve functor N : C at → C at ∞ induces a functor N : H ≤ → H ∞ (where we have used thecanonical equivalence between H ≤ and the 2-category of pseudo-functors C art op → G pd from C art op to the 2-category of groupoids). This functor, in particular, preserves final objects and products, sothat it maps smooth 2-groups to group objects in H ∞ . Our smooth string group model will be obtainedby applying this functor to the sequence (4.11). We can now state the main theorem of this section. It provides a new smooth model for smooth stringgroup extensions which fits Definition 4.2, but which lies outside the scope of the stricter definition viathe pullback (4.3). Note that applying the nerve functor N to R H ∈ H ≤ yields the familiar presheafof spaces H ∈ H ∞ , defined via H ( c ) = M fd( c, H ) for cartesian spaces c ∈ C art . Theorem 4.12
Let H be a compact, simple, simply connected Lie group. We consider the left-actionof H on itself via left multiplication. Let G ∈ G rb( H ) be a gerbe on H whose class in H ( H ; Z ) ∼ = Z isa generator (see (4.5) ). Then, the sequence N \ HLB H N \ Sym( G ) b H c Ni c Np (4.13) is a smooth string group extension of H . The proof of Theorem 4.12 will occupy the remainder of this section. By Definition 4.2 we have toshow that the sequence (4.13) is an extension of group objects in H ∞ and that its image under thefunctor S e : H ∞ → S is a string group extension in S in the sense of Definition 4.1. Proposition 4.14
The sequence (4.13) is an extension of group objects in the ∞ -topos H ∞ . roof. The nerve functor N : C at → C at ∞ is a right adjoint and hence maps products in H ≤ toproducts in H ∞ , and it maps the final object in H ≤ to the final object in H ∞ . Consequently, itpreserves group objects and group actions.We will now use the characterisation of group extensions from Theorem 3.49(4) to show that thesequence (4.13) of group objects in H ∞ is an extension of group objects. That is, we have to showthat N Sym( G ) with the N HLB H -action induced by the morphism c N i (cf. Proposition 3.24) is an N HLB M -principal ∞ -bundle over H . According to the characterisation of principal ∞ -bundles inProposition 3.31, it suffices to prove that the morphism N p is an effective epimorphism and that theaction of N HLB M on N Sym( G ) is principal.We start by showing that the morphism N p is an effective epimorphism: by [BMS, Sec. 5.1]the restriction p | c of p to any fibre is essentially surjective, hence N p | c is surjective on connectedcomponents. Since H ∞ is a presheaf ∞ -topos (in which limits and colimits are computed objectwise),a morphism in H ∞ is an effective epimorphism if and only if it is objectwise an effective epimorphismin S . The effective epimorphisms in S , however, are exactly those morphisms which are surjective onconnected components [Lur09, Cor. 7.2.1.15]. Therefore, N p is an effective epimorphism in H ∞ .The action of N HLB H on N Sym( G ) is principal with respect to N p as was shown in [BMS,Thm. 5.27]. (There, the principality condition was shown on the level of the sequence (4.11) of smooth2-groups—this suffices for the ∞ -categorical context used here because of Lemma 3.29 and becausethe nerve functor is a right adjoint.) Therefore, the sequence (4.13) is a group extension in H ∞ .It thus remains to show that the image of the sequence (4.13) under S e is a string group extensionin S . To that end, we first show the following lemma: Lemma 4.15
Let X be a connected manifold with H ( X ; Z ) ∼ = 0 .(1) The object N HLB X ∈ H ∞ is equivalent to the object B( U (1) X ) .(2) Suppose that also π ( N ) = 0 . Then, the object N HLB X ∈ H ∞ is I -locally equivalent to B U (1) ∈ H ∞ .Both equivalences are even established by morphisms of group objects in H ∞ . Since S e maps I -local equivalences in H ∞ to equivalences of spaces, applying Lemma 4.15 to X = H establishes axiom (1) of Definition 4.1 for the image of the sequence (4.13) under the functor S e . Proof.
