The character map in (twisted differential) non-abelian cohomology
TThe character map in (twisted differential) non-abelian cohomology
Domenico Fiorenza, Hisham Sati, Urs SchreiberOctober 5, 2020
Abstract
The Chern character on K-theory has a natural extension to arbitrary generalized cohomology theoriesknown as the Chern-Dold character. Here we further extend this to (twisted, differential) non-abelian cohomol-ogy theories, where its target is a non-abelian de Rham cohomology of twisted L ∞ -algebra valued differentialforms. The construction amounts to leveraging the fundamental theorem of dg-algebraic rational homotopytheory to a twisted non-abelian generalization of the de Rham theorem. We show that the non-abelian charac-ter reproduces, besides the Chern-Dold character, also the Chern-Weil homomorphism as well as its secondaryCheeger-Simons homomorphism on (differential) non-abelian cohomology in degree 1, represented by principalbundles (with connection); and thus generalizes all these to higher (twisted, differential) non-abelian cohomol-ogy, represented by higher bundles/higher gerbes (with higher connections). As a fundamental example, wediscuss the twisted non-abelian character map on twistorial Cohomotopy theory over 8-manifolds, which canbe viewed as a twisted non-abelian enhancement of topological modular forms (tmf) in degree 4. This turnsout to exhibit a list of subtle topological relations that in high energy physics are thought to govern the chargequantization of fluxes in M-theory. Contents L ∞ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Rational homotopy theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3 Non-abelian de Rham theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 A Model category theory 90 a r X i v : . [ m a t h . A T ] O c t Introduction
Generalized cohomology theories [Wh62][Ad75] – such as K-theory, elliptic cohomology, stable Cobordism andstable Cohomotopy – are rich. This makes them fascinating but also intricate to deal with. In algebraic topology, ithas become commonplace to apply filtrations by iterative localizations [Bou79] (review in [EKMM97, §V][Ba14])that allow generalized cohomology to be approximated in consecutive stages; a famous example of current interestis the chromatic filtration on complex oriented cohomology theories ([MR87], review in [Ra86][Lu10]).
The Chern-Dold character.
The primary approximation stage of generalized cohomology theories is their ratio-nalization (e.g., [Hil71][Ba14]) to ordinary cohomology (e.g., singular cohomology) with rational coefficients orreal coefficients (see Remark 3.51). This goes back to [Do65]; since on topological K-theory (Example 4.10) itreduces to the Chern character map [Hil71, Thm. 5.8], it has been called the
Chern-Dold character [Bu70]:
Chern-Doldcharacter ch nE : generalizedcohomology E n ( X ) dR ◦ ch E differential-geometricChern-Dold character (cid:47) (cid:47) rationalization (cid:47) (cid:47) E n R ( X ) Dold’s equivalence (cid:39) (cid:47) (cid:47) (cid:76) k ordinarycohomology H n + k (cid:0) X ; rationalizedhomotopy groups π k ( E ) ⊗ Z R (cid:1) (cid:39) de Rham theorem (cid:15) (cid:15) Hom (cid:16) E • ( ∗ ) ⊗ Z R , H n + • dR ( X ) de Rham cohomology (cid:17) (1)That the first map in (1) is indeed the rationalization approximation on coefficient spectra is left somewhat implicitin [Bu70] (rationalization was fully formulated only in [BK72]); a fully explicit statement is in [LSW16, §2.1].The equivalence on the right of (1) serves to make explicit how the result of that rationalization operation indeedlands in ordinary cohomology, and this was Dold’s original observation [Do65, Cor. 4] (re-derived as Prop. 4.5). At the heart of differential cohomology.
While rationalization is the coarsest of the localization approximations,it stands out in that it connects, via the de Rham theorem, to differential geometric data – when the base space X hasthe structure of a smooth manifold, and the coefficients are taken to be R instead of Q . Indeed, this “differential-geometric Chern-Dold character” shown on the bottom of (1), underlies (often without attribution to Dold orBuchstaber) the pullback-construction of differential generalized cohomology theories [HS05, §4.8] (see [BN14,p. 17][GS17b, Def. 7][GS18b, Def. 17][GS19a, Def. 1], and see Def. 4.33, Example 4.34 below). At the heart of non-perturbative field theory.
It is in this differential-geometric form that the Chern-Dold char-acter plays a pivotal role in high energy physics. Here closed differential forms encode flux densities F p ∈ Ω p dR ( X ) of generalized electromagnetic fields on spacetime manifolds X ; and the condition that these lift through (i.e., arein the image of) the Chern-Dold character (1) for E -cohomology theory encodes a charge quantization conditionin E -theory (see [Fr00][Sa10][GS19c]), generalizing Dirac’s charge quantization of the ordinary electromagneticfield in ordinary cohomology [Di31] (see [Fra97, 16.4e]): E n ( X ) dR ◦ ch nE differential-geometricChern-Dold character (cid:47) (cid:47) Hom (cid:16) E • ( ∗ ) ⊗ Z R , H n + • dR ( X ) (cid:17) [ c ] class in E -cohomology (cid:31) (cid:47) (cid:47) (cid:8)(cid:2) F p i (cid:3)(cid:9) i = , ··· , dim ( E • ( ∗ ) ⊗ Z R ) charge-quantizedflux densities (2)This idea of charge quantization in a generalized cohomology theory turned out to be fruitful for capturing muchof the expected nature of the Ramond-Ramond (RR) fields in type II/I string theory, as being charge-quantized intopological K-theory: E = KU , KO (see [FH00][Fr00][Ev06][GS19c][GS18b]).However, various further topological conditions [FSS19b, Table 1][FSS19c, p. 2][SS20a, Table 3][FSS20, p.2] in non-perturbative type IIA string theory (“M-theory”) are not captured by charge-quantization (2) in K-theory,or in any generalized cohomology theory, since they involve quadratic functions (301) in the fluxes. This motivates:
Non-abelian cohomology.
Despite their established name, generalized cohomology theories in the traditionalsense of [Wh62][Ad75] are not general enough for many purposes. Already the time-honored non-abelian coho-mology that classifies principal bundles (Example 2.3 below), being the domain of the Chern-Weil homomorphism2Ch50] (recalled as Def. 4.21, Prop. 4.23 below), falls outside the scope of “generalized” cohomology, as doesthe higher non-abelian cohomology classifying gerbes [Gi71] (Example 2.6 below). But these are just the firsttwo stages within a truly general concept of higher non-abelian cohomology (Def. 2.1 below), that classifieshigher bundles/higher gerbes (Example 2.7 below) and which fully subsumes Whitehead’s traditional generalizedcohomology as its abelian sector (Example 2.13 below).In higher non-abelian cohomology the very conceptualization of cohomology finds a beautiful culmination, asit is reduced to the pristine concept of homotopy types of mapping spaces (7), or rather, if geometric (differential,equivariant,...) structures are incorporated, of higher mapping stacks (Remark 2.27 below).In particular, the concept of twisted non-abelian co-homology is most natural from this perspective (Def.2.29 below) and naturally subsumes the traditionalconcept of twisted generalized cohomology theories(Prop. 2.38 below). twistednon-abeliancohomology H τ ( X ; A ) = X cocycle c (cid:47) (cid:47) twist τ (cid:34) (cid:34) coefficient ∞ -stack A (cid:12) G ρ localcoefficients (cid:123) (cid:123) B G (cid:39) (cid:111) (cid:119) (cid:14) homotopyrelative B G State of the literature.
It is fair to say that this transparent fundamental nature of higher non-abelian cohomologyis not easily recognized in much of the traditional literature on the topic, which is rife with unwieldy variants ofcocycle conditions presented in combinatorial n -category-theoretic language. As a consequence, the developmentof non-abelian cohomology theory has seen little and slow progress, certainly as compared to the flourishing ofgeneralized cohomology theory. In particular, the concepts of higher and of twisted non-abelian cohomology hadremained mysterious (see [Si97, p. 1]). It is the more recently established homotopy-theoretic formulation of ∞ -category theory (e.g., via model category theory, see appendix A) in its guise as ∞ -topos theory ( ∞ -stacks , recalledaround Def. A.44 below) that provides the backdrop on which twisted higher non-abelian cohomology finds its trueand elegant nature [Si97][Si99][To02][SSS12][NSS12a][NSS12b][Sch13][FSS19b][SS20b]; see §2 for details. The non-abelian character map.
From this homotopy-theoretic perspective, we observe in §4 and §5 that thegeneralization of the Chern-Dold character (1) to twisted non-abelian cohomology naturally exists (Def. 4.2),and that the non-abelian analogue of Dold’s equivalence in (1) may neatly be understood as being, up to mildre-conceptualization, the fundamental theorem of dg-algebraic rational homotopy theory (recalled as Prop. 3.60below). We highlight that this classical theorem is fruitfully recast as constituting a non-abelian de Rham theorem (Theorem 3.87 below) and, more generally, a twisted non-abelian de Rham theorem (Theorem 3.104 below). Withthis in hand, the notion of the (twisted) non-abelian character map appears naturally (Def. 4.2 and Def. 5.4 below): twistednon-abeliancharacter map (Def. 5.4) ch ρ : twistednon-abelian cohomology (Def. 2.29) H τ ( X ; A ) ( η R ρ ) ∗ rationalization (Def. 3.55, Prop. 3.60) (cid:47) (cid:47) twisted non-abelianreal cohomology (Def. 3.76) H L R τ (cid:0) X ; L R A (cid:1) (cid:39) twisted non-abeliande Rham theorem (Thm. 3.104) (cid:47) (cid:47) twisted non-abeliande Rham cohomology (Def. 3.98) H τ dR dR ( X ; l A ) . (3) Twisted differential non-abelian cohomology.
Moreover, with the (twisted) non-abelian character in hand, thenotion of (twisted) differential non-abelian cohomology appears naturally (Def. 4.33, Def. 5.13) together with theexpected natural diagrams of twisted differential non-abelian cohomology operations: differentialnon-abelian cohomology (Def. 4.33) (cid:98) H ( X ; A ) curvature (234) (cid:47) (cid:47) differentialnon-abelian character (235) (cid:37) (cid:37) characteristicclass (233) (cid:15) (cid:15) flat L ∞ -algebra valueddifferential forms (Def. 3.77) Ω dR ( X ; l A ) (cid:15) (cid:15) H ( X ; A ) non-abelian cohomology (Def. 2.1) non-abelian character (Def. 4.2) ch A (cid:47) (cid:47) H dR ( X ; l A ) non-abeliande Rham cohomology (Def. 3.84) twisted differentialnon-abelian cohomology (Def. 5.13) (cid:98) H τ diff ( X ; A ) twistedcurvature (280) (cid:47) (cid:47) twisted differentialnon-abelian character (281) (cid:38) (cid:38) twistedcharacteristicclass (279) (cid:15) (cid:15) twisted flat L ∞ -algebra valueddifferential forms (Def. 3.92) Ω τ dR dR ( X ; l A ) (cid:15) (cid:15) H τ ( X ; A ) twistednon-abelian cohomology (Def. 2.29) twistednon-abelian character (Def. 5.4) ch τ A (cid:47) (cid:47) H τ dR dR ( X ; l A ) twisted non-abeliande Rham cohomology (Def. 3.98) (4)3 nifying Chern-Dold, Chern-Weil and Cheeger-Simons. In order to show that this generalization of (twisted)character maps and (twisted) differential cohomology to higher non-abelian cohomology is sound, we proceed toprove that the non-abelian character map (Def. 4.2) specializes to:the Chern-Dold characteron generalized cohomology (Theorem 4.8),the Chern-Weil homomorphismon degree-1 non-abelian cohomology (Theorem 4.26),the Cheeger-Simons homomorphismon degree-1 differential non-abelian cohomology (Theorem 4.46).All these classical invariants are thus seen as different low-degree aspects of the higher non-abelian character map.
Examples of twisted higher character maps.
To illustrate the mechanism, we make explicit several examples ofthe (twisted) non-abelian character map on cohomology theories of relevance in high energy physics:the Chern character on complex differential K-theory (Example 4.10, 4.36),the Pontrjagin character on real K-theory (Example 4.11),the Chern character on twisted differential K-theory (Example 5.5, 5.22),the MMS-character on cohomotopy-twisted K-theory (Example 5.7),the LSW-character on twisted higher K-theory (Example 5.10),the character on integral Morava K-theory (Example 4.15),the character on topological modular forms, tmf (Example 4.12).Once incarnated this way within the more general context of non-abelian cohomology theory, we may ask fornon-abelian enhancements (Example 2.24) of these abelian character maps:
Non-abelian enhancement of the tmf -character – the cohomotopical character.
Our culminating example, in§5.3, is the character map on twistorial Cohomotopy theory [FSS19b][FSS20], over 8-manifolds X equipped withtangential Sp ( ) -structure τ (59). This may be understood (Remark 4.14) as an enhancement of the tmf-character(Example 4.12) from traditional generalized cohomology to twisted differential non-abelian cohomology: tmf-cohomologyin degree 4 (Example 4.12) tmf (cid:0) X (cid:1) (cid:39) tmf approximatessphere spectrum (Example 4.13) stable Cohomotopyin degree 4 (Example 2.16) S (cid:0) X (cid:1) ∼∼∼∼ (cid:3) non-abelianenhancement (Example 2.25) unstable/non-abelian4-Cohomotopy (Example 2.10) π (cid:0) X (cid:1) ∼∼∼∼ (cid:3) twisting byJ-homomorphism (Def. 2.29) twisted non-abelian4-Cohomotopy (Example 2.40) π τ (cid:0) X (cid:1) ∼∼∼∼∼∼∼∼ (cid:3) lift throughtwisted cohomology operationinduced by twistor fibration (Example 2.44) twistorialCohomotopy (Example 2.44) T τ (cid:0) X (cid:1) ∼∼∼∼ (cid:3) differentialenhancement (Def. 5.13) differentialtwistorialCohomotopy (Example 5.26) (cid:99) T τ (cid:0) X (cid:1) . The non-abelian character map on twistorial Cohomotopy has the striking property (Prop. 5.24, the proof ofwhich is the content of the companion physics article [FSS20, Prop. 3.9]) that the corresponding non-abelianversion of Dirac’s charge quantization (2) implies Hoˇrava-Witten’s Green-Schwarz mechanism in heterotic M-theory for heterotic line bundles F (see [FSS20, §1]) and other subtle effects expected in non-perturbative highenergy physics; these are discussed in Remark 5.29 below. Quadratic character functions from Whitehead brackets in non-abelian coefficient spaces.
The crucial ap-pearance of quadratic functions in the Cohomotopical character map (301) is brought about by the non-abeliannature of (twisted) Cohomotopy theory. These non-linearities originate in non-trivial Whitehead brackets (Remark3.66) on the non-abelian coefficient spaces S (Example 3.68) and on C P (Example 3.96). Generally, the non-abelian character map (3) involves also higher monomial terms of any order (cubic, quartic, ...), originating inhigher order Whitehead brackets on the non-abelian coefficient space (Remark 3.66).Note that the desire to conceptually grasp character-like but quadratic functions appearing in M-theory hadbeen the original motivation for developing differential generalized cohomology, in [HS05]. Here, in differentialnon-abelian cohomology, they appear intrinsically. Acknowledgements.
We thank John Lind, Chris Rogers, Carlos Simpson, Danny Stevenson, and Mathai Varghesefor comments on an earlier version of this note. 4
Non-abelian cohomology
We make explicit the concept of general non-abelian cohomology (Def. 2.1 below) and of twisted non-abeliancohomology (Def. 2.29 below), following [Si97][Si99][To02][SSS12][NSS12a][NSS12b][FSS19b][SS20b]; andwe survey how this concept subsumes essentially every notion of cohomology known.In the following, we make free use of the basic language of category theory and homotopy theory (for jointintroduction see [Rie14][Ri20]). For C a category and X , A ∈ C a pair of its objects, we write C ( X , A ) : = Hom C ( X , A ) ∈ Sets (5)for the set of morphisms from X to A . These are, of course, contravariantly and covariantly functorial in their firstand second argument, respectively: C C ( X , − ) (cid:47) (cid:47) Sets , C op C ( − , A ) (cid:47) (cid:47) Sets . (6)Basic as this is, contravariant hom-functors are of paramount interest in the case where C is the homotopy category Ho ( C ) (Def. A.14) of a model category (Def. A.3), such as the classical homotopy category of topological spacesor, equivalently, of simplicial sets (Example A.33). Definition 2.1 (Non-abelian cohomology) . For X , A ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) (Example A.33) we say thattheir hom-set (5) is the non-abelian cohomology of X with coefficients in A , or the non-abelian A-cohomology of X , to be denoted: non-abeliancohomology H ( X ; A ) : = Ho (cid:0) TopologicalSpaces Qu (cid:1) ( X , A ) = X map = cocycle c (cid:30) (cid:30) c (cid:48) map = cocycle (cid:64) (cid:64) A homotopy =coboundary (cid:11) (cid:19) (cid:14) homotopy (7)We also call the contravariant hom-functor (6) H ( − ; A ) : Ho (cid:0) TopologicalSpaces Qu (cid:1) (cid:47) (cid:47) Sets (8)the non-abelian
A-cohomology theory . Example 2.2 (Ordinary cohomology) . For n ∈ N and A a discrete abelian group, the ordinary cohomology (e.g.singular cohomology) in degree n with coefficients in A is equivalently ([Ei40, p. 243][EML54b, p. 520-521],review in [St72, §19][May99, §22][AGP02, §7.1, Cor. 12.1.20]) non-abelian cohomology in the sense of Def. 2.1 ordinarycohomology H n ( − ; A ) (cid:39) H (cid:0) − ; K ( A , n ) (cid:1) (9)with coefficients in an Eilenberg-MacLane space [EML53][EML54a]: K ( A , n ) ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) such that π k (cid:0) K ( A , n ) (cid:1) = (cid:26) A | k = n | k (cid:54) = n . (10) Example 2.3 (Traditional non-abelian cohomology) . For G a well-behaved topological group, the traditional non-abelian cohomology H ( − ; G ) classifying G -principal bundles, is equivalently ([St51, §19.3][RS12, Thm 1.], re-view in [Add07, §5]) non-abelian cohomology in the general sense of Def. 2.1 classification ofprincipal bundles H ( − ; G ) (cid:39) H ( − ; BG ) (11)5ith coefficients in the classifying space BG ([Mi56][Se68][St68][St70], review in [Ko96, §1.3][May99, §23.1][AGP02, §8.3][NSS12b, §3.7.1]). The latter may be given as the homotopy colimit (in the classical model structureof TopologicalSpaces Qu , Example A.7) over the nerve of the topological group G (e.g. [NSS12a, Rem. 2.23]): BG (cid:39) holim −! · · · G × G (cid:47) (cid:47) (cid:111) (cid:111) ( − ) · ( − ) (cid:47) (cid:47) (cid:111) (cid:111) (cid:47) (cid:47) G (cid:47) (cid:47) (cid:111) (cid:111) e (cid:47) (cid:47) ∗ . (12) Example 2.4 (Group cohomology and Characteristic classes) . Conversely, the ordinary cohomology (Example2.2) of a classifying space BG (12) is, equivalently, (i) the group cohomology of G ; (ii) the universal characteristic classes of G -principal bundles: groupcohomology H (cid:0) BG ; K ( A , n ) (cid:1) (cid:39) H n ( BG ; A ) (cid:39) H n Grp ( G ; A ) . Example 2.5 (Non-abelian cohomology in degree 2) . For a well-behaved topological 2-group, such as the string 2-group