We proceed in parallel to the proof of [BMS, Thm. 8.7]: since any c ∈ C art is contractible andsince H ( X ; Z ) ∼ = 0 , it follows that any hermitean line bundle on c × N is trivialisable. Consequently, HLB X ( c ) is equivalent to the groupoid with one object and morphisms given by the group U (1)( c ) ofsmooth maps from c to U (1) . This induces an equivalence N HLB X ≃ B( U (1) X ) in H ∞ , which extendsto a morphism of group objects in H ∞ . This proves (1).Next, since π ( X ) is trivial, there exists a smooth homotopy equivalence ev e : U (1) X → U (1) , givenby restricting a smooth map c × X → U (1) to c × { x } , where x ∈ X is any point. A homotopyinverse to ev x is given by pulling a smooth map c → U (1) back along the projection c × X → c [BMS,Lemma 8.9]. In particular, ev x is an I -local equivalence [Bunb, Cor. 3.16].Observe that ev x induces a morphism of group objects b ev x : (cid:0) U (1) X (cid:1)b −→ d U (1) . Since ev x is an I -local equivalence in H ∞ and I -local equivalences are closed under finite products(Proposition 2.4), the morphism b ev x is a levelwise I -local equivalence of simplicial objects in H ∞ .33urther, the class W I of I -local equivalences in H ∞ is strongly saturated [Lur09, Lemma 5.5.4.11].In particular, the full subcategory of F un(∆ , H ∞ ) on the I -local equivalences is stable under colimits.Therefore, taking the colimit in H ∞ of simplicial objects (i.e. taking geometric realisations), we obtainan I -local equivalence B ev x : B (cid:0) U (1) X (cid:1) −→ B U (1) in H ∞ . Composing with the morphism constructed in part (1), we now obtain the desired I -localequivalence N HLB X −→ B U (1) in H ∞ .We are thus left to show that the sequence of group objects in S obtained by applying the functor S e to the sequence (4.13) of group objects in H ∞ satisfies axiom (2) of Definition 4.1. That is, we haveto show that the principal ∞ -bundle of spaces (cid:0) S e N Sym( G ) (cid:1) // (cid:0) S e N HLB H (cid:1) −→ S e H represents a generator of H ( H ; Z ) ∼ = Z . This is best checked using Čech cohomology.Recall that a differentiably good open covering of c ∈ C art is an open covering { c a ֒ → c } a ∈ Λ suchthat every finite non-empty intersection of the images of the patches c a is again a cartesian space. Thedifferentiably good open coverings endow C art with a Grothendieck pretopology τ [FSS12, Sch]. Lemma 4.16
Let X ∈ M fd be a connected manifold, and let k ∈ N . Then,(1) In H ∞ , there is an I -local equivalence B k ( U (1) X ) ≃ B k U (1) . (2) Suppose X is simply connected. Then, the presheaf B k ( U (1) X ) satisfies descent with respect to theGrothendieck pretopology τ .Proof. Ad (1): This is an iteration of the argument in the proof of Lemma 4.15(2).Ad (2): We prove this claim by induction. For k = 0 , we have to check that the functor C art op → S , c M fd( c × X, U (1)) satisfies descent with respect to good open coverings of c . However, this followsdirectly from the fact that, for any manifold Y , the assignment functor O p( Y ) op → S et , U M fd (cid:0) U, U (1) (cid:1) defines a sheaf on Y , where O p( Y ) is the category of open subsets of Y and their inclusions.Suppose that B l ( U (1) X ) is a sheaf on C art for all l = 0 , . . . , k . Let U = { c a ֒ → c } a ∈ Λ be adifferentiably good open covering of c . We have to show that the canonical morphism q ∗ : B k +1 (cid:0) U (1) X (cid:1) ( c ) −→ lim n ∈ ∆ S H ∞ (cid:0) ˇ C U n , B k +1 U (1) X (cid:1) (4.17)is an equivalence of spaces. Here, ˇ C U ∈ F un( ∆ op , H ∞ ) is the Čech nerve of the covering U .We first show that q ∗ is essentially surjective; that is, it induces a bijection on isomorphism classesof objects. Since limits and colimits in H ∞ = F un( C art op , S ) are computed pointwise, we have isomor-phisms π B k +1 (cid:0) U (1) X (cid:1) ( c ) ∼ = π B k +1 (cid:0) U (1) X ( c ) (cid:1) = ∗ . On the other hand, we have that π lim n ∈ ∆ S H ∞ (cid:0) ˇ C U n , B k +1 U (1) X (cid:1) ∼ = ˇH k +1 (cid:0) U ; U (1) X (cid:1) ∼ = H k +2 ( c ; Z ) ∼ = ∗ , (4.18)34here on the right-hand side we have the usual Čech cohomology group with respect to the covering U of the sheaf of abelian groups on c given by O p( c ) op → A b , U M fd (cid:0) U × X, U (1) (cid:1) . By a slight abuse of notation, we also denote this sheaf by U (1) X .This Čech cohomology is indeed isomorphic to the full sheaf cohomology of U (1) X : first, we observethat since X is simply connected, there is a short exact sequence Z −→ R X −→ U (1) X . We further observe that the sheaf R X is fine (it admits partitions of unity, for instance those inducedfrom the canonical map R → R X ) when seen as a sheaf on the open subsets of any manifold. Therefore,for any manifold Y , we have a canonical isomorphism H l (cid:0) Y ; U (1) X (cid:1) ∼ = H l +1 ( Y ; Z ) for every l ≥ . From this, we see that H l (cid:0) d ; U (1) X (cid:1) ∼ = ∗ for every cartesian space d ∈ C art and l ≥ . Since the covering U of c is differentiably good, it followsfrom [Bry08, Thm. 1.3.6] that there is a canonical isomorphism ˇH k (cid:0) U ; U (1) X (cid:1) ∼ = H k (cid:0) c ; U (1) X (cid:1) . This proves the isomorphisms in (4.18), and hence that the morphism q ∗ from (4.17) is bijective onconnected components.It remains to check that q ∗ is an isomorphism on all higher homotopy groups. We will achieve thisby comparing the automorphisms of the unique object in the source and target space of q ∗ . On thesource side, this automorphism space is given as the pullback of spaces ΩB k +1 (cid:0) U (1) X (cid:1) ( c ) ∗∗ B k +1 (cid:0) U (1) X (cid:1) ( c ) and there is a canonical equivalence ΩB k +1 (cid:0) U (1) X (cid:1) ( c ) ≃ B k (cid:0) U (1) X ( c ) (cid:1) . On the target side of q ∗ , the automorphism space of the (essentially) unique object is the pullback Ω lim n ∈ ∆ S H ∞ (cid:0) ˇ C U n , B k +1 U (1) X (cid:1) ∗∗ lim n ∈ ∆ S H ∞ (cid:0) ˇ C U n , B k +1 U (1) X (cid:1) H ∞ are computed objectwise, there are canonical equivalences Ω lim n ∈ ∆ S H ∞ (cid:0) ˇ C U n , B k +1 U (1) X (cid:1) ≃ lim n ∈ ∆ S H ∞ (cid:0) ˇ C U n , Ω B k +1 U (1) X (cid:1) ≃ lim n ∈ ∆ S H ∞ (cid:0) ˇ C U n , B k U (1) X (cid:1) . However, by the induction hypothesis, the presheaf B k U (1) X is a sheaf, so that q ∗ induces an equiva-lence between the automorphism spaces. This proves that q ∗ is indeed an equivalence.Another application of [Buna, Thm. 1.2] now implies that the presheaf of spaces H ∞ (cid:0) − , B n (cid:0) U (1) H (cid:1)(cid:1) : M fd op −→ S satisfies descent with respect to open coverings (and even surjective submersions). Consequently, givenany open covering { c a ֒ → H } a ∈ Λ , whose Čech nerve we denote by V → H , the canonical morphism H ∞ ( H, B n (cid:0) U (1) H (cid:1)(cid:1) ∼ = lim ∆ S H ∞ (cid:0) V , B n (cid:0) U (1) H (cid:1)(cid:1) . is an equivalence of spaces because B n ( U (1) H ) is local with respect to the class of morphisms consistingof the Čech nerves of coverings. Therefore, there is an isomorphism π H ∞ ( H, B n (cid:0) U (1) H (cid:1)(cid:1) ∼ = ˇH n (cid:0) H ; U (1) H (cid:1) , (4.19)which can be represented explicitly by composing any morphism H → B n ( U (1) H ) with any Čech nerve V → H of an open covering of H . (Alternatively, this can be seen directly in the presentation of H ∞ by the projective model structure on simplicial presheaves on C art .) Let ev e : U (1) H −→ U (1) bethe morphism induced by pullback along the base-point inclusion ∗ ֒ → H . We obtain a commutativediagram π H ∞ ( H , B n (cid:0) U (1) H (cid:1)(cid:1) π H ∞ ( H, B n U (1) (cid:1) ˇH n (cid:0) H ; U (1) H (cid:1) ˇH n (cid:0) H ; U (1) (cid:1) ∼ = (B n ev e ) ∗ ∼ =(ev e ) ∗ (4.20)It was shown in [BMS, Prop. 8.11] that the bottom horizontal morphism is an isomorphism for all n ∈ N (with n > ). Consider the morphisms π H ∞ (cid:0) H, B n (cid:0) U (1) H (cid:1)(cid:1) −→ π S (cid:0) S e H, S e B n (cid:0) U (1) H (cid:1)(cid:1) (4.21) ∼ = π S (cid:0) H, B n (cid:0) S e U (1) H (cid:1)(cid:1) ∼ = π S (cid:0) H, B n S e U (1) (cid:1) ∼ = π S (cid:0) H, B n U (1) (cid:1) . The