String ( G ) (of a connected, simply connected semi-simple Lie group G ) [BCSS07][He08, Thm. 4.8][NSW11],the non-abelian cohomology H ( − ; String ( G )) classifying principal 2-bundles [NW11] with structure 2-groupString ( G ) is, equivalently [BS09], classification ofString-bundles H (cid:0) − ; String ( G ) (cid:1) (cid:39) H (cid:0) − ; B String ( G ) (cid:1) (13)non-abelian cohomology in the general sense of Def. 2.1 with coefficients in the classifying space B String ( G ) . Example 2.6 (Non-abelian gerbes) . For G a well-behaved topological group, a non-abelian G-gerbe [Gi71][Br09]is, equivalently [NSS12a, §4.4], a fiber 2-bundle with typical 2-fiber of homotopy type of the classifying space BG (12), associated to principal 2-bundles with structure 2-group Aut ( B G ) . Hence, as in Example 2.5, G -gerbes areclassified by non-abelian cohomology with coefficients in B Aut ( B G ) [NSS12a, Cor 4.51]: classification ofnon-abelian gerbes G Gerbes ( X ) / ∼ (cid:39) H (cid:0) X ; Aut ( B G ) (cid:1) (cid:39) H (cid:0) X ; B Aut ( B G ) (cid:1) . Example 2.7 (Non-abelian cohomology in unbounded degree) . For any ∞ -group G (see [NSS12a, §2.2][NSS12b,§3.5]), the non-abelian cohomology H (cid:0) − ; G (cid:1) classifying principal ∞ -bundles [Gl82][JL06][NSS12a][NSS12b]with structure ∞ -group G is, equivalently [We10][RS12], classification ofnon-abelian ∞ -gerbes H ( − ; G ) (cid:39) H ( − ; B G ) (14)non-abelian cohomology in the general sense of Def. 2.1 with coefficients in the classifying space B G (see also[Stv12]).Example 2.7 is, in fact, universal: Proposition 2.8 (Connected homotopy types are higher non-abelian classifying spaces [NSS12a, Thm. 2.19][NSS12b,Thm. 3.30, Cor. 3.34]) . Every connected homotopy type A ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) (338) is the classifyingspace of a topological group, namely of its loop group Ω AA (cid:39) B ( Ω A ) ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) . (15) A priori, the loop group is an A ∞ -group, for which classifying spaces are defined as in [NSS12a, Rem. 2.23], but each such is weaklyequivalent to an actual topological group, see [NSS12b, Prop. 3.35]. Remark 2.9 (Non-abelian and abelian ∞ -groups) . For A (cid:39) BG (15), the ∞ -group structure on G is reflected by itsweak homotopy equivalence G (cid:39) Ω BG with a based loop space.• There is no commutativity of loops in a generic loop space, and hence this exhibits G as a non-abelian ∞ -group.• But it may happen that A itself is already equivalent to a loop space, which by (15) means that A (cid:39) B (cid:0) BG (cid:1) = : B G is a double delooping . In this case G (cid:39) Ω (cid:0) Ω A (cid:1) = : Ω A is an iterated loop space [May72], specifically double loop space ; hence a braided ∞ -group . By the Eckmann-Hilton argument, this implies some level ofcommutativity of the group operation in G . Indeed, in the special case that such G is also 0-truncated (340),it implies that G is an ordinary abelian group.• Next, it may happen that A (cid:39) B G is a 3-fold delooping, hence that G (cid:39) Ω A is a 3-fold loop space, hencea sylleptic ∞ -group . This is one step “more abelian” than a braided ∞ -group.• In the limiting case that G is an n -fold loop space for any n ∈ N , hence an infinite loop space [May77][Ad78],it is as abelian as possible for an ∞ -group. Such abelian ∞ -groups are the coefficients of abelian cohomologytheories, namely of generalized cohomology theories in the sense of Whitehead (Example 2.13)• The fewer deloopings an ∞ -group G admits, the “more non-abelian” is the cohomology theory representedby BG . Coefficients H ( X ; BG ) Examples ∞ -group G (cid:39) Ω B G non-abeliancohomology π n ( − ) (Cohomotopy, Example 2.10)braided ∞ -group G (cid:39) Ω B G π ( − ) sylleptic ∞ -group G (cid:39) Ω B G ... G (cid:39) Ω n B n G abelian ∞ -group G (cid:39) Ω ∞ B ∞ G abelian cohomology E n ( − ) (generalized cohomology, Example 2.13)The most fundamental connected homotopy types are the n -spheres (all other are obtained by gluing n -spheres toeach other): Example 2.10 (Cohomotopy theory) . The non-abelian cohomology theory (Def. 2.1) with coefficients in thehomotopy types of n -spheres is (unstable) Cohomotopy theory [Bo36][Sp49][Pe56][Ta09][KMT12]:
Cohomotopy π n ( − ) = H ( − ; S n ) (cid:39) H ( − ; Ω S n ) for n ∈ N + . (i) By Prop. 2.8, Cohomotopy theory classifies principal ∞ -bundles (Example 2.7) with structure ∞ -group of thehomotopy type of the ∞ -group Ω S n . (ii) By Remark 2.9, Cohomotopy theory is a maximally non-abelian cohomology theory, in that S n does not admitdeloopings, for general n (it admits a single delooping for n = n = , Example 2.11 (Bundle gerbes) . The classifying space (12) of the circle group U ( ) is an Eilenberg-MacLanespace (10) B U ( ) (cid:39) K ( Z , ) ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) . Since U ( ) is abelian, this space carries itself the structure of (the homotopy type of) a 2-group, and hence has ahigher classifying space B U ( ) : = B ( B U ( )) (cid:39) K ( Z , ) ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) in the sense of Example 2.5, which is an Eilenberg-MacLane space in one degree higher. The higher princi-pal 2-bundles with structure 2-group B U ( ) are equivalently [NSS12a, Rem. 4.36] known as bundle gerbes [Mu96][SWa07]. Therefore, Example 2.7 combined with Example 2.2 gives the classification of bundle gerbesby ordinary integral cohomology in degree 3: classification ofbundle gerbes H (cid:0) − ; B U ( ) (cid:1) (cid:39) H (cid:0) − ; B U ( ) (cid:1) (cid:39) H ( − ; Z ) . xample 2.12 (Higher bundle gerbes) . In fact, Prop. 2.8 implies that, for all n ∈ N , B n + U ( ) : = B (cid:0) B n U ( ) (cid:1) (cid:39) K ( Z , n + ) ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) (16)in the sense of Example 2.7. The higher principal bundles with structure ( n + B n U ( ) [Ga97][FSSt10,§3.2.3][FSS12b, §2.6] are also known as higher bundle gerbes (for n = classification ofhigher bundle gerbes H (cid:0) − ; B n U ( ) (cid:1) (cid:39) H (cid:0) − ; B n + U ( ) (cid:1) (cid:39) H n + ( − ; Z ) . More generally, the special case of Example 2.7 where the coefficient ∞ -group happens to be abelian is “gener-alized cohomology” in the standard sense of algebraic topology (including cohomology theories such as K-theory,elliptic cohomology, stable Cobordism theory, stable Cohomotopy theory, etc.): Example 2.13 (Generalized cohomology) . For E a generalized cohomology theory [Wh62] (see [Ad75][Ad78]),Brown’s representability theorem ([Ad75, §III.6][Ko96, §3.4]) says that there is a spectrum (“ Ω -spectrum”, Ex-ample A.40) of pointed homotopy types (cid:110) E n ∈ Ho (cid:0) TopologicalSpaces ∗ / Qu (cid:1) , E n (cid:101) σ n (cid:39) (cid:47) (cid:47) Ω E n + (cid:111) n ∈ N (17)such that the generalized E -cohomology in degree n is equivalently non-abelian cohomology theory in the senseof Def. 2.1 with coefficients in E n : generalizedcohomology E n ( − ) (cid:39) H ( − ; E n ) . (18) Example 2.14 (Topological K-theory) . The classifying space (17) representing complex K-cohomology theoryKU [AH59, §2] (review in [At67]) in degree 0 is [AH61, §1.3]:KU (cid:39) Z × B U , (19)where B U : = lim −! n B U ( n ) (20)is the classifying space (12) for the infinite unitary group (e.g. [EU14]). Hence for the case of complex K-theory,Example 2.13 says that: topologicalK-theory KU ( − ) (cid:39) H ( − ; Z × B U ) . Example 2.15 (Iterated K-theory) . Given a spectrum (17) with suitable ring structure, one can form its algebraic K-theory spectrum K ( R ) [EKMM97, §VI][BGT10, §9.5][Lu14] and hence the corresponding generalized cohomol-ogy theory (Example 2.13). Much like complex topological K-theory (Example 2.14) is the K-theory of topological C -module bundles, K ( R ) -cohomology theory is the K-theory of suitable R -module ∞ -bundles [Li13]. Specifically,for R = ku the connective spectrum for topological K-theory, its algebraic K-theory K ( ku ) [Au09][AR02][AR07]has been argued to be the K-theory of certain categorified complex vector bundles [BDR03][BDRR09]. Moreover, K ( R ) is itself a suitable ring spectrum, so that the construction may be iterated to yield iterated algebraic K-theories [Ro14] K ◦ ( R ) : = K ( K ( R )) , K ◦ ( R ) : = K ( K ( K ( R ))) , et cetera. For R = ku, this generalizes the above forms ofelliptic cohomology, K ( ku ) , to higher degrees [LSW16]. By Example 2.13, we will regard these (connective)iterated algebraic K-theories K ◦ n ( ku ) of the complex topological K-theory spectrum as examples of non-abeliancohomology theories: iterated K-theory K ◦ n ( ku ) ( − ) (cid:39) H (cid:0) − ; K ◦ n ( ku ) (cid:1) . Example 2.16 (Stable Cohomotopy) . The generalized cohomology theory (Example 2.13) represented by thesuspension spectra (Example A.41) of n -spheres is called stable Cohomotopy theory (e.g. [Str81][No03]) or stableframed Cobordism theory : S n ( − ) = H (cid:0) − ; ( Σ ∞ S n ) (cid:1) . (21)8 on-abelian cohomology operations.Definition 2.17 (Non-abelian cohomology operation) . For A , A ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) (Example A.33),we say that a natural transformation in non-abelian cohomology (Def. 2.1) from A -cohomology theory to A -cohomology theory (8) is a (non-abelian) cohomology operation φ ∗ : H ( − ; A ) (cid:47) (cid:47) H ( − ; A ) . (22)By the Yoneda lemma, these are in bijective correspondence to morphisms of coefficients A φ (cid:47) (cid:47) A ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) (23)via the covariant functoriality of the hom-sets (6): φ ∗ = H ( − ; φ ) : = Ho (cid:0) TopologicalSpaces Qu (cid:1) ( − ; φ ) . (24) Example 2.18 (Cohomology of coefficient spaces parametrizes cohomology operations) . By the Yoneda lemma(24) in Ho (cid:0)
TopologicalSpaces Qu (cid:1) (Example A.33), the set of all cohomology operations (Def. 2.17) from A -cohomology theory to A -cohomology theory (22) coincides with the the non-abelian A -cohomology (Def. 2.1) of the coefficients A : non-abelian A -cohomology of A acting as cohomology operations H ( A ; A ) × H ( − ; A ) ( − ) ◦ ( − ) (cid:47) (cid:47) H ( − ; A ) (25)acting by composition composition in Ho (cid:0) TopologicalSpaces Qu (cid:1) . Example 2.19 (Cohomology operations in ordinary cohomology) . In specialization to Example 2.2 the non-abeliancohomology operations according to Def. 2.17 reduce to the classical cohomology operations in ordinary coho-mology [St72][MT08] (review in [May99, §22.5]), such as Steenrod operations [St47][SE62] (review in [Ko96,§2.5]). These operations admit refinements, involving rational/real form data, to differential cohomology opera-tions [GS18a].
Example 2.20 (Cohomology operations in generalized cohomology) . In specialization to Example 2.13, the non-abelian cohomology operations according to Def. 2.17 reduce to the traditional cohomology operations on gener-alized cohomology theories, such as the Adams operations in K-theory [Ad62] (review in [AGP02, §10]) or theQuillen operations in stable Cobordism theory (review in [Ko96, §4,5]). For differential refinements see [GS18b].
Example 2.21 (Characteristic classes of principal ∞ -bundles) . For G a topological group, the ordinary groupcohomology of G (Example 2.4) parametrizes, via Example 2.18, the cohomology operations from non-abeliancohomology classifying G -principal bundles (Examples 2.3, 2.5, 2.7) to ordinary cohomology of the base space(Example 2.2): groupcohomology H n Grp (cid:0) BG ; A (cid:1) × G -principalbundles H ( − ; G ) characteristicclasses (25) (cid:47) (cid:47) ordinarycohomology H n ( − ; A ) . (26)This is the assignment of characteristic classes to principal bundles (principal ∞ -bundles). In the case when A = R , this is equivalently the Chern-Weil homomorphism , by Chern’s fundamental theorem (see Remark 4.16 andTheorem 4.26 below).
Example 2.22 (Rationalization cohomology operation) . For fairly general non-abelian coefficients A (see Def.3.55, Def. 4.1 for details), their rationalization A η R A (cid:47) (cid:47) L R A (Def. 3.55 below) induces a cohomology operation(Def. 2.17) from non-abelian A -cohomology theory (Def. 2.1) to non-abelian real cohomology (Def. 3.72 below): non-abeliancohomology H ( − ; A ) ( η R A ) ∗ rationalization (cid:47) (cid:47) non-abelianreal cohomology H (cid:0) − ; L R A (cid:1) . (27) For definiteness, we consider rationalization over the real numbers; see Remark 3.51 below. emark 2.23 (Rationalization as character map) . Up to composition with an equivalence provided by the non-abelian de Rham theorem (Theorem 3.87 below), which serves to bring the right hand side of (27) into neatminimal form, this rationalization cohomology operation is the character map in non-abelian cohomology (Def.4.2 below).
Example 2.24 (Stabilization cohomology operation) . For A ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) , the non-abelian coho-mology operation (Def. 2.17) induced (24) by the unit of the derived stabilization adjunction (Example A.41) goesfrom non-abelian A -cohomology theory (Def. 2.1) to (abelian) generalized cohomology theory (Example 2.13)represented by the 0th component space of the suspension spectrum of A : non-abelian A -cohomology H (cid:0) − ; A (cid:1) stabilization (cid:47) (cid:47) generalized Σ ∞ A -cohomology H (cid:0) − ; ( L Σ ∞ A ) (cid:1) . Hence a lift through this operation is an enhancement of generalized cohomology to non-abelian cohomology.
Example 2.25 (Non-abelian enhancement of stable Cohomotopy) . The canonical non-abelian enhancement (in thesense of Example 2.24) of stable Cohomotopy (Example 2.16) is actual Cohomotopy theory (Example 2.10):
Cohomotopy π n ( − ) stabilization (cid:47) (cid:47) stableCohomotopy S n ( − ) . Example 2.26 (Hurewicz homomorphism and Hopf degree theorem) . By definition of Eilenberg-MacLane spaces(10) there is, for n ∈ N , a canonical map S n e ( n ) (cid:47) (cid:47) K ( Z , n ) ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) , which represents the element 1 ∈ Z (cid:39) π n (cid:0) K ( Z , n ) (cid:1) . The non-abelian cohomology operation (Def. 2.17) inducedby this, from degree n Cohomotopy (Example 2.10) to degree n ordinary cohomology (Example 2.2) π n ( − ) e ( n ) ∗ (cid:47) (cid:47) H n ( − ; Z ) is the cohomological version of the Hurewicz homomorphism. The Hopf degree theorem (e.g. [Kos93, §IX (5.8)])is the statement that the non-abelian cohomology operation e ( n ) ∗ becomes an isomorphism on connected, orientableclosed manifolds of dimension n . These maps, together with their differential refinements, are analyzed in moredetails via Postnikov towers in [GS20]. Structured non-abelian cohomology.Remark 2.27 (Structured non-abelian cohomology) . More generally, it makes sense to consider the analog of Def.2.1 for the homotopy category Ho ( H ) of a model category which is a homotopy topos [TV05][Lu09][Re10]. (i) This yields structured non-abelian cohomology [Si97][Si99][To02][SSS12][NSS12a][NSS12b][Sch13][FSS19b][SS20b]: structurednon-abelian cohomology H (cid:0) X ; A (cid:1) : = homotopy topos Ho ( H ) (cid:0) X , A ∞ -stacks (cid:1) , including the stacky non-abelian cohomology originally considered in [Gi71][Br90] (“gerbes”, see [NSS12a,§4.4]), and, more generally, differential-, ´etale-, and equivariant - nonabelian cohomology theories (see [SS20b, p.6]) based on ∞ -stacks. (ii) In good cases (cohesive homotopy toposes [Sch13][SS20b, §3.1]), the homotopy topos Ho ( H ) comes equippedwith a shape operation down to the classical homotopy category (Example A.33): homotopy topos Ho ( H ) Shp (cid:47) (cid:47) classical homotopy category Ho (cid:0) TopologicalSpaces Qu (cid:1) H (cid:0) X ; A (cid:1) structurednon-abelian cohomology (cid:31) (cid:47) (cid:47) H (cid:0) Shp ( X ) ; Shp ( A ) (cid:1) plainnon-abelian cohomology (28)10hich takes, for well-behaved group ∞ -stacks G , the classifying stacks B G of G -principal bundles to the traditionalclassifying spaces BG (cid:39) Shp ( B G ) of underlying topological groups (12). This gives a forgetful functor fromstructured non-abelian cohomology to plain non-abelian cohomology in the sense of Def. 2.1. A classical exampleis the map from non-abelian ˇCech cohomology with coefficients in a well-behaved group G to homotopy classesof maps to the classifying space of G , in which case this comparison map is a bijection (Example 2.3).All constructions on non-abelian cohomology have their structured analogues, for instance non-abelian coho-mology operations (Def. 2.17) in structured cohomology H (cid:0) X ; A (cid:1) φ ∗ (cid:47) (cid:47) H (cid:0) X ; A (cid:1) (29)are induced by postcomposition with morphisms A φ (cid:47) (cid:47) A of coefficient stacks.Ultimately, one is interested in working with structured non-abelian cohomology on the left of (28). However,since this is rich and intricate, it behooves us to study its projection into plain non-abelian cohomology on the rightof (28). This is what we are mainly concerned with here. But we provide in §4.3 a brief discussion of non-abeliandifferential cohomology on smooth ∞ -stacks, For C any category and B ∈ C any object, there is the slice category C / X , whose objects are morphisms in C to X and whose morphisms are commuting triangles over X in C . Basic as this is, hom-sets in the homotopy category Ho ( C / B ) (Def. A.14) of a slice model category C / B (Example A.10) are of paramount interest:The slicing imposes twisting on the corresponding non-abelian cohomology (Def. 2.1), in that the slicing ofthe domain space serves as a twist, the slicing of the coefficient space as a local coefficient bundle, and the slicemorphisms as twisted cocycles. Proposition 2.28 ( ∞ -Actions on homotopy types [DDK80][Pr10, §5][NSS12a, §4][Sh15][SS20b, §2.2]) . For anyA ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) (Example A.33) and G a topological group, homotopy-coherent actions of G on Aare equivalent to fibrations ρ with homotopy fiber A (Def. A.22) over the classifying space BG (12) A (cid:47) (cid:47) A (cid:12) G ρ (cid:15) (cid:15) BG . (30) Here A (cid:12) G (cid:39) (cid:0) A × EG (cid:1) / diag G is the homotopy quotient (Borel construction) of the action. Definition 2.29 (Twisted non-abelian cohomology [NSS12a, §4][FSS19b, (10)][SS20b, Rem. 2.94]) . For X , A ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) (Def. A.33) we say: (i) A local coefficient bundle for twisted A -cohomology is an A -fibration ρ over a classifying space BG (12) as inProp. 2.28: A (cid:47) (cid:47) A (cid:12) G ρ (cid:15) (cid:15) local coefficientbundle BG . (31) (ii) A twist for non-abelian A -cohomology theory on X with local coefficient bundle ρ over BG is a map X τ (cid:47) (cid:47) BG ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) . (32) (iii) The non-abelian τ -twisted A-cohomology of X with local coefficients ρ is the hom-set from τ (32) to ρ (30) twistednon-abeliancohomology H τ ( X ; A ) : = Ho (cid:16) TopologicalSpaces / BG Qu (cid:17)(cid:0) τ , ρ (cid:1) = X cocycle c (cid:47) (cid:47) twist τ (cid:32) (cid:32) A (cid:12) G ρ localcoefficients (cid:125) (cid:125) BG (cid:39) (cid:112) (cid:120) (cid:14) homotopyrelative BG (33)11n the homotopy category (Def. A.14) of the slice model category over BG (Example A.10) of the classical modelcategory on topological spaces (Example A.7). Definition 2.30 (Associated coefficient bundle [NSS12a, §4.1][SS20b, Prop. 2.92]) . Given a local coefficient A -fiber bundle ρ (31) and a twist τ (32) on a domain space X , the corresponding associated A-fiber bundle over X isthe homotopy pullback (Def. A.23) of ρ along τ , sitting in a homotopy pullback square (329) of this form: associated A -fiber bundle E (cid:47) (cid:47) R τ ∗ ρ (cid:15) (cid:15) (hpb) homotopy pullback A (cid:12) G ρ (cid:15) (cid:15) localcoefficient bundle X τ twist (cid:47) (cid:47) BG (34)We write sections ofassociated bundle Γ X ( E ) / ∼ : = Ho (cid:16) TopologicalSpaces / X Qu (cid:17)(cid:0) id X , R τ ∗ ρ (cid:1) = E associatedbundle (cid:15) (cid:15) X section σ (cid:54) (cid:54) X (cid:14) verticalhomotopy (35)for the set of vertical homotopy classes of section of the associated bundle, hence for the hom-set, from the identityon X to the associated bundle projection, in the homotopy category (Def. A.14) of the slice model category over X (Example A.10) of the classical model category on topological spaces (Example A.7). Proposition 2.31 (Twisted non-abelian cohomology is sections of associated coefficient bundle [NSS12a, Prop.4.17]) . Given a local coefficient bundle ρ (31) and a twist τ (32) , the τ -twisted non-abelian cohomology (Def.2.29) with local coefficient in ρ is equivalent to the vertical homotopy classes of sections (35) of the associatedcoefficient bundle E (Def. 2.30): twisted non-abeliancohomology H τ ( X ; A ) (cid:39) sections ofassociated bundle Γ X ( E ) / ∼ . (36) Proof.
Consider the following sequence of bijections: H τ ( X ; A ) = Ho (cid:16) TopologicalSpaces / BG Qu (cid:17)(cid:0) τ , ρ (cid:1) (cid:39) Ho (cid:16) TopologicalSpaces / BG Qu (cid:17)(cid:0) L τ ∗ id X , ρ (cid:1) (cid:39) Ho (cid:16) TopologicalSpaces / X Qu (cid:17)(cid:0) id X , R τ ∗ ρ (cid:1) = Γ X ( E ) / ∼ . Here the first line is the definition (33). Then the first step is the observation that every slice object is the derived leftbase change (Example A.18, Prop. A.20) along itself of the identity on its domain, by (320). With this, the secondstep is the hom-isomorphism (304) of the derived base change adjunction L τ ! (cid:97) R τ ∗ . The last line is (35).In twisted generalization of Example 2.2 we have: Example 2.32 (Twisted ordinary cohomology) . Let n ∈ N , let X ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) (Ex. A.33) beconnected and consider a traditional system of local coefficients [St43, §3] (see also [MQRT77][ABG10][GS18c]) Π ( X ) t (cid:47) (cid:47) AbelianGroups , namely, a functor from the fundamental groupoid of X to the category of abelian groups. Since the construction A K ( A , n ) of Eilenberg-MacLane spaces (10) is itself functorial and using the assumption that X is connected,this induces (see [BFGM03, Def. 3.1]) a local coefficient bundle (31) of the form K ( A , n ) (cid:47) (cid:47) K ( A , n ) (cid:12) π ( X ) . ρ t (cid:15) (cid:15) B π ( X ) (37)12inally, write X τ (cid:47) (cid:47) B π ( X ) for the classifying map (via Example 2.3) of the universal connected cover of X (equivalently: for the 1-truncation projection of X ). Then the τ -twisted non-abelian cohomology (Def. 2.29) of X with local coefficients in ρ t (37) is equivalently the traditional t -twisted ordinary cohomology of X in degree n : twistedordinary cohomology H n + t ( X ; A ) (cid:39) H τ (cid:0) X ; K ( A , n ) (cid:1) . This is manifest from comparing Def. 2.29 with the characterization of twisted ordinary cohomology found in[Hir79, Cor. 1.3][GJ99, p. 332][BFGM03, Lemma 4.2].