first morphism is applying the functor S e . For the second morphism we have used [Bunb, Thm. 5.1]:for every manifold X ∈ M fd , there is a canonical equivalence S e X ≃ X in S . Further, here we haveused that S e commutes with B (Proposition 3.5). For the third morphism, we have used that theinclusion U (1) ֒ → U (1) H is an I -local equivalence in H ∞ : since H is connected and simply connected,this morphism is a smooth homotopy equivalence by [BMS, Lemma 8.9], and by [Bunb, Cor. 3.16]any smooth homotopy equivalence is an I -local equivalence. The last morphism again uses [Bunb,36hm. 5.1]. Since S e preserves finite products, the equivalence S e U (1) ≃ U (1) in S is compatible withthe group structure on U (1) .We can describe the map (4.21) more explicitly as follows: we have already seen above that anyelement in π H ∞ ( H , B n ( U (1) H )) can be described as a smooth U (1) H -valued Čech cocycle with respectto a (differentiably good) open cover V of H . Under the map (4.21), these data are sent first to thesame Čech cocycle, but seen as a map of spaces, and then this resulting Čech cocycle is composed withthe evaluation U (1) H → U (1) at the unit element in H . Therefore, using the canonical isomorphism π S ( H, B n U (1)) ∼ = ˇH n ( H ; U (1)) and combining this with the maps (4.20) and (4.21) we obtain acommutative diagram of abelian groups π H ∞ ( H , B n (cid:0) U (1) H (cid:1)(cid:1) π S (cid:0) H, B n (cid:0) S e U (1) H (cid:1)(cid:1) ˇH n (cid:0) H ; U (1) H (cid:1) ˇH n (cid:0) H ; U (1) (cid:1) (ev e ) ∗ (ev e ) ∗ (4.22)In this diagram, the left-hand vertical morphism is invertible as argued before (4.19). The bottommorphism is an isomorphism by [BMS, Prop. 8.11] and the fact that Čech cohomology and abeliansheaf cohomology are isomorphic on manifolds. The right-hand vertical morphism is invertible as aconsequence of the isomorphisms in (4.20) and the fact that the map ev e : U (1) H → U (1) is an I -localequivalence.Combining diagram (4.22) with Proposition 3.48 and Lemma 4.15, we obtain that the class in H ( H ; Z ) ∼ = ˇH ( H, U (1)) defined by the N HLB H -principal ∞ -bundle (cid:0) Sym( G ) (cid:1) //N HLB H −→ H (4.23)in H ∞ agrees with the class defined by the principal ∞ -bundle (cid:0) S e N Sym( G ) (cid:1) // (cid:0) S e N HLB H (cid:1) −→ S e H ≃ H in S . Here we have used that there is an equivalence \ N HLB H ≃ B U (1) H in G rp( H ∞ ) , so that π H ∞ (cid:0) H, B N HLB H (cid:1) ≃ π H ∞ (cid:0) H, B U (1) H (cid:1) . (Again, one can alternatively see the coincidence of the cohomology classes more explicitly on the levelof Čech cocycles in the presentation of H ∞ by the simplicial model category H p ∞ : a smooth bundlerepresented by a smooth U (1) H -valued cocycle on H gets sent to the topological bundle representedby the same Čech cocycle interpreted as a collection of continuous maps.) It thus remains to computethe cohomology class associated to these bundles. In [BMS, Sec. 8] it has been shown that the classin H ( H ; Z ) of the bundle (4.23) agrees with the class in H ( H ; Z ) that classifies the gerbe G underthe isomorphism (4.5). Since we started our construction from a so-called basic gerbe, i.e. one whoseDixmier-Douady class is a generator of H ( H ; Z ) , this concludes the proof of Theorem 4.12. Remark 4.24
We conclude with the following remarks:(1) In [BMS], we suggested the smooth 2-group extension (4.13) as a model for the string groupextension of H . However, the necessary formalism to make this precise was not available then—itsdevelopment was the main goal of the present paper. This can also be seen directly: the comparison map S e M → M sends a smooth map ∆ ke → M to the restriction | ∆ k | → M , which is a k -simplex in Sing( M ) —see [Bunb, Secs. 4, 5] for details. HLB H Des L R H i p (4.25)of H ; its construction uses a connection on G as auxiliary data and relies heavily on a notion ofparallel transport on a gerbe G with connection, as developed in [BMS]. The extension (4.25)is then obtained as an explicit homotopy-coherent version of an associated bundle construction.By [BMS, Thm. 5.33], there is an equivalence (in H ≤ ) between the smooth 2-group extensionin (4.25) and (4.11), so that we automatically obtain an equivalence between the ∞ -bundles in H ∞ they induce under the nerve functor. Therefore, given the input of a basic gerbe G on H , byTheorem 4.12 the extension (4.25) also gives rise to a second (but equivalent) smooth string groupextension N \ HLB H N [ Des L b H of H , for any compact, connected and simply connected Lie group H . ⊳ A Actions and category objects