Example 2.33 (Classification of tangential structure) . Let X be a smooth manifold of dimension n . Its framebundle is an O ( n ) -principal bundle Fr ( X ) ! X , whose class (a diffeomorphism invariant of X )O ( n ) Bundles ( X ) / ∼ (cid:39) (cid:47) (cid:47) H (cid:0) X ; B O ( n ) (cid:1)(cid:2) Fr ( X ) (cid:3) ↔ (cid:2) τ fr (cid:3) (38)gives, by Example 2.3, the class of a twist τ fr (32) in the non-abelian O ( n ) -cohomology of X .Now for BG any connected homotopy type (Prop. 2.8) and for BG ρ (cid:47) (cid:47) B O ( n ) any map (equivalently thedelooping of a morphism of ∞ -groups G (cid:47) (cid:47) O ( n ) ), we get a local coefficient bundle (31) with (homotopy-)cosetspace fiber [FSS19b, Lemma 2.7]: O ( n ) (cid:12) G hofib ( ρ ) (cid:47) (cid:47) BG ρ (cid:15) (cid:15) B O ( n ) . (39)The relative homotopy class of a homotopy lift of the frame bundle classifier τ fr (2.33) through this map ρ X tangential structure (cid:47) (cid:47) τ fr (cid:34) (cid:34) BG ρ (cid:123) (cid:123) B O ( n ) g (cid:111) (cid:119) ∈ G TangentialStructures ( X ) (40)is known a topological G-structure or tangential ρ -structure on X (e.g. [Ko96, §1.4][GMTW09, §5][SS20b, Def.4.48]). For instance, for ρ a stage in the Whitehead tower of O ( n ) , this is, in turn, Orientation , Spin structure , String structure , Fivebrane structure [SSS12], etc.: ... (cid:15) (cid:15) B Fivebrane ( n ) (cid:15) (cid:15) B String ( n ) (cid:15) (cid:15) B Spin ( n ) (cid:15) (cid:15) B SO ( n ) (cid:15) (cid:15) X Orientation (cid:51) (cid:51)
Spinstructure (cid:52) (cid:52)
Stringstructure (cid:53) (cid:53)
Fivebranestructure (cid:53) (cid:53) τ fr (cid:47) (cid:47) B O ( n ) By comparison of (40) with (33) we see that tangential G -structures on X are classified by twisted non-abeliancohomology (Def. 2.29) with coefficients in (homotopy-)coset spaces O ( n ) (cid:12) G (39) and twisted by the class τ fr ofthe frame bundle (38): G TangentialStructures ( X ) (cid:39) H τ fr (cid:0) X ; O ( n ) (cid:12) G (cid:1) . (41)According to Prop. 2.8, this example is actually universal for τ fr -twisted non-abelian cohomology.13s a special case of Example 2.32 and in twisted generalization of Examples 2.11, 2.12 we have: Example 2.34 (Orientifold gerbes) . Consider the action σ U ( ) of Z on the circle group U ( ) ⊂ C × given bycomplex conjugation. This deloops (see [FSS15a, §4.4]) to an action σ B n U ( ) of Z on the classifying spaces B n U ( ) (12). By Prop. 2.28 there is a corresponding local coefficient bundle B n U ( ) (cid:47) (cid:47) B n U ( ) (cid:12) Z σ Bn U ( ) (cid:15) (cid:15) Z (42)Moreover, consider a smooth manifold X , with orientation bundle classified by X or (cid:47) (cid:47) B Z . Then the or-twistedcohomology (Def. 2.29) of X ... (i) ...with local coefficients in σ B U ( ) classifies what is equivalently known as Jandl gerbes [SSW07][GSW11] orreal gerbes [HMSV19] or orientifold B-fields; (ii) ...with local coefficients in σ B U ( ) classifies what is equivalently known as topological sectors of orientifoldC-fields [FSS15a, §4.4].More generally, one can consider twisted Deligne cohomology [GS18c] as well as higher-twisted periodic integral-and Deligne-cohomology [GS19b] (see also §4.3). Remark 2.35 (The Whitehead principle of non-abelian cohomology) . Let A ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) be con-nected, so that A (cid:39) BG (Prop. 2.8). (i) If A is also n -truncated (341), then its Postnikov tower (Prop. A.38) says that A is the total space of a localcoefficient bundle (2.29) of the form K ( π n ( A ) , n ) hfib ( p n ) (cid:47) (cid:47) A p An (cid:15) (cid:15) A ( n − ) (cid:39) B (cid:0) G ( n − ) (cid:1) with homotopy fiber an Eilenberg-MacLane space (10). (ii) Accordingly, non-abelian cohomology with coefficients in A (Def. 2.1) is equivalently the disjoint union, overthe space of twists τ n (32) in non-abelian cohohomology with coefficients in A ( n − ) , of τ -twisted non-abeliancohomology (Def. 2.29) with coefficients in K ( π n ( A ) , n ) : non-abelian cohomologyin higher degree H ( X ; A ) (cid:39) (cid:71) τ n ∈ H ( X ; A ( n − )) twist innon-abelian cohomologyof lower degree higher twistedordinary cohomology H τ n (cid:0) X ; K ( π n ( A ) , n ) (cid:1) . (43) (iii) Iterating this unravelling yields H ( X ; A ) (cid:39) (cid:71) τ n ∈ (cid:70) τ n − ∈ H ( X ; A ( n − )) H τ n − (cid:0) X ; K ( π n − ( A ) , n − ) (cid:1) H τ n (cid:0) X ; K ( π n ( A ) , n ) (cid:1) . (44)and then H ( X ; A ) (cid:39) (cid:71) τ n ∈ (cid:70) τ n − ∈ (cid:70) τ n − ∈ H (cid:0) X ; A ( n − ) (cid:1) H τ n − (cid:0) X ; K ( π n − ( A ) , n − ) (cid:1) H τ n − (cid:0) X ; K ( π n − ( A ) , n − ) (cid:1) H τ n (cid:0) X ; K ( π n ( A ) , n ) (cid:1) . (45)and then H (cid:0) X ; A (cid:1) (cid:39) (cid:71) τ n ∈ (cid:70) τ n − ∈ (cid:70) τ n − ∈ (cid:70) τ n − ∈ H (cid:0) X ; A ( n − ) (cid:1) H τ n − (cid:0) X ; K ( π n − ( A ) , n − ) (cid:1) H τ n − (cid:0) X ; K ( π n − ( A ) , n − ) (cid:1) H τ n − (cid:0) X ; K ( π n − ( A ) , n − ) (cid:1) H τ n (cid:0) X ; K ( π n ( A ) , n ) (cid:1) . (46)14nd so on. (iv) Thus non-abelian cohomology in higher degrees (Example 2.7) decomposes as a tower of consecutively highertwisted but otherwise ordinary cohomology theories, starting with a twist in non-abelian cohomology in degree 1.This phenomenon has been called the
Whitehead principle of non-abelian cohomology [To02, p. 8] and has beeninterpreted as saying that “nonabelian cohomology occurs essentially only in degree 1” [Si96, p. 1]. (v)
But the above formulas (43), (44), (45) make manifest that this phenomenon has two perspectives. On the onehand: non-abelian cohomology in higher degrees may be computed by brute force as a sequence of consecutivelyhigher twisted abelian cohomologies, with lowest twist starting in degree-1 non-abelian cohomology. On theother hand, conversely: intricate such systems of consecutively twisted abelian cohomology theories are neatly understood as unified by non-abelian cohomology. (vi)
Similarly, even though Postnikov towers do exist (Prop. A.38) in the classical homotopy category (ExampleA.33), the latter is far from being equivalent to the stable homotopy category (350) “up to twists in degree 1”.In twisted generalization of Example 2.14, we have:
Example 2.36 (Twisted topological K-theory) . The classifying space KU (cid:39) Z × B U (19) for complex topologicalK-theory (Example 2.14) is the fiber of a local coefficient bundle (31) over K ( Z , ) (cid:39) B U ( ) (16):KU (cid:47) (cid:47) KU (cid:12) B U ( ) (cid:15) (cid:15) B U ( ) (47)For X τ (cid:47) (cid:47) B U ( ) a corresponding twist (32) (hence equivalently a bundle gerbe, by Example 2.11), the corre-sponding twisted non-abelian cohomology (Def. 2.29) is twisted complex topological K-theory [Ka68][DK70]: twistedtopological K-theory KU τ ( − ) (cid:39) H τ (cid:0) − ; Z × B U (cid:1) . (48)This is manifest from comparing (33) with [FrHT08, (2.6)]. Alternatively, under Prop. 2.31, this is manifest fromcomparing the equivalent right hand side of (36) with [Ros89, Prop. 2.1] (using [NSS12a, Cor. 4.18]) or, moredirectly, with [AS04, §3][ABG10, §2.1].Generally, in twisted generalization of Example 2.13, we have: Example 2.37 (Local coefficient bundle for twisted generalized cohomology) . Let R be a suitable ring spec-trum and write GL R ( ) for its ∞ -group (as in Example 2.7) of units [Schl04, §2.3][MaSi04, §22.2][ABGHR08,§3][ABGHR14a, §2]. Its canonical action on the component space R = R Ω ∞ R (351) is given, via Prop. 2.28, bya local coefficient bundle (31) of the form R (cid:47) (cid:47) ( R ) (cid:12) GL R ( ) ρ R (cid:15) (cid:15) B GL R ( ) . (49) Proposition 2.38 (Twisted non-abelian cohomology subsumes twisted generalized cohomology) . For R a suitablering spectrum, the twisted non-abelian cohomology (Def. 2.29) with local coefficient bundle ρ R from Example 2.37is, equivalently, twisted generalized R-cohomology in the traditional sense (e.g. [MaSi04, §22.1]): twistedgeneralized cohomology R τ ( − ) (cid:39) H τ ( − ; ρ R ) . (50) Proof.
Given any twist X τ (cid:47) (cid:47) B GL R ( ) (2.29), write P ! X for the homotopy pullback (Def. A.23) along τ of theessentially unique point inclusion: P (cid:47) (cid:47) (cid:15) (cid:15) (hpb) ∗ (cid:15) (cid:15) X τ (cid:47) (cid:47) B GL R ( ) , (cid:0) P × R (cid:1) (cid:12) diag GL R ( ) (cid:39) E (cid:47) (cid:47) R τ ∗ ρ R (cid:15) (cid:15) (hpb) R (cid:12) GL R ( ) ρ R (cid:15) (cid:15) X τ (cid:47) (cid:47) B GL R ( ) (51)15his P is the GL R ( ) -principal ∞ -bundle which is classified by τ , [NSS12a, Thm. 3.17], to which the coefficientbundle E (34) is GL R ( ) -associated [NSS12a, Prop. 4.6], as shown on the right of (51). Consider then the followingsequence of natural bijections: H τ (cid:0) X ; R (cid:1) (cid:39) Γ X ( E ) (cid:39) Ho (cid:0) GL R ( ) Actions (cid:1) ( P ; R ) (cid:39) Ho (cid:0) R Modules (cid:1) ( M τ ; R ) (cid:39) R τ ( X ) . (52)Here the first step is Prop. 2.31, while the second step is [NSS12a, Cor. 4.18]. The third step is [ABGHR08,(2.15)][ABGHR14a, (3.15)], with M τ denoting the R -Thom spectrum of τ [ABGHR08, Def. 2.6][ABGHR14a,Def. 3.13]. The last step is [ABGHR08, §2.5] [ABGHR14a, §1.4][ABGHR14b, §2.7]. The composite of thesenatural bijections is the desired (50).In twisted generalization of Example 2.15, we have: Example 2.39 (Twisted iterated K-theory) . Let r ∈ N , r ≥
1. By [LSW16, Prop. 1.5, Def. 1.7] and using Prop.2.38, there is a local coefficient bundle (31) of the form (cid:0) K r − ( ku ) (cid:1) (cid:47) (cid:47) (cid:16)(cid:0) K r − ( ku ) (cid:1) (cid:17) (cid:12) B r − U ( ) ρ lsw2 r − (cid:15) (cid:15) B r U ( ) , (53)where K r − ( ku ) is the 0th space in the spectrum (17) representing iterated K-theory (Example 2.15) and B r U ( ) (cid:39) K ( Z , r + ) is the classifying space for bundle ( r − ) -gerbes (Example 2.12), such that for X τ (cid:47) (cid:47) B r U ( ) a clas-sifying map for such a higher gerbe, the τ twisted non-abelian cohomology (Def. 2.29) with local coefficients in(53) is equivalently integrally twisted iterated K-theory according to [LSW16]: twistediterated K-theory (cid:0) K ◦ r − ( ku ) (cid:1) τ ( − ) (cid:39) H τ (cid:16) − ; K ◦ r − ( ku ) (cid:17) . In twisted generalization of Example 2.10, we have:
Example 2.40 (J-Twisted Cohomotopy theory [FSS19b, §2.1]) . For n ∈ N , consider the canonical action of theorthogonal group O ( n + ) on the homotopy type of the n -sphere, via the defining action on the unit sphere in R n + . By Prop. 2.28 this corresponds to a local coefficient bundle (31) for twisting Cohomotopy theory (Example2.10): S n (cid:47) (cid:47) S n (cid:12) O ( n ) ρ J (cid:15) (cid:15) B O ( n ) . (54)The classifying map B O ( n ) J (cid:47) (cid:47) Aut ( S n ) of this fibration is the unstable J-homomorphism . For X a smooth mani-fold of dimension d ≥ k +
1, and equipped with tangential O ( k + ) -structure (e.g. [SS20b, Def. 4.48]) X T X (cid:37) (cid:37) τ (cid:47) (cid:47) B O ( k + ) Bi (cid:118) (cid:118) B O ( d ) (cid:39) (cid:109) (cid:117) the τ -twisted non-abelian Cohomology (Def. 2.29) with local coefficients in (54) is the J-twisted Cohomotopytheory of [FSS19b][FSS19c][SS20a]:
J-twistedCohomotopy π τ ( − ) : = H τ (cid:0) − ; S n (cid:1) . J-twisted Cohomotopy in degree four encodes, in particular, the shifted flux quantization condition of theC-field [FSS19b, Prop. 3.13] and the vanishing of the residual M5-brane anomaly [SS20a]; while J-twisted Coho-motopy in degree four encodes, in particular, level quantization of the Hopf-Wess-Zumino term on the M5-brane[FSS19c]. 16 wisted non-abelian cohomology operations.
In generalization of Def. 2.17, we set:
Definition 2.41 (Twisted non-abelian cohomology operation) . Given a transformation of local coefficient bundles(31) presented (under localization (317) to homotopy types (338)) as a strictly commuting diagram A (cid:12) G ρ (cid:15) (cid:15) φ t (cid:47) (cid:47) A (cid:12) G ρ (cid:15) (cid:15) BG φ b (cid:47) (cid:47) BG ∈ TopologicalSpaces Qu , (55) pasting composition induces for each twist X τ (cid:47) (cid:47) BG (32) a map φ ∗ : H τ ( X ; A ) (cid:0) φ t ◦ ( − ) (cid:1) ◦ (cid:0) ρ (cid:1) ∗ (cid:47) (cid:47) H φ b ◦ τ ( X ; A ) (56)of twisted non-abelian cohomology sets (Def. 2.29). We call these twisted non-abelian cohomology operations . Example 2.42 (Total non-abelian class of twisted cocycles) . For any coefficient bundle ρ (31) there is the tauto-logical transformation (55) to its total space regarded as fibered over the point: A (cid:12) G ρ (cid:15) (cid:15) A (cid:12) G (cid:15) (cid:15) BG (cid:47) (cid:47) ∗ . The induced twisted non-abelian cohomology operation (56) goes from twisted cohomology to non-twisted coho-mology with coefficient in the total space: H τ (cid:0) X ; A (cid:1) ρ ∗ (cid:47) (cid:47) H (cid:0) X ; A (cid:12) G (cid:1) (57) Example 2.43 (Hopf cohomology operation in J-twisted Cohomotopy [FSS19b, §2.3]) . The quaternionic Hopffibration S h H (cid:47) (cid:47) S is equivariant under the symplectic unitary group Sp ( ) (cid:39) Spin ( ) , so that after passageto classifying spaces it induces a morphism of local coefficient bundles (55) for J-twisted Cohomotopy (54) indegrees 4 and 7: S (cid:12) Sp ( ) h H (cid:12) Sp ( ) Borel-equivariantizedquaternionic Hopf fibration (cid:47) (cid:47) J (cid:15) (cid:15) S (cid:12) Sp ( ) J (cid:15) (cid:15) B Sp ( ) (cid:43) (cid:51) B Sp ( ) (58)Via (56) this induces for each Spin 8-manifold X equipped with tangential Sp ( ) -structure (Example 2.33) X T X (cid:37) (cid:37) τ (cid:47) (cid:47) B Sp ( ) Bi (cid:119) (cid:119) B O ( ) (cid:39) (cid:109) (cid:117) (59)a twisted non-abelian cohomology operation (Def. 2.41) π τ ( X ) ( h H (cid:12) Sp ( )) ∗ (cid:47) (cid:47) π τ ( X ) (60)in J-twisted non-abelian Cohomotopy theory (Example 2.40).Lifting through the twisted non-abelian cohomology transformation (60) encodes vanishing of C-field flux upto C-field background charge [FSS19b, Prop. 3.14]. We postpone discussing the details of forming pasting composites to §5, where they are provided by Lemma 5.1 with Def. 5.2. xample 2.44 (Twistorial Cohomotopy [FSS20, §3.2] ) . The equivariantized Hopf morphism (58) of coefficientbundles factors through Borel-equivariantizations of the complex Hopf fibration h C followed by that of the twistorfibration t H S (cid:12) Sp ( ) h C (cid:12) Sp ( ) Borel-equivariantizedcomplex Hopf fibration (cid:47) (cid:47) J S (cid:15) (cid:15) C P (cid:12) Sp ( ) t H (cid:12) Sp ( ) Borel-equivariantizedtwistor fibration (cid:47) (cid:47) J CP (cid:15) (cid:15) S (cid:12) Sp ( ) J S (cid:15) (cid:15) B Sp ( ) (cid:43) (cid:51) B Sp ( ) (cid:43) (cid:51) B Sp ( ) (61)The twisted non-abelian cohomology theory (Def. 2.29) with local coefficients in the bundle appearing in thisfactorization is the Twistorial Cohomotopy of [FSS20]
TwistorialCohomotopy T τ ( − ) : = H τ (cid:0) − ; C P (cid:1) . Via (56) the morphisms (61) induce, for each spin 8-manifold X equipped with tangential Sp ( ) -structure (59),twisted non-abelian cohomology operations (Def. 2.41) J-twisted7-Cohomotopy π τ ( X ) ( h C (cid:12) Sp ( )) ∗ (cid:47) (cid:47) TwistorialCohomotopy T τ ( X ) s ( t H (cid:12) Sp ( )) ∗ (cid:47) (cid:47) J-twisted4-Cohomotopy π τ ( X ) (62)between J-twisted non-abelian Cohomotopy theory (Example 2.40) and Twistorial Cohomotopy.We turn to the differential refinement of this statement in §5.3 below.18 Non-abelian de Rham cohomology
We formulate (twisted) non-abelian de Rham cohomology (Def. 3.84, Def. 3.98) of differential forms with valuesin L ∞ -algebras (Example 3.25) and prove the (twisted) non-abelian de Rham theorem (Theorem 3.87, Theorem3.104), as a consequence of the fundamental theorem of dg-algebraic rational homotopy theory, which we recall(Prop. 3.60). L ∞ -algebras Here we fix notation and conventions for the following system of categories and functors: (cid:0)
Def. 3.34 L ∞ Algebras ≥ , nil R , fin (cid:1) op (cid:127) (cid:95) (cid:15) (cid:15) (104) CE (cid:39) (cid:47) (cid:47) Def. 3.31
SullivanModels ≥ R (cid:127) (cid:95) (cid:15) (cid:15) Def. 3.25 (cid:0) L ∞ Algebras ≥ R , fin (cid:1) op (cid:31) (cid:127) (88) CE (cid:47) (cid:47) Def. 3.17
DiffGradedCommAlgebras ≥ R (cid:111) (cid:111) Def. 3.19
Sym ⊥ (cid:47) (cid:47) Def. 3.18
GrddCmmttvAlgbr (cid:15) (cid:15)
Def 3.14
CochainComplexes ≥ R Def. 3.15
GrddVctrSpc (cid:15) (cid:15)
GradedCommAlgebras ≥ R Def. 3.8 (cid:111) (cid:111)
Def. 3.10
Sym ⊥ (cid:47) (cid:47) GradedVectorSpaces ≥ R Def. 3.2 (63)
Remark 3.1 (Homotopical grading) . Our grading conventions, to be detailed in the following, are strictly homo-topy theoretic : (i) Any graded-algebraic object discussed here, corresponds, under the equivalences of rational homotopy theorylaid out in §3.2 below, to a rational space, such that algebraic generators in degree n correspond to homotopygroups in the same degree n . Since homotopy groups of spaces are in non-negative degree n ∈ N , all dg-algebraicobjects discussed here are concentrated in non-negative degree, hence are connective . (ii) In particular, our L ∞ -algebras are in non-negative degree, naturally accommodating (as in [LM95][BFM06,§2.9]) the rationalized Whitehead homotopy Lie algebras π • ( Ω X ) ⊗ Z R of connected spaces X , with their naturalnon-negative grading induced from that of the homotopy groups of Ω X . See Prop. 3.63 and Prop. 3.65 below. Graded vector spaces.Definition 3.2 (Connective graded vector spaces) . (i)
We writeGradedVectorSpaces ≥ R ∈ Categories (64)for the category whose objects are N -graded (i.e. non-negatively Z -graded) vector spaces over the real numbers;and we write GradedVectorSpaces ≥ , fin R (cid:31) (cid:127) (cid:47) (cid:47) GradedVectorSpaces ≥ R ∈ Categories (65)for its full subcategory on those objects which are of finite type , namely degree-wise finite-dimensional. (ii)
For V ∈ GradedVectorSpaces ≥ R and k ∈ N we write V k ∈ VectorSpaces R for the component vector space in degree k . Example 3.3 (The zero-object in graded vector spaces) . We write0 ∈ GradedVectorSpaces ≥ R (66)for the graded vector space which is the zero vector space in each degree. This is both the initial as well as theterminal object (hence the zero object) in GradedVectorSpaces ≥ R .19 xample 3.4 (Graded linear basis) . For n , n , · · · , n k ∈ N a finite sequence of non-negative integers, we write (cid:10) α n , α n , · · · , α n k (cid:11) ∈ GradedVectorSpaces ≥ , fin R for the graded vector space (Def. 3.2) spanned by elements α n i in degree n i , respectively. Definition 3.5 (Tensor product of graded vector spaces) . The category of GradedVectorSpaces ≥ R (Def. 3.2) be-comes a symmetric monoidal category under the graded tensor product given by ( V ⊗ W ) k : = (cid:77) n + n = k V n ⊗ W n . and the symmetric braiding isomorphism given by V ⊗ W σ V , W (cid:39) (cid:47) (cid:47) W ⊗ VV n ⊗ W n (cid:63)(cid:31) (cid:79) (cid:79) σ V , Wn , n (cid:39) (cid:47) (cid:47) W n ⊗ V n (cid:63)(cid:31) (cid:79) (cid:79) ( v , w ) ∈ ( − ) n n · ( w , v ) ∈ (67)We denote this by (cid:0) GradedVectorSpaces ≥ R , ⊗ , σ (cid:1) ∈ SymmetricMonoidalCategories . (68) Definition 3.6 (Degreewise linear dual) . For V ∈ GradedVectorSpaces ≥ , fin R (Def. 3.2) we write V ∨ ∈ GradedVectorSpaces ≥ , fin R for its degree-wise linear dual: ( V ∨ ) k : = ( V k ) ∗ . (69) Definition 3.7 (Degree shift) . For V ∈ GradedVectorSpaces ≥ R (Def. 3.2) we write b V ∈ GradedVectorSpaces ≥ R (70)for the result of shifting degrees up by 1: ( b V ) k : = (cid:26) V k − | k ≥ , | k = . Graded-commutative algebras.Definition 3.8 (Graded-commutative algebras) . We writeGradedCommAlgebras ≥ R : = CommMonoids (cid:0)
GradedVectorSpaces ≥ R , ⊗ , σ (cid:1) ∈ Categories (71)for the category whose objects are non-negatively Z -graded, graded-commutative unital algebras over the realnumbers (hence commutative unital monoids with respect to the braided tensor product of Def. 3.5); and we writeGradedCommAlgebras ≥ , fin R (cid:31) (cid:127) (cid:47) (cid:47) GradedCommAlgebras ≥ R ∈ Categories (72)for its full sub-category in those objects which are of finite type , namely degree-wise finite dimensional.
Definition 3.9 (Underlying graded vector space) . We writeGradedCommAlgebras ≥ R GrddVctrSpc (cid:47) (cid:47)
GradedVectorSpaces ≥ R (73)for the functor on graded algebras (Def. 3.8) that forgets the algebra structure and remembers only the underlyinggraded vector space (Def. 3.2). Example 3.10 (Free graded-commutative algebras) . For V ∈ GradedVectorSpaces ≥ R (Def. 3.2), we writeSym ( V ) ∈ GradedCommAlgebras ≥ R (74)for the graded-commutative algebra (Def. 3.8) freely generated by V , hence that whose underlying graded vectorspace (73) is GrddVctrSpc (cid:0) Sym ( V ) (cid:1) = R ⊕ V ⊕ (cid:0) V ⊗ V (cid:1) / Sym ( ) ⊕ (cid:0) V ⊗ V ⊗ V (cid:1) / Sym ( ) ⊕ · · · , where the symmetric groups Sym ( n ) act via the braiding (67). This is in contrast to the intrinsic duality ( − ) ∗ in the monoidal category of graded vector spaces in unbounded degree (not consideredhere), which instead goes along with inversion of the degree: ( V ∗ ) k = ( V − k ) ∗ . xample 3.11 (Graded Grassmann algebra) . For V ∈ GradedVectorSpaces ≥ R (Def. 3.2), we write ∧ • V : = Sym (cid:0) b V (cid:1) ∈ GradedCommAlgebras ≥ R for the free graded-commutative algebra (Def. 3.10) on V shifted up in degree (Def. 3.7); and we call this the graded Grassmann-algebra on V . Example 3.12 (Graded polynomial algebra) . For n , n , · · · , n k ∈ N a finite sequence of non-negative integers, wewrite R (cid:2) α n , α n , · · · , α n k (cid:3) : = Sym (cid:16)(cid:10) α n , α n , · · · , α n k (cid:11)(cid:17) ∈ GradedCommAlgebras ≥ , fin R for the free graded-commutative algebras (Def. 3.10) the graded vector space spanned by the α n i (Def. 3.4). Remark 3.13 (Incarnations of Grassmann algebras) . With these notation conventions from Examples 3.10, 3.11,3.12, an ordinary Grassmann algebra on k generators is equivalently: ∧ • (cid:0) R k (cid:1) = Sym (cid:0) b R k (cid:1) = R (cid:2) θ ( ) , θ ( ) , · · · , θ ( k ) (cid:3) . Cochain complexes.Definition 3.14 (Connective cochain complexes) . We writeCochainComplexes ≥ R ∈ Categoriesfor the category of cochain complexes (i.e. with differential of degree +
1) of real vector spaces in non-negativedegree.
Definition 3.15 (Underlying graded vector space) . We writeCochainComplexes ≥ R GrddVctrSpc (cid:47) (cid:47)
GradedVectorSpaces ≥ R (75)for the forgetful functor on connective cochain complexex (Def. 3.14) which forgets the differential and remembersonly the underlying connective graded vector space (Def. 3.2). Definition 3.16 (Tensor product on cochain complexes) . The tensor product and braiding of graded vector spacesfrom Def. 3.5 lifts, through (75), to a tensor product and braiding on CochainComplexes ≥ R (Def. 3.14), making ita symmetric monoidal category: (cid:0) CochainComplexes ≥ R , ⊗ , σ (cid:1) ∈ SymmetricMonoidalCategories . (76) Differential graded commutative algebras.Definition 3.17 (Connective differential graded commutative algebras [GM96, V.3.1]) . We writeDiffGradedCommAlgebras ≥ R : = CommMonoids (cid:0)
CochainComplexes ≥ R , ⊗ , σ (cid:1) ∈ Categoriesfor the category whose objects are differential-graded, graded-commutative, unital algebras over the real numbersconcentrated in non-negative degrees (hence commutative unital monoids in the symmetric monoidal category ofDef. 3.16).
Definition 3.18 (Underlying graded-commutative algebra) . We writeDiffGradedCommAlgebras ≥ R GrddCmmttvAlgbr (cid:47) (cid:47)
GradedCommAlgebras ≥ R (77)for the functor on dgc-algebras (Def. 3.17) that forgets the differential and remembers only the underlying graded-commutative algebra (Def. 3.8). Definition 3.19 (Free differential graded algebras) . For V • in CochainComplexes ≥ R (Def. 3.14) we writeSym ( V • ) ∈ DiffGradedCommAlgebras ≥ R for the free differential graded-commutative algebra on V • , (Def. 3.17), hence whose underlying graded-commutativealgebra algebra (77) is as in Example 3.10. 21 xample 3.20 (Initial algebra) . The real algebra of real numbers, regarded as concentrated in degree-0 R ∈ GradedCommAlgebras ≥ R (cid:31) (cid:127) (cid:47) (cid:47) DiffGradedCommAlgebras ≥ R is the initial object: For any other A ∈ GradedCommAlgebras ≥ R (Def. 71) or ∈ DiffGradedCommAlgebras ≥ R (Def.3.17) there is a unique morphism R (cid:31) (cid:127) i R (cid:47) (cid:47) A (because our algebras are unital and homomorphims need to preserve the unit element). Example 3.21 (The terminal algebra) . We write0 ∈ GradedCommAlgebras ≥ R (cid:31) (cid:127) (cid:47) (cid:47) DiffGradedCommAlgebras ≥ R (78)for the unique graded-commutative algebra (Def. 3.8) or dgc-algebra (Def. 3.17) whose underlying graded vectorspace (Def. 3.9) is the zero-vector space (66). This is the terminal object in GradedCommAlgebras ≥ R : For every A ∈ GradedCommAlgebras ≥ R , there is a unique morphism A ∃ ! (cid:47) (cid:47) . Example 3.22 (Product and co-product algebras) . In the categories GradedCommAlgebras ≥ R (Def. 3.8) andDiffGradedCommAlgebras ≥ R (Def. 3.17): (i) the coproduct is given by the tensor product (Def. 3.5), (ii) the product is given by the direct sumon underlying graded vector spaces (Def. 3.9).(The first follows by [Joh02, p. 478, Cor. 1.1.9], while the second holds since (73) is a right adjoint.) Example 3.23 (Smooth de Rham complex (e.g. [BT82])) . For X be a smooth manifold, its de Rham algebra ofsmooth differential forms is a dgc-algebra in the sense of Def. 3.17, to be denoted here: Ω • dR ( X ) ∈ DiffGradedCommAlgebras ≥ R . Example 3.24 (Chevalley-Eilenberg algebras of Lie algebras) . For ( g , [ − , − ]) a finite-dimensional real Lie algebra,its Chevalley-Eilenberg algebra is a dgc-algebra (Def. 3.17):CE ( g ) : = (cid:0) ∧ • g ∗ , d | ∧ g ∗ = [ − , − ] ∗ (cid:1) ∈ DiffGradedCommAlgebras ≥ R with underlying graded-commutative algebra (Def. 3.8) the Grassmann algebra on the linear dual space g ∗ (Def.3.11, Remark 3.13), and with differential given on ∧ g ∗ by the linear dual of the Lie bracket. More explicitly, for { v a } dim R ( g ) a = a linear basis for the underlying vector space of the Lie algebra g (cid:39) (cid:104) v , v , · · · , v dim ( g ) (cid:105) , (79)with Lie brackets [ v a , v b ] = f cab v c , for structure constants f cab ∈ s R (80)we have CE ( g ) (cid:39) R (cid:2) θ ( ) , θ ( ) , · · · θ ( dim ( g )) (cid:3)(cid:14)(cid:0) d θ ( c ) = f abc θ ( b ) ∧ θ ( a ) (cid:1) . (81)One observes that the Jacobi identity on [ − , − ] is equivalent to the condition that the differential d : = [ − , − ] ∗ squares to zero, so that (81) being a dgc-algebra is actually equivalent to ( g , [ − , − ]) being a Lie algebra.This construction is evidently contravariantly functorial and constitutes a full subcategory inclusionLieAlgebras R , fin (cid:31) (cid:127) CE (cid:47) (cid:47) (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) op , (82)meaning that, in addition, homomorphisms of Lie algebras are in natural bijection to dgc-algebra morphismsbetween their CE-algebras. Notice that the algebra 0 (78) is indeed a unital algebra (71). Beware that the corresponding statement in [GM96, p. 335] is incorrect. ∞ -algebras.Definition 3.25 (Chevalley-Eilenberg algebras of L ∞ -algebras [LM95, Thm . 2.3][SSS09a, Def. 13][BFM06, §2]) . In direct generalization of (82), consider those A ∈ DiffGradedCommAlgebras ≥ R (Def. 3.17) whose underlyinggraded-commutative algebra (77) is free (Example 3.10, Remark 3.13) on the degreewise dual bg ∨ (Def. 3.6) ofthe degree shift bg (Def. 3.7) of some connective finite-type graded vector space (Def. 3.2) g ∈ GradedVectorSpaces ≥ , fin R (83)in that A : = (cid:0) ∧ • g ∨ , d (cid:1) : = (cid:0) Sym ( bg ∨ ) , d (cid:1) ∈ DiffGradedCommAlgebras ≥ R . (84)In this case the differential d restricted to ∧ g ∨ defines, under linear dualization, a sequence of n -ary graded-symmetric multilinear maps {− , · · · , −} on g : d | ∧ g ∨ ( − ) = {−} ∗ + {− , −} ∗ + {− , − , −} ∗ + · · ·∧ g ∨ d (cid:47) (cid:47) ∧ g ∨ ⊕ ∧ g ∨ ⊕ ∧ g ∨ ⊕ · · · = ∧ • g ∨ = Sym (cid:0) bg ∨ (cid:1) , (85)and the condition d ◦ d = n -ary brackets ([LS93, (3)]) [ a , · · · , a n ] : = ( − ) n + ∑ i ≤ n / deg ( a i ) { a , · · · , a n } (86)subject to these conditions give g the structure of an L ∞ -algebra (or strong homotopy Lie algebra ): (cid:16) g , [ − ] , [ − , − ] , [ − , − , − ] , · · · (cid:17) ∈ L ∞ Algebras ≥ R , fin , (87)which makes A in (84) its Chevalley-Eilenberg algebra:CE ( g ) : = (cid:0) ∧ • g ∨ , d = {−} ∗ + {− , −} ∗ + {− , − , −} ∗ + · · · (cid:1) = (cid:0) Sym ( bg ∨ ) , d CE (cid:1) . (88)This construction constitutes a full subcategory inclusion L ∞ Algebras ≥ R , fin (cid:31) (cid:127) CE (cid:47) (cid:47) (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) op . (89)of the category of connective finite-type L ∞ -algebras into that of connective dgc-algebras. Example 3.26 (Differential graded Lie algebras) . A differential graded Lie algebra is an L ∞ -algebra (87) whoseonly possibly non-vanishing brackets are the unary bracket ∂ : = [ − ] (its differential) and the binary bracket [ − , − , ] (its graded Lie bracket). In further specialization, a plain Lie algebra (Example 3.24) is an L ∞ -algebra/dg-Liealgebra concentrated in degree 0:LieAlgebras R , fin (cid:31) (cid:127) (cid:47) (cid:47) DiffGradedLieAlgebras ≥ R , fin (cid:31) (cid:127) (cid:47) (cid:47) L ∞ Algebras ≥ R , fin . (90) Example 3.27 (Line Lie n -algebra) . For n ∈ N we say that the line Lie ( n + ) -algebra is the L ∞ -algebra (Def.3.25) b n R ∈ L ∞ Algebras ≥ R , fin (91)whose Chevalley-Eilenberg algebra (88) is the polynomial dgc-algebra (Example 3.29) on a single closed generatorin degree n +
1: CE (cid:0) b n R (cid:1) : = R [ c n + ] (cid:14) ( d c n + = ) . (92)23 xample 3.28 (String Lie 2-algebra [BCSS07, §5][He08, §1.2][FSS12a, App.]) . Let g ∈ LieAlgebras R , fin be semisim-ple (such as g = su ( n + ) , so ( n + ) , for n ∈ N ), hence equipped with a non-degenerate, symmetric, g -invariantbilinear form (“Killing form”) g ⊗ g (cid:104)− , −(cid:105) (cid:47) (cid:47) R . (93)Then the element µ : = (cid:10) − , [ − , − ] (cid:11) ∈ CE ( g ) in the Chevalley-Eilenberg (3.24) is closed (is a Lie algebra cocycle) d µ = . In terms of a linear basis { v a } (79) with structure constants { f cab } (80) and inner product k ab : = (cid:104) v a , v b (cid:105) we have,in terms of (81): µ : = f abc (cid:48) k c (cid:48) c θ ( c ) ∧ θ ( b ) ∧ θ ( a ) . Hence we get an L ∞ -algebra (Def. 3.25) string g ∈ L ∞ Algebras ≥ R , fin (94)with the following Chevalley-Eilenberg algebra (88):CE (cid:0) string g (cid:1) : = R (cid:34) { θ a } , b (cid:35)(cid:14) d θ ( c ) = f cab θ ( b ) ∧ θ ( a ) d b = f c (cid:48) ab k c (cid:48) c θ ( c ) ∧ θ ( b ) ∧ θ ( a ) (cid:124) (cid:123)(cid:122) (cid:125) = µ . (95)This is known as the string Lie 2-algebra . Sullivan models and nilpotent L ∞ -algebras.Example 3.29 (Polynomial dgc-algebras) . For A ∈ DiffGradedCommAlgebras ≥ R (Def. 3.17), and µ ∈ A n + ⊂ A , d µ = n +
1, we write A (cid:2) α n (cid:3)(cid:14)(cid:0) d α n = µ (cid:1) ∈ DiffGradedCommAlgebras ≥ R (97)for the dgc-algebra obtained by adjoining a generator α n of degree n to the underlying graded-commutative algebra(77) of A and extending the differential from A to A (cid:2) α n (cid:3) by taking its value on the new generator to be µ . Thepolynomial dgc-algebras (97) receives a canonical algebra inclusion of A (the unique A -algebra homomorphism): A (cid:31) (cid:127) i A (cid:47) (cid:47) A [ α n ] (cid:14) ( d α n = µ ) . (98) Example 3.30 (Multivariate polynomial dgc-algebras) . Let A ∈ DiffGradedCommAlgebras ≥ R (Def. 3.17), µ ( ) ∈ A n + , d µ ( ) =
0, with corresponding polynomial dgc-algebra (97) as in Example 3.29. Then, for µ ( ) ∈ A (cid:2) α ( ) n (cid:3)(cid:14)(cid:0) d α ( ) n = µ ( ) (cid:1) , d µ ( ) = n +
1, in the new algebra (97) we may iterate the constructionof Example 3.29 to obtain the bivariate polynomial dgc-algebra over A , to be denoted: A α ( ) n + α ( ) n + , (cid:46) d α ( ) n = µ ( ) n + , d α ( ) n = µ ( ) n + : = (cid:16) A (cid:2) µ ( ) n + (cid:3)(cid:14)(cid:0) d α ( ) n + = µ ( ) (cid:1)(cid:17)(cid:2) α ( ) (cid:3)(cid:14)(cid:0) d α ( ) n + = µ ( ) (cid:1) . Iterating further, we have multivariate polynomial dgc-algebras over A , to be denoted as follows: A α ( k ) n k + , ... α ( ) n + α ( ) n + , (cid:46) d α ( k ) n k , = µ ( k ) ... d α ( ) n = µ ( ) , d α ( ) n = µ ( ) ∈ DiffGradedCommAlgebras ≥ R (99)24ith µ r ∈ A α ( r − ) n r − + , ... α ( ) n + , , for 1 ≤ r ≤ k .These multivariate polynomial algebras (99) receive the canonical inclusion (98) of A : A (cid:31) (cid:127) i A (cid:47) (cid:47) A α ( k ) n k + , ... α ( ) n + α ( ) n + , (cid:46) d α ( k ) n k = µ ( k ) , ... d α ( ) n = µ ( ) , d α ( ) n = µ ( ) , (100)these being the composites of the stage-wise inclusions (98). Definition 3.31 (Semifree dgc-Algebras/Sullivan models/FDAs) . The multivariate polynomial dgc-algebras ofExample 3.30 are sometimes called (i) semi-free dgc-algebras over A (since their underlying graded-commutativealgebra (77) is free, as in Example 3.10), but they are traditionally known (ii) in rational homotopy theory as relativeSullivan models (due to [Su77], review in [FHT00, II][Me13][FH17]), or, (iii) in supergravity theory (following[vN82][D’AF82]), as FDAs [CDF91], (for translation see [FSS13b][FSS16a][FSS16b][HSS18][BMSS19]). Herewe write: SullivanModels ≥ R (cid:31) (cid:127) (cid:47) (cid:47) SullivanModels R (cid:31) (cid:127) (cid:47) (cid:47) DiffGradedCommAlgebras ≥ R (101)for, from right to left, (a) the full subcategory of connective dgc-algebras (Def. 3.17) on those which are isomorphicto a multivariate polynomial dgc-algebra over R , as in Example 3.30 (i.e., the ordering of the generators in (99) isnot part of the data of a Sullivan model, only the resulting dgc-algebra); and (b) for the further full subcategory onthose Sullivan model that are generated in positive degree ≥ Example 3.32 (Polynomial dgc-algebras as pushouts) . For A ∈ DiffGradedCommAlgebras ≥ R (Def. 3.17) the poly-nomial dgc-algebras over A (Def. 3.29) are pushouts in DiffGradedCommAlgebras ≥ R of the following form: A (cid:2) α n (cid:3)(cid:14)(cid:0) d α n = µ (cid:1) (po) (cid:79) (cid:79) i A (cid:31) (cid:63) (cid:111) (cid:111) α n (cid:91) α n µ (cid:91) c n + R (cid:20) α n , c n + (cid:21)(cid:14)(cid:32) d α n = c n + d c n + = (cid:33) (cid:79) (cid:79) c n + ! c n + (cid:31) (cid:63) A (cid:111) (cid:111) µ (cid:91) c n + R (cid:2) c n + (cid:3)(cid:14)(cid:0) d c n + = (cid:1) (102)Here on the right we have multivariate polynomial dgc-algebras (Example 3.30) over R (Example 3.20) as shown.The horizontal morphisms encode the choice of µ ∈ A (96) and the left vertical morphism is the canonical inclusion(98). Example 3.33 (Chevalley-Eilenberg algebras of nilpotent Lie algebras) . Beware that not every Lie algebra g hasChevalley-Eilenberg algebra (Example 3.24) which satisfies the stratification in the Definition 3.30 of multivariatepolynomial dg-algebras. (i) For instance, the Lie algebra su ( ) hasCE (cid:0) su ( ) (cid:1) = R (cid:2) θ , θ , θ (cid:3)(cid:14)(cid:16) d θ i = ∑ j , k ε i jk θ j ∧ θ k (cid:17) and no ordering of { , , } brings this into the iterative form required in (99). (ii) Instead, those Lie algebras whose CE-algebra is of the form (99) are precisely the nilpotent Lie algebras. Beware that “FDA” in the supergravity literature is meant to be short-hand for “free differential algebra”, which is misleading, becausewhat is really meant are not free dgc-algebras as in Example 3.19 (in general) but just “semi-free” dcg-algebras, only whose underlyinggraded-commutative algebras (77) is required to be free (Example 3.10).
25n generalization of Example 3.33, we say (see also [Ber15, Theorem 2.3]):
Definition 3.34 (Nilpotent L ∞ -algebras) . An L ∞ -algebra (87) is nilpotent if its CE-algebra (Def. 3.25) is a multi-variate polynomial dgc-algebra (Example 3.30), hence is in the sub-category of SullivanModels R (101): L ∞ Algebras ≥ , nil R , fin (cid:31) (cid:127) CE (cid:47) (cid:47) (cid:127) (cid:95) (cid:15) (cid:15) (pb) (cid:0) SulivanModels (cid:1) op (cid:127) (cid:95) (cid:15) (cid:15) L ∞ Algebras ≥ R , fin (cid:31) (cid:127) CE (cid:47) (cid:47) (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) op (103)In fact, from (84) it is clear that every connected Sullivan model, hence with generators in degrees ≥
1, is theChevalley-Eilenberg algebra of a unique nilpotent L ∞ -algebra, so that the defining inclusion at the top of (103)further restricts to an equivalence of homotopy categories: L ∞ Algebras ≥ , nil R , fin CE (cid:39) (cid:47) (cid:47) (cid:0) SullivanModels ≥ R (cid:1) op . (104) Homotopy theory of connective dgc-Algebras.
We recall the homotopy theory of connective differential graded-commutative algebras. We make free use of the language of model categories [Qu67]; for review see [Ho99][Lu09,A.2] and appendix A.
Definition 3.35 (Homotopical structure on connective dgc-algebras [BG76, §4.2][GM96, V.3.4]) . Consider thefollowing sub-classes of morphisms in the category of DiffGradedCommAlgebras ≥ R (Def. 3.17): (i) W – weak equivalences are the quasi-isomorphisms; (ii)
Fib – fibrations are the degreewise surjections;We call this the projective homotopical structure on dgcAlgebras ≥ R . Proposition 3.36 (Projective model structure connective on dgc-algebras [BG76, §4.3][GM96, V.3.4]) . Equippedwith the projective homotopical structure from Def. 3.35, the category of
DiffGradedCommAlgebras ≥ R (Def. 3.17)becomes a model category (Def. A.3). We denote this as: (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj ∈ ModelCategories . (105) Remark 3.37 (All dgc-algebras are projectively fibrant) . Every object A ∈ (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj (105) is fibrant: By Example 3.21 the terminal morphism is to the 0-algebra, and this is clearly surjective, hence isa fibration, by Def. 3.35: A ∈ Fib (cid:47) (cid:47) . Cofibrant dgc-algebras.
In order to identify cofibrant dgc-algebras, it is useful to first consider the following:
Definition 3.38 (Homotopical structure on connective cochain complexes) . Consider the following sub-classes ofmorphisms in the category CochainComplexes ≥ R (Def. 3.14): (i) W – weak equivalences are the quasi-isomorphisms; (ii)
Fib – fibrations are the degreewise surjections; (iii)
Cof – cofibrations are the injections in positive degrees.We call this the injective homotopical structure on CochainComplexes ≥ R . Proposition 3.39 (Injective model structure on connective cochain complexes [He07, p. 6]) . Equipped with theinjective homotopical structure of Def. 3.38 the category of
CochainComplexes ≥ R (Def. 3.14) becomes a modelcategory (Def. A.3). We denote this: (cid:0) CochainComplexes ≥ R (cid:1) inj ∈ ModelCategories . Proof.
This is formally dual to the proof of the projective model structure on connective chain complexes [Qu67,II.4][GoS06, Thm. 1.5]; see, for instance, [Dun10, Thm. 2.4.5].26 emark 3.40 (Other model categories of chain complexes) . Prop. 3.39 is usually stated in the generality ofcochain complexes of abelian groups, in which case the fibrations are only those degreewise surjections that havedegreewise injective kernel, a condition that becomes trivial for abelian groups that are vector spaces.
Proposition 3.41 (Quillen adjunction between dgc-algebras and cochain complexes) . The adjunction (63) between
DiffGradedCommAlgebras ≥ R (Def. 3.17) and CochainComplexes ≥ R (Def. 3.14) is a Quillen adjunction (Def. A.17)with respect to the model category structures from Def. 3.39 and that from Def. 3.36. (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj (cid:111) (cid:111) Sym ⊥ Qu (cid:47) (cid:47) (cid:0) CochainComplexes ≥ R (cid:1) inj . Proof.
It is immediate from the definitions that the forgetful right adjoint preserves the classes W and Fib.
Lemma 3.42 (Generating cofibrations) . The following inclusions of multivariate polynomial dgc-algebras (Exam-ple 3.30) are cofibrations in (cid:0)
DiffGradedCommAlgebras ≥ R (cid:1) proj (Def. 3.36) R [ c n + ] (cid:14) ( d c n + = ) (cid:31) (cid:127) c n + c n − ∈ Cof (cid:47) (cid:47) R (cid:20) α n , c n + (cid:21)(cid:14)(cid:32) d α n = c n + , d c n + = (cid:33) for n ∈ N . (106) Proof.