In this appendix, we prove Theorem 3.19; that is, we show that group actions
P //G in ∞ -topoi (as inDefinition 3.15) are automatically groupoid objects. Definition A.1
Let C be an ∞ -category. A category object in C is a simplicial object X ∈ F un( ∆ op , C ) such that for every n ∈ N the pullback X × X · · · × X X exists in C and the morphism X n −→ X × X · · · × X X , induced by the spine decomposition [ n ] ∼ = [1] ⊔ [0] · · · ⊔ [0] [1] of finite ordered sets, is an equivalence. Suppose C has a final object. In analogy with Definition 3.3, a monoid object in C is a categoryobject ∗ //M ∈ F un( ∆ op , C ) such that ( ∗ //M ) ≃ ∗ is a final object in C . As for group objects, weset M := ( ∗ //M ) , and it follows that there are canonical natural equivalences ( ∗ //M ) n ≃ M n − .Therefore, by Lemma 3.12 we may assume, without loss of generality, that we have ( ∗ //M ) n = M n − for any n ∈ N . We set M //M := Dec ( ∗ //M ) ∈ F un( ∆ op , C ) . Monoid objects can act on objects intheir ambient ∞ -category. A monoid action is defined precisely like a group action (Definition 3.15),but for the reader’s convenience, we spell out the definition: Definition A.2
Let C be an ∞ -category with a final object, and let ∗ //M be a monoid object in C . Let P ∈ C be an object in C . An action of ∗ //M on P is a simplicial object P //M ∈ F un( ∆ op , C ) such that(1) for each n ∈ N , we have ( P //M ) n = P × M n ,(2) the morphism d : P × M → P agrees with the canonical projection onto P , the morphism s : P → P × M agrees with the morphism P × ( ∗ → M ) , and(3) the collapse morphism P → ∗ induces a morphism P //M → ∗ //M in F un( ∆ op , C ) . We set a := d : P × M → P . The pasting law for pullbacks implies that there are canonicalequivalences of morphisms between d : P × M n → P × M n − and a × M n − : P × M n → P × M n − ,and similarly between d n : P × M n → P × M n − and the projection onto the first n factors.38 roposition A.3 Let C be an ∞ -category with pullbacks and a final object, let ∗ //M ∈ F un( ∆ op , C ) bea monoid object in C , and let P //M ∈ F un( ∆ op , C ) be an action of ∗ //M on an object P ∈ C . Then, P //M is a category object in C .Proof. Consider the diagram ( P //M ) ( P //M ) ( P //M ) d d = P × M P × MP a pr We use the following notational convention: let I be a set, and consider a product Q i ∈ I C i of objectsin C . For a subset J ⊂ I , we let pr J : Q i ∈ I C i → Q j ∈ J C j denote the canonical projection. If J = { i , . . . , i n } is finite, we also write pr i ...i n instead of pr { i ,...,i n } .We can augment the above diagram to a diagram P × M × M P × M MP × M P ∗ pr a × M pr pr a Here, the right and the outer rectangle are pullback diagrams, and hence the left square is a pullbackdiagram as well by the pasting law. It follows that the canonical morphism ( P //M ) −→ ( P //M ) × ( P //M ) ( P //M ) is an equivalence in C .We now proceed by induction: suppose that the canonical morphism ( P //M ) k −→ ( P //M ) × ( P //M ) · · · × ( P //M ) ( P //M ) is an equivalence, for each ≤ k ≤ n . By this assumption, it now suffices to show that the morphism ( P //M ) n +1 −→ ( P //M ) n × ( P //M ) ( P //M ) (A.4)induced by the partition [ n + 1] = [ n ] ⊔ [0] [1] is an equivalence. We again have an augmented diagram P × M n +1 P × M MP × M n P ∗ pr a ( n ) × M pr pr a ( n ) where the morphism a ( n ) is, up to canonical equivalence, the morphism a ◦ ( a × M ) ◦ · · · ◦ ( a × M n ) : P × M n → P .