Consider the following morphisms of cochain complexes, for n ∈ N : ...0 " d " d (cid:104) c n + (cid:105) " d " d " d ... " d (cid:31) (cid:127) i n (cid:47) (cid:47) ...0 " d " d (cid:104) c n + (cid:105) " d (cid:104) α n (cid:105) " d " d ... " d with d α n = c n + . (107)Since these are injections, they are cofibrations in (cid:0) CochainComplexes ≥ R (cid:1) inj (Prop. 3.39), by Def. 3.38. Thus alsotheir images under Sym (Def. 3.19) are cofibrations in (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj (Prop. 3.36) becauseSym is a left Quillen functor, by Prop. 3.41. But Sym ( i n ) manifestly equals (106), and so the claim follows. Proposition 3.43 (Relative Sullivan algebras are cofibrations) . For a multivariate polynomial dgc-algebra from Ex-ample 3.30, the canonical inclusion (108) of the base algebra is a cofibration in (cid:0)
DiffGradedCommAlgebras ≥ R (cid:1) proj (Prop. 3.36): A (cid:31) (cid:127) i A ∈ Cof (cid:47) (cid:47) A α ( k ) n k + , ... α ( ) n + , (cid:46) d α ( k ) n k = µ ( k ) , ...d α ( ) n = µ ( ) . (108) In particular, since R ∈ DiffGradedCommAlgebras ≥ R is the initial object (Example 3.20), all multivariate polyno-mial dgc-algebras over R (the Sullivan models, Def. 3.31) are cofibrant objects in (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj .Proof. By Lemma 3.42, the right vertical morphisms in the pushout diagram (102) are cofibrations. Since theclass of cofibrations is preserved under pushout, so are hence the left vertical morphisms in (102), which arethe base algebra inclusions (98) of polynomial dgc-algebras. The base algebra inclusions into general multivariatepolynomial dgc-algebras are composites of these, and since the class of cofibrations is presered under composition,the claim follows. 27 emma 3.44 (Pushout along relative Sullivan algebras preserves quasi-isomorphisms [FHT00, Prop. 6.7 (ii),Lemma 14.2]) . The operation of pushout (306) along the canonical inclusion (108) of a base dgc-algebra intoa multivariate polynomial dgc-algebra (Example 3.30) preserves quasi-isomorphisms. In fact, it sends quasi-isomorphism between base algebras to quasi-isomrophisms of multivariate polynomial dgc-algebras:A (cid:31) (cid:127) i A ∈ Cof (cid:47) (cid:47) f ∈ W (cid:15) (cid:15) (po) A α ( k ) n k + , ... α ( ) n + , (cid:46) d α ( k ) n k = µ ( k ) , ...d α ( ) n = µ ( ) ( i A ) ∗ f (cid:15) (cid:15) A (cid:48) (cid:31) (cid:127) i A ∈ Cof (cid:47) (cid:47) A (cid:48) α ( k ) n k + , ... α ( ) n + , (cid:46) d α ( k ) n k = µ ( k ) , ...d α ( ) n = µ ( ) ⇒ ( i a ) ∗ f ∈ W . (109) Lemma 3.45 (Weak equivalences of nilpotent L ∞ -algebras [FHT00, Prop. 14.13]) . A morphism between Chevalley-Eilenberg algebras (Def. 3.25) of nilpotent L ∞ -algebras (Def. 3.34), is a quasi-isomorphism of dgc-algebras(hence a weak equivalence according to Def. 3.35) precisely if the corresponding morphism (82) of L ∞ -algebras isa quasi-isomorphism between the chain complexes given by the unary bracket operation ∂ : = [ − ] (86) : CE ( g ) (cid:111) (cid:111) CE ( φ ) ∈ W CE ( h ) ⇔ (cid:0) g , [ − ] g (cid:1) φ ∈ W (cid:47) (cid:47) (cid:0) h , [ − ] h (cid:1) . Remark 3.46 (Homotopy theory of nilpotent L ∞ -algebras inside all L ∞ -algebras) .(i) Prop. 3.43, with Remark 3.37 and Def. 3.25, allows to identify the homotopy category of finite-type nilpotentconnective L ∞ -algebras (Def. 3.34), with a full subcategory of the homotopy category (Def. A.14) of the opposite(Example A.9) of dgc-algebras (Prop. 3.36):Ho (cid:0) L ∞ Algebras ≥ , nil R , fin (cid:1) (cid:31) (cid:127) CE (cid:47) (cid:47) Ho (cid:0)(cid:0) DiffGradedCommAlgebras ≥ R (cid:1) opproj (cid:1) . (110) (ii) There is also the homotopy theory of more general L ∞ -algebras [Hi01][Pr10][Va14][Ro20], whose weak equiv-alences are the quasi-isomorphisms on chain complexes formed by the unary bracket [ − ] (86). Lemma 3.45 saysthat the homotopy theory (110) of finite-type, nilpotent connective L ∞ -algebras that we are concerned with here isfully faithfully embedded into this more general L ∞ homotopy theory:Ho (cid:0) L ∞ Algebras ≥ , nil R , fin (cid:1) (cid:31) (cid:127) (cid:47) (cid:47) Ho (cid:0) L ∞ Algebras R (cid:1) . Minimal Sullivan modelsDefinition 3.47 (Minimal Sullivan models [BG76, Def. 7.2][He07, Def. 1.10]) . A connected (relative) Sullivanmodel dgc-algebra A ∈ SullivanModels ≥ R (Def. 3.31) is called minimal if it is given by a multivariate polynomialdgc-algebras as in (99) the degrees n i of whose generators α ( i ) n i are monotonically increasing i < j ⇒ n j ≤ n j . Example 3.48 (Minimal models of simply connected dgc-algebras [BG76, Prop. 7.4]) . If A ∈ SullivanModels ≥ R (Def. 3.31) is trivial in degree 1, then it is minimal (Def. 3.47) precisely if the unary bracket [ − ] (85) of thecorresponding L ∞ -algebra (104) vanishes: A = ⇒ (cid:0) A is minimal ⇔ [ − ] = (cid:1) . roposition 3.49 (Existence of minimal Sullivan models [BG76, Prop. 7.7, 7.8]) . If A ∈ DiffGradedCommAlgebras ≥ R is cohomologically connected, in that H ( A ) = R , then: (i) There exists a minimal Sullivan model A min (Def. 3.47) with weak equivalence in (cid:0)
DiffGradedCommAlgebras ≥ R (cid:1) proj (105) to A A min p min A ∈ W (cid:47) (cid:47) A . (111) (ii) This A min is unique up to isomorphism of
DiffGradedCommAlgebras ≥ R . More generally:
Proposition 3.50 (Existence of minimal relative Sullivan models [FHT00, Thm. 14.12]) . Let B φ (cid:47) (cid:47) A be a mor-phism in
DiffGradedCommAlgebras ≥ R (Def. 3.17) such that (a) A and B are cohomologically connected, in that H ( A ) = R and H ( B ) = R , (b) H ( φ ) : H ( B ) −! H ( A ) is an injection.Then: (i) There exists a minimal relative Sullivan model B (cid:44) −! A min B (Def. 3.47) equipped with a weak equivalence to φ in (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj (Def. 105): A min B ∈ W (cid:47) (cid:47) AB φ (cid:54) (cid:54) (cid:87)(cid:55) (cid:105) (cid:105) (ii) This A min B is unique up to isomorphism in DiffGradedCommAlgebras ≥ R . We recall fundamental facts of dg-algebraic rational homotopy theory [Su77][BG76][GM13] (review in [FHT00][He07][FOT08][FH17]), streamlined towards the application to non-abelian de Rham theory below in §3.3.
Remark 3.51 (Rational homotopy theory over the real numbers) . Throughout, we consider rational homotopytheory over the real numbers R (as in [GM13]), instead of over the rational numbers Q . This is the version inwhich rational homotopy theory connects to differential geometry (e.g. [FOT08]), since the smooth de Rhamcomplex is not defined over Q but over R (see Lemma 3.90). The original account [BG76] of rational homotopytheory is, for the most part, formulated over an arbitrary field k of characteristic zero; and [BG76, Lem. 11.7]makes explicit that the choice of this base field does not change the form of the classical theorems. For example,the “real-ified” homotopy groups of a space X π • ( X ) ⊗ Z R (cid:39) (cid:0) π • ( X ) ⊗ Z Q (cid:1) ⊗ Q R form a real vector space with real dimension equal to the rational dimension of the corresponding rationalizedhomotopy groups dim Q (cid:0) π • ( X ) ⊗ Z Q (cid:1) = dim R (cid:0) π • ( X ) ⊗ Z R (cid:1) , and hence the rational Whitehead L ∞ -algebras (Prop. 3.63 below) have the same collection of generators andtheir CE-alberas/minimal Sullivan models (Prop. 3.49 below) have the same differential relations, irrespective ofwhether they come as algebras over Q or over R . For technical reasons, we focus on the following class of homotopy types (with little to no restriction in prac-tice):
Definition 3.52 (Connected nilpotent spaces of finite rational type [BG76, 9.2]) . WriteHo (cid:0)
TopologicalSpaces Qu (cid:1) fin R ≥ , nil (cid:31) (cid:127) (cid:47) (cid:47) Ho (cid:0) TopologicalSpaces Qu (cid:1) for the full subcategory of homotopy types of topological spaces X (338) on those which are: (i) connected : π ( X ) (cid:39) ∗ ; (ii) nilpotent : π ( X ) ∈ NilpotentGroups, and π n ≥ ( X ) are nilpotent π ( X ) -modules (e.g. [Hil82]); (iii) finite rational type : dim R (cid:0) H n ( X ; R ) (cid:1) < ∞ , for all n ∈ N . While in homotopy theory Q and R coefficients behave similarly yet seem a priori not directly comparable, differential refinementsmight provide such comparison (with coefficients R / Q naturally arising; see, e.g., [GS19a][GS19c]). We leave this for a future discussion. emark 3.53 (Technical assumptions) . The connectedness assumption in Def. 3.52 is a pure convenience; fornon-connected spaces all of the following applies just by iterating over connected components. On the other hand,the nilpotency and R -finiteness condition in Def. 3.52 are strictly necessary for the plain dg-algebraic formulationof rational homotopy theory (due to [BG76][Su77]) to satisfy the fundamental theorem (Theorem 3.60 below).The generalizations required to drop these assumptions are known, but considerably more unwieldy: (i) To drop the nilpotency assumption, all dgc-algebra models need to be equipped with the action of the funda-mental group (see [FHT15]). (ii)
To drop the finite-type assumption one needs dgc-coalgebras in place of dgc-algebras, as in the original [Qu69].Therefore, we expect that the construction of the (twisted) non-abelian character map, below in sections §4and §5, works also without imposing these technical assumptions, but a discussion in that generality is beyond thescope of the present article.
Example 3.54 (Examples of nilpotent spaces [Hil82, §3][MP12, §3.1]) . Such examples (Def. 3.52) include: (i) every simply connected space X , π ( X ) = (ii) every simple space X , i.e. with abelian fundamental group acting trivially, such as tori; (iii) hence every connected H-space; (iv) hence every loop space X (cid:39) Ω Y , and hence every ∞ -group (Prop. 2.8); (v) hence every infinite-loop space, i.e., every component space E n of a spectrum E (17); (vi) the classifying spaces BG (12) of nilpotent Lie groups G ; (vii) the mapping spaces Maps ( X , A ) out of manifolds X into nilpotent spaces A .Rational homotopy theory is concerned with understanding the following notion: Definition 3.55 (Rationalization [BK72, p. 133][BG76, §11.21][He07, §1.4, §1.7]) .(i)
A connected nilpotent homotopy type X ∈ Ho (cid:0) TopologicalSpaces Q (cid:1) ≥ , nil (Def. 3.52) is called rational if all itshomotopy groups admit the structure of real vector spaces. (ii) A rationalization of X is a map X η R X (cid:47) (cid:47) L R ( X ) ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) ≥ , nil (112)such that: (a) L R is rational: π n ( X ) ∈ VectorSpaces R ! Groups for n ≥ (b) the map η R X induces an isomorphism on rational cohomology groups: H • ( X ; R ) H • ( η R X ; R ) (cid:39) (cid:47) (cid:47) H • (cid:0) L R X ; , R (cid:1) . Rationalization exists essentially uniquely, and defines a reflective subcategory inclusion connected, nilpotent, R -finite,rational homotopy types Ho (cid:0) TopologicalSpaces Qu (cid:1) Q ≥ , nil (cid:111) (cid:111) L R (cid:31) (cid:127) ⊥ (cid:47) (cid:47) connected, nilpotent, R -finite,homotopy types Ho (cid:0) TopologicalSpaces Qu (cid:1) ≥ , nil (113)whose adjunction unit is (112). PL de Rham theory.
At the heart of dg-algebraic rational homotopy theory is the observation that a variant of thede Rham dg-algebra of a smooth manifold (Example 3.23) applies to general topological spaces: the
PL de Rhamcomplex (Def. 3.56). This satisfies an appropriate PL de Rham theorem (Prop. 3.57) and makes dg-algebras ofPL differential forms detect rational homotopy type (Prop. 3.60). At the same time, over a smooth manifold thePL de Rham complex is suitably equivalent to the smooth de Rham complex (Lemma 3.90). The terminology “PL” or “P.L.” for this construction seems to have been silently introduced in [BG76], as shorthand for “piecewiselinear”, and has become widely adopted (e.g. [GM13, §9]). But beware that this refers to the piecewise-linear structure that a choice oftriangulation induces on a topological space, while the actual differential forms in the PL de Rham complex are piecewise polynomial withrespect to this piecewise linear structure. efinition 3.56 (PL de Rham complex and PL de Rham cohomology [BG76, pp. 1-7][GM13, §9.1]) . Write Ω • pdR ( ∆ • ) : ∆ op (cid:47) (cid:47) (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) op (114)for the simplicial dgc-algebras of polynomial differential forms on the standard simplices. (i) For S ∈ SimplicialSets, its
PL de Rham complex is the hom-object of simplicial objects from S to Ω • ( ∆ • ) , henceis the following end in DiffGradedCommAlgebras ≥ R : Ω • PLdR ( S ) : = (cid:90) [ k ] ∈ ∆ ⊕ S k Ω • pdR ( ∆ k ) , (115)hence an element in ω ∈ Ω • PLdR ( S ) is a polynomial differential form ω ( n ) σ ∈ Ω • pdR ( ∆ n ) on each n -simplex σ ∈ S n for all n ∈ N , such that these are compatible under pullback along all simplex face inclusions δ i and along alldegenerate simplex projections σ i : Ω • PLdR ( S ) = ... ... S δ ∗ (cid:15) (cid:15) (cid:79) (cid:79) σ ∗ δ ∗ (cid:15) (cid:15) (cid:79) (cid:79) σ ∗ δ ∗ (cid:15) (cid:15) ω ( ) (cid:47) (cid:47) Ω • pdR ( ∆ ) δ ∗ (cid:15) (cid:15) (cid:79) (cid:79) σ ∗ δ ∗ (cid:15) (cid:15) (cid:79) (cid:79) σ ∗ δ ∗ (cid:15) (cid:15) S ω ( ) (cid:47) (cid:47) δ ∗ (cid:15) (cid:15) (cid:79) (cid:79) σ ∗ δ ∗ (cid:15) (cid:15) Ω • pdR ( ∆ ) δ ∗ (cid:15) (cid:15) (cid:79) (cid:79) σ ∗ δ ∗ (cid:15) (cid:15) S ω ( ) (cid:47) (cid:47) Ω • pdR ( ∆ ) (ii) For X ∈ TopologicalSpaces, its
PL de Rham complex is that of its simgular simplicial set, according to (115): Ω • PLdR ( X ) : = Ω • PLdR (cid:0)
Sing ( X ) (cid:1) . (116)By pullback of differential forms, this extends to a functorSimplicialSets Ω • PLdR (cid:47) (cid:47) (cid:0)
DiffGradedCommAlgebras ≥ R (cid:1) op . (117) (iii) We write H • PLdR ( − ) : = H Ω • PLdR ( − ) (118)for PL de Rham cohomology , the cochain cohomology of the PL de Rham complex.
Proposition 3.57 (PL de Rham theorem [BG76, Thm. 2.2][GM13, Thm. 9.1]) . The evident operation of integrat-ing differential forms over simplices induces a quasi-isomorphism Ω • PLdR ( − ) ∈ qIso (cid:47) (cid:47) C • ( − ; R ) from the PL de Rham complex (Def. 3.56) to the cochain complex of ordinary singular cohomology with coefficientsin R . Hence on cochain cohomology this induces an isomorphismH • PLdR ( − ) (cid:39) (cid:47) (cid:47) H • ( − ; R ) between PL de Rham cohomology (118) and ordinary real cohomology. But, in fact, before passing to cochain cohomology, the PL de Rham complex captures full rational homotopytype:
Lemma 3.58 (Extension lemma for polynomial differential forms [GM13, Lemma 9.4]) . For n ∈ N , the operationof pullback of piecewise polynomial differential forms (Def. 3.90) along the boundary inclusion of the n-simplex ∂ ∆ n i n (cid:47) (cid:47) ∆ n is an epimorphism: Ω • PLdR ( ∆ n ) i ∗ n (cid:47) (cid:47) (cid:47) (cid:47) Ω • PLdR ( ∂ ∆ n ) . roposition 3.59 (PL de Rham Quillen adjunction [BG76, 8]) . The PL de Rham complex functor (Def. 3.56) isthe left adjoint in a Quillen adjunction (Def. A.17) (cid:0)
DiffGradedCommAlgebras ≥ R (cid:1) opproj (cid:111) (cid:111) Ω • PLdR ⊥ Qu exp PS (cid:47) (cid:47) SimplicialSets Qu (119) between the opposite (Def. A.9) of the model category of dgc-algebras (Prop. 3.36) and the classical modelstructure on simplicial sets (Prop. A.8); where the right adjoint sends a dgc-algebra A to exp PS ( A ) = ∆ [ k ] DiffGradedCommAlgebras ≥ R (cid:0) Ω • PLdR ( ∆ k ) , A (cid:1) ∈ SimplicialSets . (120) Proof.
That the right adjoint exists and is give as in (120) follows by general nerve/realization theory [Kan58], orelse by direct inspection.For the left adjoint to preserve cofibrations means to take injections of simplicial sets to degreewise surjectionsof dgc-algebras. This follows from the extension lemma (Lemma 3.58). Moreover, the left adjoint preserves infact all weak equivalences, by the PL de Rham theorem (Prop. 3.57).
Proposition 3.60 (Fundamental theorem of dgc-algebraic rational homotopy theory) . The derived adjunction(Prop. A.20) Ho (cid:0)(cid:0) DiffGradedCommAlgebras ≥ R (cid:1) opproj (cid:1) (cid:111) (cid:111) L Ω • PLdR R exp ⊥ (cid:47) (cid:47) Ho (cid:0) SimplicialSets Qu (cid:1) (121) of the Quillen adjunction (119) from Prop. 3.59 is such that: (i) on connected, nilpotent, R -finite homotopy types (Def. 3.52) the derived PLdR-adjunction unit (325) is the unit (112) of rationalization (Def. 3.55):X D η PLdR X derived unit ofPL de Rham adjunction (cid:47) (cid:47) R exp ◦ Ω • PLdR ( X ) (cid:39) X η R X rationalization unit (cid:47) (cid:47) L R X ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil . (122) (ii) For X , A nilpotent, connected, R -finite homotopy types (Def. 3.52), the PL de Rham space functor (117) fromDef. 3.56 induces natural bijections Ho (cid:0) TopologicalSpaces Qu (cid:1)(cid:0) X , L R A (cid:1) (cid:39) Ω • PLdR (cid:47) (cid:47) Ho (cid:16)(cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj (cid:17)(cid:16) Ω • PLdR ( A ) , Ω • PLdR ( X ) (cid:17) . (123) Proof. (i)
This is [BG76, Thm 11.2]. (ii)
This follows via [BG76, Thm 9.4(i)], which says that the derived adjunction (121) restricts on connected,nilpotent, R -finite (Def. 3.52) rational homotopy types (Def. 3.55) to an equivalence of categories:Ho (cid:0)(cid:0) DiffGradedCommAlgebras ≥ R (cid:1) opproj (cid:1) ≥ (cid:111) (cid:111) L Ω • PLdR R exp (cid:39) (cid:47) (cid:47) Ho (cid:0) SimplicialSets Qu (cid:1) R , fin R ≥ , nil . (124)In detail, consider the following sequence of natural isomorphisms:Ho (cid:0) TopologicalSpaces Qu (cid:1) (cid:0) X , L R A (cid:1) (cid:39) Ho (cid:0) SimplicialSets Qu (cid:1) (cid:0) Sing ( X ) , L R Sing ( A ) (cid:1) (cid:39) Ho (cid:0) SimplicialSets Qu (cid:1) (cid:0) L R Sing ( X ) , L R Sing ( A ) (cid:1) (cid:39) Ho (cid:0) SimplicialSets Qu (cid:1) (cid:0) R exp ◦ Ω • PLdR ( X ) , R exp ◦ Ω • PLdR ( A ) (cid:1) (cid:39) Ho (cid:16)(cid:0) DiffGradedCommAlgebras ≥ R (cid:1) opproj (cid:17) (cid:0) Ω • PLdR ◦ R exp ◦ Ω • PLdR ( X ) , Ω • PLdR ◦ R exp ◦ Ω • PLdR ( A ) (cid:1) (cid:39) Ho (cid:16)(cid:0) DiffGradedCommAlgebras ≥ R (cid:1) opproj (cid:17) (cid:0) Ω • PLdR ( X ) , Ω • PLdR ( A ) (cid:1) (cid:39) Ho (cid:16)(cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj (cid:17) (cid:16) Ω • PLdR ( A ) , Ω • PLdR ( X ) (cid:17) . (125)32ere the first step is (338); the second step uses that rationalization is a reflection (113); the third step uses (122);the fourth is the equivalence (124) along L Ω • PLdR (using, with Example A.21, that every simplicial set is alreadycofibrant, Example A.8); the fifth step is the statement from (124) that R exp is the inverse equivalence. The laststep is just the definition of the opposite of a category. The composite of the bijections (125) is the desired bijection(123). PS de Rham theory.