Again, the right-hand square in this diagram is a pullback square, and the top left object is constructedas the pullback of P × M n → ∗ ← M . It follows by the pasting law that the left-hand square is a39ullback as well, which then implies that the top left morphism is (canonically equivalent to) a ( n ) × M .Since ( P //M ) n +1 = P × M n +1 , and the morphisms P × M n +1 → P × M n and P × M n +1 → P × M in (A.4) are canonically equivalent to the morphisms induced from the partition [ n + 1] = [ n ] ⊔ [0] [1] this completes the proof.We recall a criterion from (the proof of) [Lur, Prop. 1.1.8] for when a category object is a groupoidobject. Given a simplicial object X ∈ F un( ∆ op , C ) in an ∞ -category C and a simplicial set K ∈ S et ∆ ,we define an object X ( K ) ∈ C as the limit (if it exists) of the diagram N ( ∆ /K ) op −→ N ∆ op X −→ C . Proposition A.5 [Lur]
Let C be an ∞ -category with finite limits. A category object X in C is agroupoid object in C if and only if the inclusion Λ ֒ → ∆ induces an equivalence X ≃ −→ X (Λ ) . Let I be the span category, depicted as { , } ← { } → { , } . Consider the functor D : I → ∆ / Λ ,which sends the object { } ∈ I to the canonical inclusion ∆ { } ֒ → Λ and the object { , i } to thecanonical inclusion ∆ { ,i } ֒ → Λ , for i = 0 , . Lemma A.6
Let D : I → ∆ / Λ be defined as above, and let C be an ∞ -category with finite limits. Thefollowing statements hold true:(1) The functor D : I → ∆ / Λ is cofinal.(2) For any X ∈ F un( ∆ op , C ) , the diagram X (Λ ) X X X ι ∗ , ι ∗ , d d is a pullback diagram in C , where ι ∗ ,i denotes the morphism X (∆ { ,i } ֒ → Λ ) .(3) A category object X ∈ F un( ∆ op , C ) in C is a groupoid object precisely if the diagram X X X X d d d d is a pullback diagram in C .Proof. For claim (1), note that an object of ∆ / Λ is a pair ([ n ] , ϕ ) of an object [ n ] ∈ ∆ and a morphismof simplicial sets ϕ : ∆ n → Λ . We show that, for each object ([ n ] , ϕ ) of ∆ / Λ , the slice category ([ n ] , ϕ ) /D is contractible.Since the horn Λ fits into a pushout diagram ∆ { } ∆ { , } ∆ { , } Λ d d S et ∆ , the morphism ϕ : ∆ n → Λ is either the constant map at the apex of the horn, i.e. ϕ factorsas ϕ : ∆ n → ∆ { } ֒ → Λ , or it factors through a unique map ∆ n → ∆ { ,i } , for i = 0 or i = 2 , but notthrough the apex ∆ { } ֒ → Λ . (One can see this either by writing ∆ n = N [ n ] and Λ = N I and usingthat the nerve functor is fully faithful, or by using that S et ∆ (∆ n , − ) = ( − ) n preserves colimits.)In the first case, the slice category ([ n ] , ϕ ) /D is the category describing spans; in other words, it isisomorphic to I , and we have | N I | ∼ = | ∆ ⊔ ∆ ∆ | ∼ = | ∆ | ⊔ | ∆ | | ∆ | ≃ ∗ . In the other cases, the slicecategory ([ n ] , ϕ ) /D is the final category, and hence contractible as well.Claim (2) now follows directly from the definition of X ( K ) , for K ∈ S et ∆ , together with part (1)(after taking opposites), and claim (3) then follows by combining claim (2) with Proposition A.5. Lemma A.7
Let K be a simplicial set, let C be an ∞ -category, and let C ∈ C be an object. Let c : C → F un( K, C ) denote the constant-diagram functor.(1) If K is contractible, i.e. K ≃ ∗ in S et ∆ with the Kan-Quillen model structure, and colim C K ( c C ) exists in C , then the canonical morphism colim C K ( c C ) → C in C is an equivalence.(2) Dually, if K is contractible and lim C K ( c C ) exists in C , then the canonical morphism C → lim C K ( c C ) in C is an equivalence.Proof. By the definition of c , there is a commutative diagram K C ∗ coll c CC in S . By [Lur09, Cor. 4.1.2.6, Thm. 4.1.3.1], the morphism coll is cofinal if and only if the simplicialset K × ∗ ∼ = K is contractible, i.e. precisely if coll : K → ∗ is an equivalence in S et ∆ (in the Kan-Quillen model structure). The first claim then follows from the fact that cofinal morphisms preservecolimits [Lur09, Prop. 4.1.1.8]. The second statement follows by duality. Example A.8
We need the following two specific cases in which Lemma A.7 applies:(1) The nerve N I ∈ S et ∆ is contractible, as already seen in the proof of Lemma A.6.(2) The inclusion { [0] } ֒ → ∆ is the inclusion of a final object. Thus, the nerve N ∆ ∈ S et ∆ is contractiblein S et ∆ . ⊳ Lemma A.9
Let K ∈ S et ∆ be contractible (in the Kan-Quillen model structure) and let C be an ∞ -category admitting limits of shape K . Let P ∈ C be any object.(1) The constant diagram functor c : C → F un( K, C ) is fully faithful.(2) If C admits finite products, then the functor P × ( − ) : C → C preserves limits of shape K .Proof. Let
C, D ∈ C be any objects. To see (1), we use the adjunction c ⊣ lim C K and Lemma A.7,which yield canonical equivalences C K ( c C, c D ) ≃ C ( C, lim C K c D ) ≃ C ( C, D ) . For claim (2), let
C, P ∈ C be objects, and let F : K → C be a diagram. We now have canonicalequivalences C (cid:0) C, lim C K ( c P × F ) (cid:1) ≃ C K ( c C, c P × F ) ≃ C K ( c C, c P ) × C K ( c C, F ) C ( C, P ) × C ( C, lim C K F ) ≃ C ( C, P × lim C K F ) . In the third equivalence we have used part (1), i.e. that c is fully faithful here. Then, the statementfollows from the Yoneda Lemma. Remark A.10
If we were working with categories instead of ∞ -categories, it would suffice to have anindexing category J (instead of K ) which is connected, rather than contractible, to prove an analogueof Lemma A.9. ⊳ We can now prove Theorem 3.19:
Proof of Theorem 3.19.