The point of using piecewise polynomial differential forms in the PL de Rham complex(Def. 3.56) is that these, but not the piecewise smooth differential forms, can be defined over the field Q of rationalnumbers. But since we may and do use the real numbers as the rational ground field (Remark 3.51), it is expedientto also consider piecewise smooth de Rham complexes: Definition 3.61 (PS de Rham complex) . For n ∈ N , we write Ω • dR ( R n × ∆ • ) : ∆ op (cid:47) (cid:47) (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) op for the simplicial dgc-algebra of smooth differential forms on the product manifold of n -dimensional Cartesianspace with the standard simplices, i.e., of smooth differential forms on an ambient Cartesian space (Example 3.23),restricted to the simplex. As in Def. 3.90, this induces for each S ∈ SimplicialSets the corresponding piecewisesmooth de Rham complexes Ω • PSdR ( R n × S ) : = (cid:90) [ k ] ∈ ∆ ⊕ S n Ω • dR ( R n × ∆ n ) (126)and by pullback of differential forms these extend to functorsSimplicialSets Ω • PSdR ( R n × ( − )) (cid:47) (cid:47) (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) op . (127) Proposition 3.62 (Fundamental theorem for piecewise smooth de Rham complexes) . For all n ∈ N the piecewisesmooth de Rham complex functors (Def. 3.61) participate in a Quillen adjunction analogous to the PL de Rhamadjunction (Prop. 3.59) (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) opproj (cid:111) (cid:111) Ω • PSdR ( R n × ( − )) ⊥ Qu exp PS , n (cid:47) (cid:47) SimplicialSets Qu (128) with right adjoint given as in (120) : exp PS , n ( A ) = ∆ [ k ] DiffGradedCommAlgebras ≥ R (cid:0) Ω • PLdR ( R n × ∆ k ) , A (cid:1) ∈ SimplicialSets (129) whose derived functors (Prop. A.20) are naturally equivalent to those of the PL de Rham adjunction (121) : L Ω • PSdR ( R n × ( − )) (cid:39) L Ω • PSdR (cid:39) L Ω • PLdR , (130) R exp PS , n (cid:39) R exp PS (cid:39) R exp PL . (131) Proof. (i)
The proofs of the PL de Rham theorem (Prop. 3.57) as well as of the extension Lemma (Lemma 3.58)apply essentially verbatim also to piecewise-smooth differential forms ([GM13, Prop. 9.8]) and hence so does theproof of the PL de Rham Quillen adjunction in the form given in Prop. 3.59. (ii)
We have evident natural transformations Ω • PLdR ( S ) ∈ W (cid:47) (cid:47) Ω • PSdR ( S ) ∈ W (cid:47) (cid:47) Ω • PSdR ( R n × S ) given by inclusion of polynomial differential forms into smooth differential forms, and by pullback of differentialforms along the projections R n × ∆ k (cid:47) (cid:47) ∆ k . The corresponding component morphisms are quasi-isomorphisms,hence are weak equivalences in (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj ([GM13, Cor. 9.9]). Under passage to homo-topy categories (Def. A.14) and derived functors (Example A.21), these natural weak equivalences become thenatural isomorphisms (130) (by Prop. A.15). By essential uniqueness of adjoint functors, this implies the naturalisomorphisms (131). 33 hitehead L ∞ -algebras.Proposition 3.63 (Rational Whitehead L ∞ -algebras) . For X ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def. 3.52), thereexists a nilpotent L ∞ -algebra (Def. 3.34) l X ∈ L ∞ Algebras ≥ , nil R , fin , (132) unique up to isomorphism, whose Chevalley-Eilenberg algebra (Def. 3.25) is the minimal model (Def. 3.47) of thePL de Rham complex of X (Def. 3.56): CE ( l X ) : = (cid:0) Ω • PLdR ( X ) (cid:1) min ∈ W p min X (cid:47) (cid:47) Ω • PLdR ( X ) . (133) Proof.
By the PL de Rham theorem (Prop. 3.57) and the assumption that X is connected, it follows that we have H Ω ( X ) = R . Therefore Prop. 3.49 applies and says that (cid:0) Ω • PLdR ( X ) (cid:1) min ∈ SullivanModels ≥ R exists, and isunique up to isomorphism. With this, the equivalence (104) says that l X exists and is unique up to isomorphism.Notice the immediate corollary: Proposition 3.64 (Rational Whitehead L ∞ -algebra encodes rational homotopy type) . The rational Whitehead L ∞ -algebra l X in Prop. 3.63 encodes the rationalized homotopy type (Def. 3.55) of X ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil ,in that: L R X (cid:39) exp ◦ CE (cid:0) l X (cid:1) ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (134) with exp from (119) .Proof. This is the composite of the following sequence of isomorphisms in the homotopy category: L R X (cid:39) R exp ◦ Ω • PLdR ( X ) (cid:39) R exp ◦ (cid:0) Ω • PLdR ( X ) (cid:1) min (cid:39) R exp ◦ CE ( l X ) (cid:39) exp ◦ CE ( l X ) . Here the first step is the fundamental theorem (Prop. 3.60), the second step is the existence of the minimal model(Prop. 3.49) for the PL de Rham complex (using that it is cohomologically connected, by the PL de Rham theorem,Prop. 3.57), the third step is the existence of the rational Whitehead L ∞ -algebra (Prop. 3.63), and the last step usesthat Sullivan models are cofibrant (Prop. 3.43), hence fibrant in (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) opproj , so that on theseobjects the right derived functor R exp is given by the plain functor exp. Proposition 3.65 (Rational homotopy groups in the rational Whitehead L ∞ algebra) . Let X ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def. 3.52). (i) If X is simply connected, π ( X ) = (Example 3.54), then there is an isomorphism of graded vector spaces(Def. 3.2) between the graded vector space underlying (83) the Whitehead L ∞ -algebra l X (Prop. 3.63) and therationalized homotopy groups of the based loop space Ω X :
Whitehead L ∞ -algebra l X (cid:39) rationalizedhomotopy groups π • ( Ω X ) ⊗ Z R ∈ GradedVectorSpaces ≥ R . (ii) If π ( X ) is not necessarily trivial but abelian, then this statement still holds with l X replaced by its homologywith respect to the unary differential [ − ] (85) . (iii) If π ( X ) is not abelian, then (ii) still holds in degrees ≥ .Proof. Under translation through Prop. 3.63 and Def. 3.25, and using π • ( Ω X ) (cid:39) π • + ( X ) , claim (i) is equivalentto the existence of a dual isomorphism:CE ( l X ) (cid:14) CE ( l X ) (cid:39) Hom Z (cid:0) π • ( X ) , R (cid:1) ∈ GradedVectorSpaces ≥ R , (135)where the left hand side denotes the graded vector space of indecomposable elements in the Chevalley-Eilenbergalgebra (the α ( i ) n i in (99)). In this form, this is the statement of [BG76, Theorem 11.3 with Def. 6.12], in the specialcase when, with π ( X ) =
1, the unary differential [ − ] in l X vanishes (Example 3.48). The generalizations followanalogously. 34 emark 3.66 (Equivalent L ∞ -structures on Whitehead products) . The original discussion of the Whitehead algebrastructure on the homotopy groups of a space is in terms of differential-graded Lie algebras ([Hil55, Theorem B]),as are the Quillen models of rational homotopy theory [Qu69]. (i)
Notice that dg-Lie algebras (Example 3.26) and L ∞ -algebras with minimal CE-algebra (Def. 3.47) are twoopposite classes of L ∞ -algebras: The former has k -ary brackets (85) only for k ≤
2, the latter only for k ≥ L ∞ -algebras is described in [BBMM16, Theorem 2.1]; that from L ∞ - to dg-Lie-algebras in [FRS13, §1.0.2]. (ii) The minimal L ∞ -algebra structure on l X that we obtained in Prop. 3.63, 3.65, has the property that its k -arybrackets are, up to possibly a sign, equal to the order- k higher Whitehead products on X [BBMM16, Prop. 3.1]. Example 3.67 (Rationalization of Eilenberg-MacLane spaces) . Since the homotopy types of Eilenberg-MacLane-spaces K ( A , n ) = B n + A (see (10)) are fully characterized by their homotopy groups (for discrete abelian groups A ,e.g. [AGP02, §6])) π k (cid:0) B n + A (cid:1) (cid:39) (cid:26) A | k = n + | k (cid:54) = n + n ∈ N : (i) The rationalization (Def. 3.55) of the integral EM-space is the real EM-space L R (cid:0) B n + Z (cid:1) (cid:39) B n + R . (136) (ii) Their Whitehead L ∞ -algebra (Prop. 3.63) is the line Lie n -algebra b n R (Def. 3.27), by Prop. 3.65: l B n + Z (cid:39) b n R . (137) (iii) Hence, by Prop. 3.64, B n + R (cid:39) exp ◦ CE (cid:0) b n R (cid:1) ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) . (138) Example 3.68 (Rationalization of n -spheres) . The Serre finiteness theorem (see [Ra86, Thm. 1.1.8]) says that thehomotopy groups of n -spheres for n ≥ π n + k (cid:0) S n (cid:1) (cid:39) Z | k = Z ⊕ fin | k = m and n = m − | otherwisewhere “fin” stands for some finite group. Since finite groups are pure torsion, hence have trivial rationalization,this means that the rational homotopy groups of spheres are: π n + k (cid:0) S n (cid:1) ⊗ Z R (cid:39) R | k = R | k = m and n = m − | otherwise . Moreover, the fact that ordinary cohomology is represented by Eilenberg-MacLane spaces (Example 2.2) impliesthat H k (cid:0) S n ; R (cid:1) ∼ (cid:26) R | k = n | otherwise . With this, Prop. 3.65 together with Prop. 3.57 implies that the Whitehead L ∞ -algebras of spheres (Prop. 3.63) areas follows: CE (cid:0) l S n (cid:1) (cid:39) R (cid:2) ω n (cid:3)(cid:14)(cid:0) d ω n = . (cid:1) if n is odd (139)and CE (cid:0) l S n (cid:1) (cid:39) R (cid:20) ω n − ω n (cid:21)(cid:14)(cid:32) d ω n − = − ω n ∧ ω n d ω n = (cid:33) if n > xample 3.69 (Rationalization of loop spaces) . The minimal Sullivan model (Def. 3.47) of a loop space A (cid:39) Ω A (cid:48) has vanishing differential (e.g. [FHT00, p. 143]). Therefore, Prop. 3.65 implies that the rational Whitehead L ∞ -algebra l A (Prop. 3.63) of A is the direct sum of line Lie n -algebras b n R (Example 3.27): l A (cid:39) (cid:77) n ∈ N b n (cid:0) π n + ( A ) ⊗ Z R (cid:1) ∈ L ∞ Algebras ≥ , nil R , fin . Accordingly, its Chevalley-Eilenberg algebra (Def. 3.24) is the tensor product of those of the summands:CE (cid:0) l A (cid:1) (cid:39) (cid:79) n ∈ N CE (cid:16) b n (cid:0) π n + ( A ) ⊗ Z R (cid:1)(cid:17) ∈ DiffGradedCommAlgebras ≥ R . Relative rational Whitehead L ∞ -algebras. In generalization of Prop. 3.63 we have:
Proposition 3.70 (Relative rational Whitehead L ∞ -algebras) . For A , B , F ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def.3.52) and p a Serre fibration (Example A.7) from A to B with fiber FF fib ( p ) (cid:47) (cid:47) A p ∈ Fib (cid:15) (cid:15)
Bthere exists a nilpotent L ∞ -algebra (Def. 3.34) l B A ∈ L ∞ Algebras ≥ , nil R , fin , (141) unique up to isomorphism, whose Chevalley-Eilenberg algebra (Def. 3.25) is the relative minimal model (Def.3.47, Prop. 3.50) of the PL de Rham complex of p (Def. 3.56), relative to CE ( l B ) (from Prop. 3.63): CE ( l B A ) : = (cid:0) Ω • PLdR ( A ) (cid:1) min CE ( l B ) ∈ W p min BA (cid:47) (cid:47) Ω • PLdR ( A ) CE ( l B ) (cid:55) (cid:87) relative minimal model CE ( l p ) (cid:106) (cid:106) ∈ W p min B (cid:47) (cid:47) Ω • PLdR ( B ) . Ω • PLdR ( p ) (cid:79) (cid:79) (142) Proof.
By the PL de Rham theorem (Prop. 3.57) and the assumption that A and B are connected, it follows thatwe have H Ω ( A ) = R and H Ω ( B ) = R . Moreover, by the assumption that p is a Serre fibration withconnected fiber, it follows that H ( Ω • PLdR ( p )) is injective (e.g. [FHT00, p. 196]).Therefore Prop. 3.50 applies and says that (cid:0) Ω • PLdR ( A ) (cid:1) min B ∈ SullivanModels ≥ R exists, and is unique up toisomorphism. With this, the equivalence (104) says that l B A exists and is unique up to isomorphism. Lemma 3.71 (Minimal relative Sullivan models preserve homotopy fibers [FHT00, §15 (a)][FHT15, Thm. 5.1]) . Consider F , A , B ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def. 3.52) and let p be a Serre fibration from A to B (ExampleA.7) such that the homology groups H • ( F , R ) of the fiber are nilpotent as π ( B ) -modules (for instance in that B issimply-connected or that the fibration is principal). Then the cofiber of the minimal relative Sullivan model for p (142) is the minimal Sullivan model (133) for the homotopy fiber F (Def. A.22):F fib ( p ) (cid:47) (cid:47) A p ∈ Fib (cid:15) (cid:15) CE ( l F ) (cid:111) (cid:111) cofib ( CE ( l p )) CE (cid:0) l B A (cid:1) (cid:79) (cid:79) CE ( l p ) (cid:31) (cid:63) B CE ( l B ) (143)See Prop. 3.75 below for the key application of Lemma 3.71. Non-abelian real cohomology.Definition 3.72 (Non-abelian real cohomology) . Let X , A ∈ TopologicalSpaces Then the non-abelian real co-homology of X with coefficients in A is the non-abelian cohomology (Def. 2.1) of X with coefficients in therationalization L R A (Def. 3.55) H ( X ; L R A ) : = Ho (cid:0) TopologicalSpaces Qu (cid:1) ( X , L R A ) . (144)36 emark 3.73 (Non-abelian real cohomology subsumes ordinary real cohomology) . For n ∈ N , non-abelian realcohomology (Def. 3.72) with coefficients in the rationalized classifying space (Example 3.67) L R (cid:0) B n + Z (cid:1) (cid:39) B n R is naturally equivalent, by Example 2.2, to ordinary real cohomology in degree n : H (cid:0) X ; B n + R (cid:1) (cid:39) H m + ( X ; R ) . More generally:
Proposition 3.74 (Non-abelian real cohomology with coefficients in loop spaces) . Let A ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def. 3.52) such that it admits loop space structure, hence such that thereexists A (cid:48) with A (cid:39) Ω A (cid:48) ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) . Then the non-abelian real cohomology (Def. 3.72) with coefficients in L R A is naturally equivalent to ordinary realcohomology with coefficients in the rationalized homotopy groups of A:H (cid:0) X ; L R A (cid:1) (cid:39) (cid:77) n ∈ N H n + (cid:0) X ; π n + ( A ) ⊗ Z R (cid:1) . (145) Proof.
By Example 3.69 the we haveCE (cid:0) l A (cid:1) (cid:39) (cid:79) n ∈ N CE (cid:16) b n (cid:0) π n + ( A ) ⊗ Z R (cid:1)(cid:17) Noticing that the tensor product of dgc-algebras is the coproduct in the category of DiffGradedCommAlgebras ≥ R (Example 3.22), and hence the Cartesian product in the opposite category, the right adjoint functor exp (119)preserves this, so thatexp ◦ CE (cid:0) l A (cid:1) (cid:39) ∏ n ∈ N (cid:16) exp ◦ CE (cid:0) b n ( π n + ( A ) ⊗ Z R ) (cid:1)(cid:17) ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) . But, by Prop. 3.64 and by (138) in Example 3.67, this says that: L R A (cid:39) ∏ n ∈ N (cid:16) B n + (cid:0) π n + ( A ) ⊗ Z R (cid:1)(cid:17) ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) . Using this, and that cohomology preserves products of coefficients, we get the following sequence of naturalbijections: H (cid:0) X ; L R A (cid:1) (cid:39) H (cid:16) X ; ∏ n ∈ N B n + (cid:0) π n + ( A ) ⊗ Z R (cid:1)(cid:17) (cid:39) ∏ n ∈ N H (cid:16) X ; B n + (cid:0) π n + ( A ) ⊗ Z R (cid:1)(cid:17) = ∏ n ∈ N H n + (cid:0) X : π n + ( A ) ⊗ Z R (cid:1) = (cid:77) n ∈ N H n + (cid:0) X : π n + ( A ) ⊗ Z R (cid:1) . The composite of these is the desired (145).
Twisted non-abelian real cohomology.Proposition 3.75 (Rationalization of local coefficients – “fiber lemma” [BK72, §II]) . LetA (cid:47) (cid:47) A (cid:12) G ρ (cid:15) (cid:15) BG e a local coefficient bundle (Def. 2.29) such that all spaces are connected, nilpotent and of R -finite tupe,A , BG , A (cid:12) G ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def. 3.52) and such that the action of π ( BG ) on H • ( A , R ) isnilpotent (for instance in that BG is simply connected). Then: (i) Component-wise rationalization (Def. 3.55) yields a natural transformation to as rational local coefficientbundle as shown in the middle here: minimal Sullivan model CE ( l A ) (cid:111) (cid:111) cofib ( CE ( l p )) CE (cid:0) l BG (cid:0) A (cid:12) G (cid:1)(cid:1) (cid:79) (cid:79) CE ( l ρ ) (cid:31) (cid:63) rationalization L R A hofib ( L R ρ ) (cid:47) (cid:47) (cid:0) L R A (cid:1) (cid:12) ( L R G ) L R ρ (cid:15) (cid:15) local coefficient bundle A hofib ( ρ ) (cid:47) (cid:47) η R A (cid:53) (cid:53) A (cid:12) G ρ (cid:15) (cid:15) η R A (cid:12) G (cid:52) (cid:52) CE (cid:0) l BG (cid:1) B (cid:0) L R G (cid:1) BG η R BG (cid:52) (cid:52) (146) (ii) with minimal (relative) Sullivan model (Def. 3.47) as shown on the right.Proof. First, since forming classifying spaces shifts homotopy groups up in degree, it follows that BG B η R G (cid:47) (cid:47) B ( L R G ) induces an isomorphism on rationalized homotopy groups and hence is the rationalization map (Def. 3.55) on BG .Moreover, Lemma 3.71 says that the homotopy fiber (Def. A.22) of the rationalized fibration has Sullivanmodel CA ( l A ) , this being the cofiber of a relative Sullivan model for the rationalized fibrations, as shown on theright of (146). Since relative Sullivan models are cofibrations in (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj (Prop. 3.43),hence fibrations in the opposite model structure (Example A.9), this means, with the fundamental theorem (124)that CE ( l A ) is in fact a Sullivan model for the homotopy fiber (Def. A.22) of the rationalized fibration. Hencethe homotopy fiber of the rationalized fibration is the rationalization L R A of the homotopy fiber of the originalfibration, as shown in (146).Together these say that the rationalized fibration is an L R A -fibration over B (cid:0) L R G (cid:1) . With this, Prop. 2.28implies that the total space of the rationalized fibration is a homotopy quotient (cid:0) L R A (cid:1) (cid:12) (cid:0) L R G (cid:1) , which is hence therationalization of A (cid:12) G , as shown in the middle of (146).Due to Prop. 3.75 it makes sense to say, in generalization of Def. 3.72: Definition 3.76 (Twisted non-abelian real cohomology) . Let X ∈ TopologicalSpaces and let A (cid:12) G ρ (cid:47) (cid:47) BG be alocal coefficient bundle (Prop. 2.28, Def. 2.29) in Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def. 3.52). Then the twistednon-abelian real cohomology of X with local coefficients ρ is the twisted non-abelian L R A -cohomology (Def.2.29) of X with coefficients in the rationalized local coefficient bundle L R ρ from Prop. 3.75: H τ (cid:0) X ; L R A (cid:1) : = Ho (cid:16)(cid:0) TopologicalSpaces / L R BG Qu (cid:1)(cid:17)(cid:0) τ , L R ρ (cid:1) . We establish non-abelian de Rham theory for differential forms with values in (nilpotent) L ∞ -algebras, following[SSS09a][FSSt10]. The main result is the non-abelian de Rham theorem, Theorem 3.87, and its generalization tothe twisted non-abelian de Rham theorem, Theorem 3.104. L ∞ -Algebra valued differential forms.Definition 3.77 (Flat L ∞ -algebra valued differential forms [SSS09a, §6.5][FSSt10, §4.1]) .(i) For X ∈ SmoothManifold and g ∈ L ∞ Algebras ≥ R , fin (Def. 3.25), a flat g -valued differential form on X is a mor-phism of dgc-algebras (Def. 3.17) Ω • dR ( X ) (cid:111) (cid:111) A CE ( g ) ∈ DiffGradedCommAlgebras ≥ R (147)38o the smooth de Rham dgc-algebra of X (Example 3.23) from the Chevalley-Eilenberg dgc-algebra of g (Def.3.25). (ii) We write Ω dR ( X ; g ) flat : = DiffGradedCommAlgebras ≥ R (cid:0) CE ( g ) , Ω • dR ( X ) (cid:1) (148)for the set of all flat g -valued forms on X . Example 3.78 (Flat Lie algebra valued differential forms) . Let g ∈ LieAlgebras R , fin be a Lie algebra (90) with Liebracket [ − , − ] . Then a flat g -valued differential form in the sense of Def. 3.77 is the traditional concept: a g -valued1-form satisfying the Maurer-Cartan equation: Ω • dR ( X ; g ) flat (cid:39) (cid:110) A ∈ Ω ( X ) ⊗ g (cid:12)(cid:12) dA + [ A ∧ A ] = (cid:111) . (149)One way to see this is to appeal to the classical fact that the Chevalley-Eilenberg algebra of a finite-dimensional Liealgebra (Example 3.24) is isomorphic to the dgc-algebra of left invariant differential forms on the corresponding Liegroup G , which is generated from the Maurer-Cartan form θ ∈ Ω ( G ) ⊗ g satisfying θ | T e G = id g and d θ = [ θ ∧ θ ] .More explicitly, for { v a } a linear basis for g (79) with structure constants { f cab } (80), we see from (81) that adgc-algebra homomorphims (147) has the following components (second line) and constraints (third line): Ω • dR ( X ) (cid:111) (cid:111) A flat Lie algebra valued differential form R (cid:2) { θ ( a ) } (cid:3)(cid:14)(cid:16) d θ ( c ) = f cab θ ( b ) ∧ θ ( a ) (cid:17) (cid:39) CE ( g ) . A ( c ) (cid:111) (cid:111) components (cid:31)(cid:95) d (cid:15) (cid:15) θ ( c ) (cid:95) d (cid:15) (cid:15) dA ( c ) constraints f cab A ( b ) ∧ A ( a ) (cid:111) (cid:111) (cid:31) f cab θ ( a ) ∧ θ ( b ) (150) Example 3.79 (Ordinary closed forms are flat line L ∞ -algebra valued forms) . For n ∈ N , consider g = b n R theline Lie ( n + ) -algebra (Example 3.27). Then the corresponding flat g -valued differential forms (Def. 3.77) are innatural bijection to ordinary closed ( n + ) -forms: Ω dR ( X ; b n R ) flat (cid:39) Ω n + ( X ) closed . (151)That is, by (92), we see that the elements on the left of (151) have the following component (second line) subjectto the follows constraint (third line): Ω • dR ( X ) flatline Lie ( n + ) -algebra-valueddifferential form (cid:47) (cid:47) R [ c n + ] (cid:14) ( d c n + = ) (cid:39) CE ( b n R ) . C n + (cid:111) (cid:111) component (cid:95) d (cid:15) (cid:15) c n + (cid:95) d (cid:15) (cid:15) dC n + constraint (cid:111) (cid:111) (cid:31) Example 3.80 (Flat String Lie 2-algebra valued differential forms) . Flat L ∞ -algebras valued forms (Def. 3.77) withvalues in a String Lie 2-algebra string g (Example 3.28) are pairs consisting of a flat g -valued 1-form A (Example3.78) and a coboundary 2-form B for its Chern-Simons form CS ( A ) : = c (cid:10) A ∧ [ A ∧ A ] (cid:11) : Ω dR (cid:0) X ; string g (cid:1) flat (cid:39) (cid:40) B , A ∈ Ω • dR ( X ) (cid:12)(cid:12)(cid:12)(cid:12) d B = c CS ( A ) , d A = − [ A ∧ A ] (cid:41) . Namely, from (95) we see that in degree 1 the components of and constraints on such a differential form datum areexactly as in (150), while in degree 2 they are as follows: Ω • dR ( X ) (cid:111) (cid:111) flat String Lie 2-algebra valued form R (cid:34) b , { θ ( a ) } (cid:35)(cid:14)(cid:32) d b = µ abc θ ( c ) ∧ θ ( b ) ∧ θ ( a ) d θ ( c ) = f cab θ ( b ) ∧ θ ( a ) (cid:33) (cid:39) CE (cid:0) string g (cid:1) . B (cid:111) (cid:111) component in degree 2 (cid:31)(cid:95) d (cid:15) (cid:15) b (cid:95) d (cid:15) (cid:15) dB constraint µ abc A ( c ) ∧ A ( b ) ∧ A ( a ) (cid:111) (cid:111) (cid:31) µ abc θ ( c ) ∧ θ ( b ) ∧ θ ( a ) (153)39 xample 3.81 (Flat sphere-valued differential forms) . Flat L ∞ -algebras valued forms (Def. 3.77) with values inthe rational Whitehead L ∞ -algebra (Prop. 3.63) of a sphere (Example 3.68) of positive even dimension 2 k are pairsconsisting of a closed differential 2 k -form and a ( k − ) -form whose differential equals minus the wedge squareof the 2 k -form: Ω dR (cid:0) − ; l S k (cid:1) (cid:39) (cid:40) G k − , G k ∈ Ω • dR ( X ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d G k − = − G k ∧ G k , d G k = (cid:41) . Namely, from (140) one sees that the components of and the constraints on an l S k -valued form are as follows: Ω • dR (cid:0) X (cid:1) (cid:111) (cid:111) flat l S k -valued form R (cid:20) ω k − , ω k (cid:21)(cid:14)(cid:32) d ω k − = − ω k ∧ ω k , d ω k = (cid:33) = CE (cid:0) l S k (cid:1) G kd (cid:15) (cid:15) (cid:111) (cid:111) component in degree k (cid:31) ω k (cid:95) d (cid:15) (cid:15) d G k constraint (cid:111) (cid:111) (cid:31) G k − (cid:95) d (cid:15) (cid:15) (cid:111) (cid:111) component in degree k − (cid:31) ω k − (cid:95) (cid:15) (cid:15) d G k − constraint − G k ∧ G k (cid:111) (cid:111) (cid:31) − ω k ∧ ω k (154)For 2 k = G = (cid:63) G ) [Sa13, §2.5]. Example 3.82 (PL de Rham right adjoint via L ∞ -algebra valued forms) . For n ∈ N , the right adjoint functor in thePS de Rham adjunction (128) sends the Chevalley-Eilenberg algebra (Def. 3.25) of any g ∈ L ∞ Algebras ≥ , nil R , fin (Def.3.34) to a simplicial set of flat g -valued differential forms (Def. 3.77): (cid:91) exp ( g )( R n ) : = exp PS , n (cid:0) CE ( g ) (cid:1) : [ k ] Ω dR (cid:0) R n × ∆ k ; g (cid:1) flat ∈ SimplicialSets(by direct comparison of (129) with (148)). Regarded as a simplicial presheaf over CartesianSpaces (Def. 353),this construction is the moduli ∞ -stack of flat L ∞ -algebra valued differential forms (see §4.3 below). Non-abelian de Rham cohomology.Definition 3.83 (Coboundaries between flat L ∞ -algebra valued forms) . Let X ∈ SmoothManifolds and (from Def.3.25) g ∈ L ∞ Algebras ≥ R , fin . For A ( ) , A ( ) ∈ Ω dR ( X ; g ) flat a pair of flat g -valued differential forms on X (Def. 3.77), we say that a coboundary between them is a flat g -valueddifferential form on the cylinder manifold over X (its Cartesian product manifold with the real line): (cid:101) A ∈ Ω ( X × R ; g ) flat (155)such that its restrictions along X (cid:39) X × { } (cid:31) (cid:127) i X (cid:47) (cid:47) X × R (cid:111) (cid:111) i X (cid:63) (cid:95) X × { } (cid:39) X are equal to A ( ) and to A ( ) , respectively: ( i X ) ∗ (cid:101) A = A ( ) and ( i X ) ∗ (cid:101) A = A ( ) . (156)If such a coboundary exists, we say that A ( ) and A ( ) are cohomologous , to be denoted A ( ) ∼ A ( ) . efinition 3.84 (Non-abelian de Rham cohomology) . Let X ∈ SmoothManifolds and g ∈ L ∞ Algebras ≥ R , fin (Def.3.25). Then the non-abelian de Rham cohomology of X with coefficients in g is the set H dR ( X ; g ) : = (cid:0) Ω dR ( X ; g ) flat (cid:1) / ∼ (157)of equivalence classes with respect to the coboundary relation from Def. 3.83 on the set of flat g -valued differentialforms on X (Def. 3.77).We recall the following basic facts (e.g. [GT00, Rem 3.1]): Lemma 3.85 (Fiberwise Stokes theorem and Projection formula) . Let X be a smooth manifold and let F be acompact smooth manifold with corners, e.g. F = ∆ k a standard k-simplex, which for k = is the interval F = [ , ] .Then fiberwise integration over F of differential forms on the Cartesian product manifold X × F Ω • dR ( X × F ) (cid:82) F (cid:47) (cid:47) Ω •− dim ( F ) dR ( X ) e.g. Ω • dR ( X × R ) (cid:82) [ , ] (cid:47) (cid:47) Ω •− ( X ) satisfies, for all α ∈ Ω • dR ( X × F ) and β ∈ Ω • dR ( X ) : (i) The fiberwise Stokes formula: (cid:90) F d α = ( − ) dim ( F ) d (cid:90) F α + (cid:90) ∂ F α e . g . d (cid:90) [ , ] α = ( i X ) ∗ α − ( i X ) ∗ α − (cid:90) [ , ] d α , (158) where X (cid:39) X × { } (cid:31) (cid:127) i X (cid:47) (cid:47) X × R (cid:111) (cid:111) i X (cid:63) (cid:95) X × { } (cid:39) Xare the boundary inclusions. (ii)
The projection formula (cid:90) F (cid:0) pr ∗ X β (cid:1) ∧ α = ( − ) dim ( F ) deg ( β ) β ∧ (cid:90) F α , e.g. (cid:90) [ , ] (cid:0) pr ∗ X β (cid:1) ∧ α = ( − ) deg ( β ) β ∧ (cid:90) [ , ] α , (159) where X × F pr X (cid:47) (cid:47) Xis projection on the first factor.
Proposition 3.86 (Non-abelian de Rham cohomology subsumes ordinary de Rham cohomology) . For any n ∈ N ,let g = b n R be the line Lie ( n + ) -algebra (Example 3.27). Then the non-abelian de Rham cohomology withcoefficients in g (Def. 3.84) is naturally equivalent to ordinary de Rham cohomology in degree n + :H dR ( − ; b n R ) (cid:39) H n + ( − ) . (160) Proof.
From Example 3.79, we know that the canonical cocycle sets are in natural bijection Ω dR ( X ; b n R ) flat (cid:39) Ω n + ( X ) closed . Therefore, it just remains to see that the coboundary relations in both cases coincide. By the explicit nature (152)of the above natural bijection and by the Definition 3.83 of non-abelian coboundaries, we hence need to see that apair of closed forms C ( ) n + , C ( ) n + ∈ Ω n + ( X ) closed has a de Rham coboundary, i.e., ∃ h n ∈ Ω n dR ( X ) , such that C n + + dh n = C ( ) n + , (161)precisely if the pair extends to a closed ( n + ) -form on the cylinder over X , as in (155) (156): ∃ (cid:101) C n + ∈ Ω n + ( X × R ) closed , such that (cid:0) i X (cid:1) ∗ (cid:101) C n + = C ( ) n + and (cid:0) i X (cid:1) ∗ (cid:101) C n + = C ( ) n + . (162)That (161) ⇔ (162) is a standard argument: Let t denote the canonical coordinate function on R . In one direction,given h n as in (161), the choice (cid:101) C n + : = ( − t ) pr ∗ X (cid:0) C ( ) n + (cid:1) + t pr ∗ X (cid:0) C ( ) n + (cid:1) + dt ∧ pr ∗ X (cid:0) h n (cid:1) clearly satisfies (162). In the other direction, given (cid:101) C n + as in (162), the choice h n : = (cid:90) [ , ] (cid:101) C n + satisfies (161), by the fiberwise Stokes theorem (Lemma 3.85).41 he non-abelian de Rham theorem.Theorem 3.87 (Non-abelian de Rham theorem) . Let X , A ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def. 3.52), and letX admit the structure of a smooth manifold. Then the non-abelian de Rham cohomology (Def. 3.84) of X withcoefficients in the real Whitehead L ∞ -algebra l A (Prop. 3.63) is in natural bijection with the non-abelian realcohomology (Def. 3.72) of X with coefficients in L R A (Def. 3.55):H (cid:0) X ; L R A (cid:1) (cid:39) H dR ( X ; l A ) . (163) Proof.
Consider the following sequence of natural bijections: H (cid:0) X ; L R A (cid:1) = Ho (cid:0) TopologicalSpaces Qu (cid:1)(cid:0) X , L R A (cid:1) (cid:39) Ho (cid:0)(cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj (cid:1)(cid:0) Ω • PLdR ( A ) , Ω • PLdR ( X ) (cid:1) (cid:39) Ho (cid:0)(cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj (cid:1)(cid:0) CE ( l A ) , Ω • PL ( X ) (cid:1) (cid:39) H dR ( X ; l A ) . (164)Here the first line is the definition Def. 3.72. Then the first step is the fundamental theorem of rational homotopytheory (Prop. 3.60). The second step uses the following isomorphisms: CE ( l A ) (cid:39) Ω • PLdR ( A ) (Prop. 3.63) and Ω • PLdR ( X ) (cid:39) Ω • dR ( X ) (Lemma 3.90) in the homotopy category. The last step is Lemma 3.89. The composite ofthese natural bijections gives the desired bijection (163).We now prove the three lemmas used in the proof of Theorem 3.87: Lemma 3.88 (De Rham complex over cylinder of manifold is path space object) . For X ∈ SmoothManifolds ,consider the following morphisms of dgc-algebras (Def. 3.17) Ω • dR ( X ) ( pr X ) ∗ (cid:47) (cid:47) Ω • dR (cid:0) X × R (cid:1) ( i ∗ , i ∗ ) (cid:47) (cid:47) Ω • dR ( X ) ⊕ Ω • dR ( X ) (165) (from the de Rham complex of X (Example 3.23) to that of its cylinder manifold X × R , to its Cartesian productwith itself, by Example 3.22), given by pullback of differential forms along these smooth functions:X (cid:111) (cid:111) pr X X × R (cid:111) (cid:111) ( i , i ) (cid:63) (cid:95) (cid:0) X × { } (cid:1) (cid:116) (cid:0) X × { } (cid:1) (cid:39) X (cid:116) X . This is a path space object (Def. A.11) for Ω • dR ( X ) in (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj (Prop. 3.36).Proof. (i) It is clear by construction that the composite morphism is the diagonal. (ii)
That ( pr X ) ∗ is a weak equivalence, hence a quasi-isomorphism, follows from the de Rham theorem, using thatordinary cohomology is homotopy invariant: H • ( X × R ; R ) (cid:39) H • ( X ; R ) . (iii) That ( i ∗ , i ∗ ) is a fibration, namely degreewise surjective, is seen from the fact that any pair of forms on theboundaries X × { } , X × { } may be smoothly interpolated to zero along any small enough positive parameterlength, and then glued to a form on X × R . Lemma 3.89 (Non-abelian de Rham cohomology via the dgc-homotopy category) . Let X ∈ SmoothManifolds and g ∈ L ∞ Algebras ≥ , nil R , fin (Def. 3.34). Then the non-abelian de Rham cohomology of X with coefficients in g (Def. 3.84)is in natural bijection with the hom-set in the homotopy category of (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj (Prop. 3.36)from CE ( g ) (Def. 3.25) to Ω • dR ( X ) (Example 3.23):H dR (cid:0) X ; g (cid:1) (cid:39) Ho (cid:0)(cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj (cid:1)(cid:0) CE ( g ) , Ω • dR ( X ) (cid:1) . (166) Proof.
Consider a pair of dgc-algebra homomorphisms A ( ) , A ( ) ∈ DiffGradedCommAlgebras ≥ R (cid:0) CE ( g ) , Ω • dR ( X ) (cid:1) (167)hence of flat g -valued differential forms, according to Def. 3.77. Observe that: (i) CE ( g ) is cofibrant in (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj (105). (by Prop. 3.43, and since g is assumed to benilpotent (103)); 42 ii) Ω • dR ( X ) is fibrant in (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj (105). (by Remark 3.37); (iii) A right homotopy (Def. A.12) between the pair (167) of morphisms, with respect to the path space object Ω • dR ( X × R ) from Lemma 3.88, namely a morphism (cid:101) A making the following diagram commute Ω • dR ( X ) (cid:79) (cid:79) i ∗ (cid:106) (cid:106) A ( ) Ω • dR ( X × R ) i ∗ (cid:15) (cid:15) (cid:111) (cid:111) (cid:101) A CE ( g ) Ω • dR ( X ) (cid:116) (cid:116) A ( ) (168)is manifestly the same as a coboundary (cid:101) A between the corresponding flat g -valued forms according to Def.3.83.Therefore, Prop. A.16 says that the quotient set (157) defining the non-abelian de Rham cohomology is in naturalbijection to the hom-set in the homotopy category. Lemma 3.90 (PL de Rham complex on smooth manifold is equivalent to smooth de Rham complex) . Let X be asmooth manifold. Then (i)
There exists a zig-zag of weak equivalences (Def. 3.35) in (cid:0)
DiffGradedCommAlgebras ≥ R (cid:1) proj (105) between thesmooth de Rham complex of X (Example 3.23) and the PL de Rham complex of its underlying topological space(Def. 3.56). (ii) In particular, both are isomorphic in the homotopy category:X smooth manifold ⇒ Ω • dR ( X ) (cid:39) Ω • PLdR ( X ) ∈ Ho (cid:16)(cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj (cid:17) . Proof.
Let Ω • PSdR ( − ) (for “piecewise smooth”) be defined as the PL de Rham complex in Def. 3.56, but withsmooth differential forms on each simplex. Notice that this comes with the canonical natural inclusion Ω • PLdR ( − ) (cid:31) (cid:127) i poly (cid:47) (cid:47) Ω • PSdR ( − ) . Let then Tr ( X ) ∈ SimplicialSets be any smooth triangulation of X . This means that we have a homeomorphism outof its geometric realization to X | Tr ( X ) | p homeo (cid:47) (cid:47) X , (169)which restricts on the interior of each simplex to a diffeomorphism onto its image; and that we have an inclusionTr ( X ) (cid:31) (cid:127) η Tr ( X ) ∈ W (cid:47) (cid:47) Sing (cid:0) | Tr ( X ) | (cid:1) Sing ( p ) ∈ Iso (cid:47) (cid:47)
Sing ( X ) , (170)which is a weak equivalence (by Example A.35). In summary, this gives us the following zig-zag of dgc-algebrahomomorphisms: Ω • PLdR (cid:0) Tr ( X ) (cid:1) i poly (cid:37) (cid:37) Ω • dR ( X ) p ∗ (cid:124) (cid:124) Ω • PLdR ( X ) = Ω • PLdR (cid:0)
Sing ( X ) (cid:1) ( η S ) ∗ (cid:56) (cid:56) Ω • PSdR (cid:0) Tr ( X ) (cid:1) Here the two morphisms on the right are quasi-isomorphisms by [GM13, Cor. 9.9] (as in Prop. 3.62). Themorphism on the left is a quasi-isomorphism because i is a weak homotopy equivalence (345) and since Ω • PLdR preserves weak equivalences, by Ken Brown’s Lemma (Lemma A.19), since it is a Quillen left adjoint, by Prop.3.59, and since every simplicial set is cofibrant (Example A.8).
Flat twisted L ∞ -algebra valued differential forms. We generalize the above discussion to include twistings.43 efinition 3.91 (Local L ∞ -algebraic coefficients) . We say that a local L ∞ -algebraic coefficient bundle is a fibration g (cid:47) (cid:47) (cid:98) b p (cid:15) (cid:15) b (171)in L ∞ Algebras ≥ R , fin (Def. 3.25), hence a morphism such that under passage to Chevalley-Eilenberg algebras (89) wehave a cofibration CE ( g ) (cid:111) (cid:111) cofib ( CE ( p )) CE (cid:0)(cid:98) b (cid:1) (cid:79) (cid:79) CE ( p ) ∈ Cof CE ( b ) (172)in (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj (Prop. 3.36).In generalization of Def. 3.77, we say: Definition 3.92 (Flat twisted L ∞ -algebra valued differential forms) .(i) Let X ∈ SmoothManifolds and (cid:98) b (171) a local L ∞ -algebraic coefficient bundle (Def. 3.91). For τ dR ∈ Ω dR ( X ; b ) flat (173)a flat b -valued differential form on X (Def. 3.77), we say that a flat τ -twisted g -valued differential form on X is amorphism of dgc-algebras (Def. 3.17) in the slice over CE ( b ) Ω • dR ( X ) (cid:106) (cid:106) twist τ dR (cid:111) (cid:111) flat τ dR -twisted g -valued differential form A CE (cid:0)(cid:98) b (cid:1) (cid:53) (cid:53) CE ( p ) local L ∞ -algebraiccoefficients (cid:7)(cid:39) CE ( b ) (174) (ii) We write Ω τ dR dR ( X ; g ) flat : = (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) / CE ( b ) ( τ dR , p ) for the set of all flat τ dR -twisted g -valued differential forms on X . Remark 3.93 (Underlying flat forms of flat twisted forms) . Let X ∈ SmoothManifolds, let g (cid:47) (cid:47) (cid:98) b p (cid:47) (cid:47) b be a local L ∞ -algebraic coefficient bundle (Def. 3.91), and let τ dR ∈ Ω dR (cid:0) X ; b (cid:1) . Then there is a canonical forgetful naturaltransformation Ω τ dR ( X ; g ) flat (cid:47) (cid:47) Ω (cid:0) X ; (cid:98) b (cid:1) flat (175)from flat τ dR -twisted g -valued differential forms (Def. 3.92) to flat (cid:98) b -valued differential forms (Def. 3.77), givenby remembering only the top morphism in (174). Example 3.94 ( L ∞ -coefficient bundle for H -twisted differential forms [FSS16a, §4][FSS16b, §4][BMSS19, Lem.2.31]) . Consider the local L ∞ -algebraic coefficient bundle (Def. 3.91) given by the following multivariate polyno-mial dgc-algebras (Def. 3.30):CE (cid:0) l ku (cid:1) = R ... f , f , f , (cid:46) ... d f = d f = d f = (cid:111) (cid:111) ω k + (cid:91) ω k + R ... f , f , f , h (cid:46) ... d f = h ∧ f , d f = h ∧ f , d f = , d h = = CE (cid:16) l (cid:0) ku (cid:12) B U ( ) (cid:1)(cid:17) (cid:79) (cid:79) h ! h R (cid:2) h (cid:3)(cid:0) d h = (cid:1) = CE (cid:0) b R (cid:1) for complex topological K-theory in degree 1 and for itstwisted version is as in [FSS16a, §4][FSS16b, §4][BMSS19, Lem. 2.31]. In this case: (i) A twist (173) is equivalently an ordinary closed 3-form form (by Example 3.79): H ∈ Ω dR (cid:0) X ; b R (cid:1) flat (cid:39) Ω ( X ) closed . (176) (ii) The flat τ dR ∼ H -twisted l ku -valued differential forms according to Def. 3.92 are equivalently sequences ofodd-degree differential forms F k + ∈ Ω k + ( X ) satisfying the H -twisted de Rham closure condition (see [RW86,(23)][GS19c]): Ω τ dR (cid:0) X ; l ku (cid:1) flat (cid:39) (cid:26) F • + ∈ Ω • + (cid:12)(cid:12)(cid:12) d ∑ k F k + = H ∧ ∑ k F k − (cid:27) (177)(where we set F k − : = k − <
0, for convenience of notation).In direct generalization of Example 3.94, we have:
Example 3.95 ( L ∞ -coefficient bundle for higher twisted differential forms [FSS18, Def. 2.14]) . For r ∈ N , r ≥ L ∞ -algebraic coefficient bundle (Def. 3.91) given by the following multivariate polynomialdgc-algebras (Def. 3.30):CE (cid:16) ⊕ k ∈ N b rk R (cid:17) CE (cid:16)(cid:16) ⊕ k ∈ N b rk R (cid:17) (cid:12) B r − U ( ) (cid:17) R ... f r + , f r + , f , (cid:46) ... d f r + = d f r + = d f = (cid:111) (cid:111) f rk + (cid:91) f rk + R ... f r + , f r + , f , h r + (cid:46) ... d f r + = h r + ∧ f r + , d f r + = h r + ∧ f , d f = , d h r + = (cid:79) (cid:79) h r + ! h r + R (cid:2) h r + (cid:3)(cid:0) d h r + = (cid:1) CE (cid:0) b r R (cid:1) (178)In this case: (i) A twist (173) is equivalently an ordinary closed ( r + ) -form form (by Example 3.79): H r + ∈ Ω dR (cid:0) X ; b r R (cid:1) flat (cid:39) Ω r + ( X ) closed . (179) (ii) The flat τ dR ∼ H r + -twisted ⊕ k ∈ N b rk R -valued differential forms according to Def. 3.92 are equivalently se-quences of differential forms F r • + ∈ Ω k • + ( X ) satisfying the H ( r + ) -twisted de Rham closure condition (189): Ω τ dR (cid:0) X ; ⊕ k ∈ N b rk R (cid:1) flat (cid:39) (cid:26) F r • + ∈ Ω r • + (cid:12)(cid:12)(cid:12) d ∑ k F rk + = H r + ∧ ∑ k F rk − (cid:27) (180)(where we set F rk − : = rk − <
0, for convenience of notation).In twisted generalization of Example 3.81, we have the following:
Example 3.96 (Flat twisted differential forms with values in Whitehead L ∞ -algebras of spheres and twistor space) . The L ∞ -algebraic local coefficient bundles (Def. 3.91) given as the relative Whitehead L ∞ -algebras (Prop. 3.70) ofthe local coefficient bundles (61) for twisted and twistorial Cohomotopy (Example 2.44) are as shown on the rightof the following diagram [FSS19b, Lemma 3.19][FSS20, Thm. 2.14]:45E (cid:0) l ( B Sp ( )) ( C P (cid:12) Sp ( )) (cid:1) (cid:79) (cid:79) ( t H (cid:12) Sp ( )) ∗ = CE ( l B Sp ( )) h , f , ω , ω (cid:14) d h = ω − p − f ∧ f d f = d ω = − ω ∧ ω + ( p ) − χ d ω = (cid:79) (cid:79) (cid:31) (cid:63) Ω • dR ( X ) (cid:111) (cid:111) ( G , G ) (cid:86) (cid:86) τ dR (cid:122) (cid:122) ( G , G , F , H ) CE (cid:0) l ( B Sp ( )) ( S (cid:12) Sp ( )) (cid:1) (cid:64) (cid:64) = CE ( l B Sp ( )) (cid:20) ω , ω (cid:21)(cid:14)(cid:32) d ω = − ω ∧ ω + ( p ) − χ d ω = (cid:33) (cid:57) (cid:57) (cid:11)(cid:43) CE (cid:0) l B Sp ( ) (cid:1) = R (cid:20) χ , p (cid:21)(cid:14)(cid:32) d χ = d p = (cid:33) Therefore, given a smooth 8-dimensional spin manifold X equipped with tangential Sp ( ) -structure τ (59), theflat τ dR -twisted l S - and l C P -valued differential forms (Def. 3.92) are of the following form [FSS19b, Prop.3.20][FSS20, Prop. 3.9]: Ω τ dR dR (cid:0) X ; l S (cid:1) = (cid:40) G , G ∈ Ω • dR ( X ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d G = − (cid:0) G − p ( ∇ ) (cid:1) ∧ (cid:0) G + p ( ∇ ) (cid:1) − χ ( ∇ ) , d G = (cid:41) Ω (cid:101) τ dR dR (cid:0) X ; l C P (cid:1) = H , F , G , G ∈ Ω • dR ( X ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d H = G − p ( ∇ ) − F ∧ F , d F = , d G = − (cid:0) G − (cid:1) ∧ (cid:0) G + (cid:1) − χ ( ∇ ) , d G = , (181)Notice: (a) Here we are using (Example 4.27) that the de Rham image τ dR of the rationalization L R τ of the twist τ is givenby evaluating characteristic forms (Def. 4.19) on any Sp ( ) -connection ∇ . (b) In the second equation of (181) we are using the above minimal model for C P (cid:12) Sp ( ) relative to S (cid:12) Sp ( ) (instead of relative to B Sp ( ) ). Twisted non-abelian de Rham cohomology.