Since every group object in C is in particular a monoid object in C , it followsfrom Proposition A.3 that P //G is a category object in C . We now use Lemma A.6 to show that it iseven a groupoid object. By that lemma, it suffices to check that the diagram ( P //G ) ( P //G ) ( P //G ) ( P //G ) d d d d = P × G P × GP × G P d =pr d d =pr d =pr (A.11)is a pullback diagram in C , where we have used axioms (1) and (2) of Definition A.2 and their conse-quences pointed out after Definition 3.15.Our goal now is to split off the factor P in diagram (A.11). To that end, consider the diagram P PP × G P × GG G pr ≃ d d pr d pr = d pr d G (A.12)The bottom rectangle in diagram (A.12) commutes by axiom (3) of Definition 3.15. The top rectanglecommutes because P //G is a simplicial object in C , so we have a canonical equivalence d d ≃ d d .This establishes the morphism d as a morphism of binary products in C . As such, it is induced bythe morphisms P : P → P and d G : G → G . Thus, there is a canonical equivalence d ≃ P × d G , of morphisms ( P //G ) → ( P //G ) in H . We thus have an equivalence of diagrams P × G P × GP × G P d =pr d ≃ P × d G d =pr d =pr ≃ P × G GG ∗ d G d G
42y the definition of b G as a groupoid object with G ≃ ∗ , we have that the diagram G GG ∗ d G d G is a pullback diagram in C by Lemma A.5. It now follows from Lemma A.9 that the functor P × ( − ) sends this pullback diagram to a pullback diagram. Consequently, the square (A.11) is a pullbackdiagram in H . References [AHR] M. Ando, M. J. Hopkins, and C. Rezk. Multiplicative orientations of KO -theory and of the spectrum of topological modular forms. URL: https://faculty.math.illinois.edu/~mando/papers/koandtmf.pdf .[BEBdBP] D. Berwick-Evans, P. Boavida de Brito, and D. Pavlov. Classifying spaces of infinity-sheaves. arXiv:1912.10544 .[BMS] S. Bunk, L. Müller, and R. J. Szabo. Smooth 2-Group Extensions and Symmetries of Bundle Gerbes. arXiv:2004.13395 .[Bry08] J.-L. Brylinski. Loop spaces, characteristic classes and geometric quantization . Modern BirkhäuserClassics. Birkhäuser Boston, Inc., Boston, MA, 2008. Reprint of the 1993 edition.[BS17] S. Bunk and R. J. Szabo. Fluxes, bundle gerbes and 2-Hilbert spaces.
Lett. Math. Phys. ,107(10):1877–1918, 2017. arXiv:1612.01878 .[BSCS07] J. C. Baez, D. Stevenson, A. S. Crans, and U. Schreiber. From loop groups to 2-groups.
HomologyHomotopy Appl. , 9(2):101–135, 2007. arXiv:math/0504123 .[BSS18] S. Bunk, C. Sämann, and R. J. Szabo. The 2-Hilbert space of a prequantum bundle gerbe.
Rev.Math. Phys. , 30(1):1850001, 101 pp., 2018. arXiv:1608.08455 .[Buna] S. Bunk. Sheaves of Higher Categories and Presentations of Smooth Field Theories. arXiv:2003.00592 .[Bunb] S. Bunk. The R -local Homotopy Theory of Smooth Spaces. arXiv:2007.06039v2 .[Bun17] S. Bunk. Categorical Structures on Bundle Gerbes and Higher Geometric Prequantisation . PhDthesis, Heriot-Watt University, Edinburgh, 2017. arXiv:1709.06174 .[Cis19] D.-C. Cisinski.
Higher categories and homotopical algebra , volume 180 of
Cambridge Studies inAdvanced Mathematics . Cambridge University Press, Cambridge, 2019.[DHH11] C. L. Douglas, A. G. Henriques, and M. A. Hill. Homological obstructions to string orientations.
Int.Math. Res. Not. IMRN , (18):4074–4088, 2011. arXiv:0810.2131 .[FRS16] D. Fiorenza, C. L. Rogers, and U. Schreiber. Higher U (1) -gerbe connections in geometric prequan-tization. Rev. Math. Phys. , 28(6):1650012, 72, 2016. arXiv:1304.0236 .[FSS12] D. Fiorenza, U. Schreiber, and J. Stasheff. Čech cocycles for differential characteristic classes: an ∞ -Lie theoretic construction. Adv. Theor. Math. Phys. , 16(1):149–250, 2012. arXiv:1011.4735 .[Hen08] A. Henriques. Integrating L ∞ -algebras. Compos. Math. , 144(4):1017–1045, 2008. arXiv:math/0603563 . Höh] G. Höhn. Komplexe elliptische Geschlechter und S -äquivariante Kobordismustheorie. arXiv:math/0405232 .[Kil87] T. P. Killingback. World-sheet anomalies and loop geometry. Nuclear Phys. B , 288(3-4):578–588,1987.[LM89] H. B. Lawson, Jr. and M.-L. Michelsohn.
Spin geometry , volume 38 of
Princeton MathematicalSeries . Princeton University Press, Princeton, NJ, 1989.[Lur] J. Lurie. ( ∞ , -Categories and the Goodwillie Calculus I. URL: .[Lur09] J. Lurie. Higher topos theory , volume 170 of
Annals of Mathematics Studies . Princeton UniversityPress, Princeton, NJ, 2009.[NSS15] T. Nikolaus, U. Schreiber, and D. Stevenson. Principal ∞ -bundles: general theory. J. HomotopyRelat. Struct. , 10(4):749–801, 2015. arXiv:1207.0248 .[NSW13] T. Nikolaus, C. Sachse, and C. Wockel. A smooth model for the string group.
Int. Math. Res. Not.IMRN , (16):3678–3721, 2013. arXiv:1104.4288 .[Rie14] E. Riehl.
Categorical homotopy theory , volume 24 of
New Mathematical Monographs . CambridgeUniversity Press, Cambridge, 2014.[Sch] U. Schreiber. Differential Cohomology in a Cohesive ∞ -Topos. arXiv:1310.7930v1 .[SP11] C. J. Schommer-Pries. Central extensions of smooth 2-groups and a finite-dimensional string 2-group. Geom. Topol. , 15(2):609–676, 2011. arXiv:0911.2483 .[ST] S. Stolz and P. Teichner. The Spinor Bundle on Loop Space. URL: https://people.mpim-bonn.mpg.de/teichner/Math/ewExternalFiles/MPI.pdf .[ST04] S. Stolz and P. Teichner. What is an elliptic object? In
Topology, geometry and quantum fieldtheory , volume 308 of
London Math. Soc. Lecture Note Ser. , pages 247–343. Cambridge Univ. Press,Cambridge, 2004. arXiv:1108.0189 .[Ste12] D. Stevenson. Décalage and Kan’s simplicial loop group functor.
Theory Appl. Categ. , 26, 2012. arXiv:1112.0474 .[Sto96] S. Stolz. A conjecture concerning positive Ricci curvature and the Witten genus.
Math. Ann. ,304(4):785–800, 1996.[Wal07] K. Waldorf.
Algebraic structures for bundle gerbes and the Wess-Zumino termin conformal field theory . PhD thesis, Universität Hamburg, 2007. URL: http://ediss.sub.uni-hamburg.de/volltexte/2008/3519/ .[Wal12] K. Waldorf. A construction of string 2-group models using a transgression-regression technique. In
Analysis, geometry and quantum field theory , volume 584 of
Contemp. Math. , pages 99–115. Amer.Math. Soc., Providence, RI, 2012. arXiv:1201.5052 .[Wal15] K. Waldorf. String geometry vs. spin geometry on loop spaces.
J. Geom. Phys. , 97:190–226, 2015. arXiv:1403.5656 .[Wit] E. Witten. The index of the Dirac operator in loop space. In
Elliptic curves and modular forms inalgebraic topology (Princeton, NJ, 1986) , volume 1326 of
Lecture Notes in Math. , pages 161–181.Universität Hamburg, Fachbereich Mathematik, Bereich Algebra und Zahlentheorie,Bundesstraße 55, 20146 Hamburg, [email protected], pages 161–181.Universität Hamburg, Fachbereich Mathematik, Bereich Algebra und Zahlentheorie,Bundesstraße 55, 20146 Hamburg, [email protected]