In generalization of Def. 3.83, we set:
Definition 3.97 (Coboundaries between flat twisted L ∞ -algebraic forms) . Let X ∈ SmoothManifolds, let g (cid:47) (cid:47) (cid:98) b p (cid:47) (cid:47) b be a local L ∞ -algebraic coefficient bundle (Def. 3.91), and let τ dR ∈ Ω dR ( X ; b ) . Then for A ( ) , A ( ) ∈ Ω τ dR dR ( X ; g ) a pair of flat τ dR -twisted g -valued differential forms on X (Def. 3.92) a coboundary between them is a coboundary (cid:101) A ∈ Ω dR (cid:0) X × R ; (cid:98) b (cid:1) (182)in the sense of Def. 3.83 between the underlying flat (cid:98) b -valued forms (via Remark 3.93), such that the underling b -valued form of H equals the pullback of the twist τ dR along X × R pr X (cid:47) (cid:47) X p ∗ ( H ) = pr ∗ X ( τ dR ) . (183)If such a coboundary exists, we say that A ( ) and A ( ) are cohomologous , to be denoted A ( ) ∼ A ( ) .
46n generalization of Def. 3.84, we set:
Definition 3.98 (Twisted non-abelian de Rham cohomology) . Let X ∈ SmoothManifolds, let g (cid:47) (cid:47) (cid:98) b p (cid:47) (cid:47) b be alocal L ∞ -algebraic coefficient bundle (Def. 3.91), and let τ dR ∈ Ω dR (cid:0) X ; b (cid:1) . Then the τ dR -twisted non-abelian deRham cohomology of X with coefficients in g is the set H τ dR dR ( X ; g ) : = (cid:0) Ω τ dR dR ( X ; g ) flat (cid:1) / ∼ (184)of equivalence classes with respect to the coboundary relation from Def. 3.97 on the set of flat τ dR -twisted g -valueddifferential forms on X (Def. 3.92).Twisted de Rham cohomology is traditionally familiar in the form of degree-3 twisted cohomology of even/odddegree differential forms [RW86, §III, Appendix][BCMMS02, §9.3][MaSt03, §3][FrHT08, §2][Te04, Prop. 3.7][Cav05, §I.4][Sa10][MW11][GS19b] (which is the target of the twisted Chern character in degree-3 twisted K-theory, see Prop. 5.5): Definition 3.99 (Degree-3 twisted abelian de Rham cohomology) . For X ∈ SmoothManifolds, and H ∈ Ω ( X ) closed a closed differential 3-form, the H -twisted de Rham cohomology of X is the cochain cohomology H • + H dR ( X ) : = ker • (cid:0) d − H ∧ ( − ) (cid:1) im • (cid:0) d − H ∧ ( − ) (cid:1) (185)of the following 2-periodic cochain complex: · · · (cid:47) (cid:47) (cid:76) k Ω ( n − )+ k dR ( X ) ( d − H ∧ ( − )) (cid:47) (cid:47) (cid:76) k Ω n + k dR ( X ) ( d − H ∧ ( − )) (cid:47) (cid:47) (cid:76) k Ω ( n + )+ k dR ( X ) (cid:47) (cid:47) · · · . We show that this is a special case of twisted non-abelian de Rham cohomology according to Def. 3.98:
Proposition 3.100 (Twisted non-abelian de Rham cohomology subsumes H -twisted abelian de Rham cohomol-ogy) . Given a twisting 3-form as in (176) τ dR (cid:111) (cid:111) (cid:47) (cid:47) ∈ H ∈ Ω (cid:0) X ; b R (cid:1) flat (cid:39) Ω ( X ) closed the τ dR -twisted non-abelian de Rham cohomology (Def. 3.98) of flat twisted l ku -valued differential forms (Exam-ple 3.94) is naturally equivalent to H -twisted abelian de Rham cohomology (Def. 3.99) in odd degree b R -twisted l ku -valuednon-abelian de Rham cohomology H τ dR dR ( X ; l ku ) (cid:39) traditional H -twistedde Rham cohomology H + H dR ( X ) Proof.
By (177) in Example 3.94 the cocycle sets on both sides are in natural bijection. Hence it is sufficient to seethat the coboundary relations on the cocycle sets coincide, under this identification. In one direction, consider acoboundary in the sense of twisted non-abelian de Rham cohomology (Def. 3.97) with coefficients as in Example3.94: (cid:101) F • + ∈ Ω dR (cid:0) X × R ; l ku (cid:1) . We claim that h • : = (cid:90) [ , ] (cid:101) F • + (186)satisfies the coboundary condition (185): (cid:0) d − H ∧ (cid:1) ∑ k h k = ∑ k (cid:0) F ( ) k + − F ( ) k + (cid:1) . (187)To see this, we may compute as follows: The notation “ H ” for the twist (and of “ H r + ” for the higher twists later) originates in the physics literature and has made it as aconvention in differential geometry as well. Of course, not to be confused with homology. The discussion for even degrees is directly analogous and we omit it for brevity. ∑ k h k = ∑ k (cid:32) F ( ) k + − F ( ) k + − (cid:90) [ , ] d (cid:101) F k + (cid:33) = ∑ k (cid:32) F ( ) k + − F ( ) k + − (cid:90) [ , ] (cid:0) pr ∗ X H (cid:1) ∧ (cid:101) F k − (cid:33) = ∑ k (cid:32) F ( ) k + − F ( ) k + + H ∧ (cid:90) [ , ] (cid:101) F k − (cid:33) = ∑ k (cid:16) F ( ) k + − F ( ) k + + H ∧ h k − (cid:17) , where the first step is the fiberwise Stokes formula (158) together with the defining restrictions (156) of (cid:101) F • + ;the second step is the cocycle condition (177) on (cid:101) F • + using the constraint (183); the third step is the projectionformula (159); and the last step uses again the definition (186).Conversely, given h • satisfying (187), we claim that (cid:101) F • + ; = ( − t ) pr ∗ (cid:0) F ( ) • + (cid:1) + t pr ∗ (cid:0) F ( ) • + (cid:1) + dt ∧ pr ∗ X ( h • ) (188)is a coboundary of twisted non-abelian cocycles, in the sense of Def. 3.97: It is immediate that (188) has therequired restrictions (156). We check by direct computation that it satisfies the required differential equation: d ∑ k (cid:101) F k + = ∑ k (cid:16) − dt ∧ pr ∗ X (cid:0) F ( ) k + (cid:1) + ( − t ) pr ∗ X (cid:0) H (cid:1) ∧ pr ∗ X (cid:0) F ( ) k − (cid:1) + dt ∧ pr ∗ X (cid:0) F ( ) k + (cid:1) + t pr ∗ X (cid:0) H (cid:1) ∧ pr ∗ X (cid:0) F ( ) k − (cid:1) − dt ∧ pr ∗ X (cid:0) d h k (cid:124)(cid:123)(cid:122)(cid:125) = F ( ) k + − F ( ) k + + H ∧ h k (cid:1) (cid:17) = ∑ k (cid:16) pr ∗ X ( H ) ∧ (cid:101) F k − (cid:17) . In generalization of Def. 3.99, there are twisted abelian Rham complexes with twist any odd-degree closedform [Te04][Sa09][MW11][Sa10][GS19b] (these serve as the targets for the Chern character [MMS20] on higher-twisted ordinary K-theory [Te04][Go08][DP13][Pen16], see Example 5.7 below; and for the LSW-character ontwisted higher K-theories [LSW16, §2.1], see Prop. 5.10 below):
Definition 3.101 (Higher twisted abelian de Rham cohomology) . For X ∈ SmoothManifolds, r ∈ N , r ≥
1, and H r + ∈ Ω r + ( X ) closed a closed differential ( r + ) -form, the H r + -twisted de Rham cohomology of X is thecochain cohomology Ω • + H r + dR ( X ) : = ker • (cid:0) d − H r + ∧ ( − ) (cid:1) im • (cid:0) d − H r + ∧ ( − ) (cid:1) (189)of the following 2 r -periodic cochain complex: · · · (cid:47) (cid:47) (cid:76) k Ω ( n − )+ rk dR ( X ) ( d − H r + ∧ ( − )) (cid:47) (cid:47) (cid:76) k Ω n + rk dR ( X ) ( d − H r + ∧ ( − )) (cid:47) (cid:47) (cid:76) k Ω ( n + )+ rk dR ( X ) (cid:47) (cid:47) · · · . In direct generalization of Prop. 3.100, we find:
Proposition 3.102 (Twisted non-abelian de Rham cohomology subsumes higher twisted abelian de Rham coho-mology) . For r ∈ N , r ≥ , consider a twisting ( r + ) -form as in (179) τ dR (cid:111) (cid:111) (cid:47) (cid:47) ∈ H r + ∈ Ω (cid:0) X ; b r R (cid:1) flat (cid:39) Ω r + (cid:0) X (cid:1) closed he τ dR -twisted non-abelian de Rham cohomology (Def. 3.98) of flat twisted l K r − ( ku ) -valued differential forms(Example 3.95) is naturally equivalent to H r + -twisted abelian de Rham cohomology (Def. 3.101) in degree r. twistednon-abelian de Rham cohomology H τ dR dR (cid:16) X ; ⊕ k ∈ N b rk R (cid:17) (cid:39) higher H r + -twistedde Rham cohomology H + H r + dR ( X ) . Proof.
By Example 3.95, the cocycle sets on both sides are in natural bijection. Hence it remains to see that thecoboundary relations correspond to each other, under this identification. This proceeds verbatim, up to degreeshifts, as in the proof of Prop. 3.100 (which is the special case of r = Example 3.103 (Cohomology operation in (higher-) twisted de Rham cohomology) . Degree-3 twisted de Rhamcohomology (Def. 3.99) supports the following twisted cohomology operations (Def. 2.41): (i) wedge product with H : H • + H dR ( X ) (cid:47) (cid:47) H • + + H dR ( X ) ∑ k F k ∑ k F k ∧ H (ii) wedge square : (cid:76) r H r + H dR ( X ) (cid:47) (cid:47) (cid:76) r H r + H dR ( X ) ∑ k F k (cid:16) ∑ k F k (cid:17) ∧ (cid:16) ∑ k F k (cid:17) (iii) compositions of these : (cid:76) r H r + H dR ( X ) (cid:47) (cid:47) (cid:76) r H r + + H dR ( X ) ∑ k F k (cid:16) ∑ k F k (cid:17) ∧ (cid:16) ∑ k F k (cid:17) ∧ H In type IIA string theory, terms of the form (iii) arise, together with terms of the form I ∪ [ H ] with I a polynomialin the Pontrjagin classes (cf. Example 4.27). See [GS19c] for extensive discussions.This evidently generalizes to higher twisted de Rham cohomology (Def. 3.101) and higher twisted real coho-mology in the sense of [GS19b], with H replaced by H r + for r ∈ N . The twisted non-abelian de Rham theorem.Theorem 3.104 (Twisted non-abelian de Rham theorem) . Let X ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def. 3.52),equipped with the structure of a smooth manifold, and letA (cid:47) (cid:47) local coefficient bundle A (cid:12) G ρ (cid:15) (cid:15) BG (190) be a local coefficient bundle (31) in Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def. 3.52) such that the action of π ( BG ) = π ( G ) on the real homology groups of A is nilpotent. Consider, via Prop. 3.75, the rationalized coefficient bundleL R ρ with corresponding L ∞ -algebraic coefficient bundle l ρ (Def. 3.91) of the relative real Whitehead L ∞ -algebra(Prop. 3.70): L R A (cid:47) (cid:47) rationalizedlocal coefficient bundle (cid:0) L R A (cid:1) (cid:12) (cid:0) L R G (cid:1) L R ρ (cid:15) (cid:15) L R BG l A (cid:47) (cid:47) L ∞ -algebraic coefficient bundleof Whitehead L ∞ -algebras l BG (cid:0) A (cid:12) G (cid:1) . l ρ (cid:15) (cid:15) l BG The discussion for other degrees is directly analogous, and we omit it for brevity. oreover, let X τ (cid:47) (cid:47) L R BG ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) be such that H ( τ ; R ) is injective (for instance in that BG is simply connected).Then the τ -twisted non-abelian real cohomology (Def. 3.76) of X with local coefficients in L R ρ (Prop. 3.75) isin natural bijection with the twisted non-abelian de Rham cohomology (Def. 3.98) of X with local coefficients in l ρ , τ -twisted non-abelianreal cohomology H τ (cid:0) X ; L R A (cid:1) (cid:39) τ dR -twisted non-abeliande Rham cohomology H τ dR dR ( X ; l A ) , (191) where the twists are related by the plain non-abelian de Rham theorem (Theorem 3.87): [ τ ] (cid:111) (cid:111) (cid:47) (cid:47) ∈ [ τ dR ] ∈ H (cid:0) X ; L R BG (cid:1) (cid:39) H dR (cid:0) X ; l BG (cid:1) Proof.
Consider the following sequence of natural bijections H τ (cid:0) X ; L R A (cid:1) = Ho (cid:16)(cid:0) TopologicalSpaces / L R BG Qu (cid:1)(cid:17)(cid:0) τ , L R ρ (cid:1) (cid:39) Ho (cid:16)(cid:0) DiffGradedCommAlgebras ≥ R (cid:1) Ω • PLdR ( BG ) / proj (cid:17)(cid:0) Ω • PLdR ( ρ ) , Ω • PLdR ( τ ) (cid:1) (cid:39) Ho (cid:16)(cid:0) DiffGradedCommAlgebras ≥ R (cid:1) CE ( b ) / proj (cid:17)(cid:0) CE ( l ρ ) , τ ∗ dR (cid:1) (cid:39) H τ dR dR ( X ; l A ) . Here the first line is the definition (Def. 3.76). Then the first step is the fundamental theorem (Prop. 3.60) in theco-slice category. The substitutions in the second step are: (a)
Lemma 3.108 in the first argument (this is where the H -injectivity is needed); (b) Lemma 3.90 with Theorem 3.87 in the second argument (as in the second step of (164)).The last step is Lemma 3.106. The composite of these equivalences is the desired (191).We now establish the remaining four lemmas which enter the proof of Theorem 3.104.
Lemma 3.105 (Pullback to de Rham complex over cylinder of manifold is relative path space object) . Let X ∈ SmoothManifolds , let b ∈ L ∞ Algebras ≥ R , fin (Example 3.24) with Chevalley-Eilenberg algebra CE ( b ) ∈ DiffGradedCommAlgebras ≥ R (88) , and let (cid:8) Ω • dR ( X ) (cid:111) (cid:111) τ ∗ dR CE ( b ) (cid:9) ∈ (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) CE ( b ) / proj bea morphism of dgc-algebras to the de Rham complex of X (Example 3.23), regarded as an object in the coslicemodel category (Example A.10) of (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj (Prop. 3.36) under CE ( b ) . Then a pathspace object (Def. A.11) for τ ∗ dR is given by this diagram: Ω • dR ( X ) pr ∗ X ∈ W (cid:47) (cid:47) (cid:107) (cid:107) τ ∗ dR Ω • dR ( X × R ) ( i ∗ , i ∗ ) ∈ Fib (cid:47) (cid:47) (cid:79) (cid:79) pr ∗ X ◦ τ ∗ dR Ω • dR ( X ) ⊕ Ω • dR ( X ) (cid:51) (cid:51) ( τ ∗ dR , τ ∗ dR ) CE ( b ) , where the top morphisms are from (165) .Proof. It is clear that the diagram commutes, by construction. Moreover, the top morphisms are a weak equivalencefollowed by a fibration in (cid:0)
DiffGradedCommAlgebras ≥ R (cid:1) proj , by Lemma 3.88. Therefore, by the nature of thecoslice model structure (Example A.10) the total diagram constitutes a factorization of the diagonal on τ ∗ dR througha weak equivalence followed by a fibration, as required (314). (To see that the composite really is still the diagonalmorphism in the coslice, observe that Cartesian products in any coslice category are reflected in the underlyingcategory.) It only remains to observe that τ ∗ dR is actually a fibrant object in the coslice model category. But theterminal object in the coslice is clearly the unique morphism from CE ( b ) to the zero-algebra (Example 3.21), sothat in fact every object in the coslice is still fibrant 50 • dR ( X ) ∈ Fib (cid:47) (cid:47) (cid:107) (cid:107) τ ∗ dR (cid:51) (cid:51) CE ( b ) (192)as in Remark 3.37. Lemma 3.106 (Twisted non-abelian de Rham cohomology via the coslice dgc-homotopy category) . ConsiderX ∈ SmoothManifolds , let g (cid:47) (cid:47) (cid:98) b p (cid:15) (cid:15) b ∈ L ∞ Algebras ≥ , nil R , fin be an L ∞ -algebraic local coefficient bundle (Def. 3.91) of nilpotent L ∞ -algebras (Def. 3.34) with Chevalley-Eilenberg algebra CE ( (cid:98) b ) , CE ( b ) ∈ DiffGradedCommAlgebras ≥ R (88) , and let Ω • dR ( X ) (cid:111) (cid:111) τ ∗ dR CE ( b ) ∈ (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) CE ( b ) / proj (193) be a morphism of dgc-algebras to the de Rham complex of X (Example 3.23), hence a flat b -valued differentialform (Def. 3.77) τ dR ∈ Ω dR ( X ; b ) , equivalently regarded as an object in the coslice model category (Example A.10) of (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj (Prop. 3.36) under CE ( b ) . Then the τ dR -twisted non-abelian de Rham cohomology of X with coefficients in g (Def. 3.98) is in natural bijection with the hom-set in the homotopy category (Def. A.14) of the coslice modelcategory (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) CE ( b ) proj (Example A.10) of the projective model structure on dgc-algebras(Prop. 3.36) from CE ( p ) (172) to τ ∗ dR (193) :H τ dR dR (cid:0) X ; g (cid:1) (cid:39) Ho (cid:16)(cid:0) DiffGradedCommAlgebras ≥ R (cid:1) CE ( b ) / proj (cid:17)(cid:0) CE ( p ) , τ ∗ dR (cid:1) . (194) Proof.
Consider a pair of dgc-algebra homomorphisms in the coslice Ω • dR (cid:0) X (cid:1) (cid:114) (cid:114) A ( ) (cid:108) (cid:108) A ( ) (cid:103) (cid:103) τ ∗ dR CE ( (cid:101) b ) (cid:55) (cid:55) CE ( p ) CE ( b ) ∈ (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) CE ( b ) / proj (cid:0) CE ( p ) , τ ∗ dR (cid:1) , (195)hence of flat τ dR -twisted g -valued differential forms, according to Def. 3.92. Observe that: (i) CE ( p ) is cofibrant in (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) CE ( p ) / proj , since: (a) the initial object in the coslice is CE ( b ) (cid:111) (cid:111) id CE ( b ) , (b) the unique morphism from this object to CE ( p ) isCE ( b ) CE ( p ) ∈ Cof (cid:47) (cid:47) (cid:105) (cid:105) id CE (cid:0)(cid:98) b (cid:1) (cid:53) (cid:53) CE ( p ) CE ( b ) (196) (c) CE ( p ) is a cofibration in (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj , by (172), so that the diagram (196) is acofibration in the coslice model category, by Example A.10. (ii) pr ∗ X ◦ τ ∗ dR is fibrant in (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) CE ( b ) / proj , by (192); (iii) A right homotopy (Def. A.12) between the pair (195) of coslice morphisms, with respect to the path spaceobject from Lemma 3.105, namely a (cid:101) A that makes the following diagram commute Ω • dR ( X ) (cid:79) (cid:79) i ∗ (cid:106) (cid:106) A ( ) Ω • dR ( X × R ) (cid:101) (cid:101) pr ∗ X , τ ∗ dR i ∗ (cid:15) (cid:15) (cid:111) (cid:111) (cid:101) A CE ( (cid:98) b ) (cid:59) (cid:59) CE ( p ) Ω • dR ( X ) (cid:116) (cid:116) A ( ) CE ( (cid:98) b ) (197)51s manifestly the same as a coboundary (cid:101) A between the corresponding flat twisted g -valued forms accordingto Def. 3.97: (a) The top part of (197) is, just as in (168), the flat twisted (cid:98) g -valued form on the cylinder over X that isrequired by (182); (b) the bottom part of (197) is the condition (183) on the extension of the twist to the cylinder over X .Therefore, Prop. A.16 says that the quotient set (184) defining the twisted non-abelian de Rham cohomology is innatural bijection to the hom-set in the coslice homotopy category. Lemma 3.107 (Derived cobase change along quasi-isomorphism is equivalence on H -injectives) . LetB φ ∈ W (cid:47) (cid:47) B ∈ (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj be a quasi-isomorphism of dgc-algebras (Def. 3.17), hence a weak equivalence in the projective model structure(Prop. 3.36). Assume that either, hence both, dgc-algebras are cohomologically connected (H ( B ) = R , H ( B ) = R ). Then the derived adjunction (Prop. A.20) of the base change Quillen adjunction (Example A.18) between thecorresponding co-slice model categories (Example A.10) of the opposite model category of dgc-algebras (ExampleA.9) restricts to an equivalence on the full subcategories of the homotopy categories (Def. A.14) on those co-sliceobjects which are connected in H ( − ) and injective on H ( − ) : Ho (cid:16)(cid:16)(cid:0) DiffGradedCommAlgebras ≥ R (cid:1) B / proj (cid:1) op (cid:17) H − conn H − inj (cid:111) (cid:111) L ( φ op ) ! R ( φ op ) ∗ (cid:39) (cid:47) (cid:47) Ho (cid:16)(cid:16)(cid:0) DiffGradedCommAlgebras ≥ R (cid:1) B / proj (cid:1) op (cid:17) H − conn H − inj Proof.
Notice that if (cid:0)
DiffGradedCommAlgebras ≥ R (cid:1) proj were a left proper model category (Def. A.5), so that (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) opproj were right proper, the statement would directly follow as a special case of Prop.A.31, without any restriction to subcategories.While (cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj is (apparently) not left proper, it comes close: Lemma 3.44 says thatquasi-isomorphisms are preserved by pushout along at least those cofibrations that are relative Sullivan algebras(i.e. the relative cell complexes, but possibly not their retracts). Hence we adapt the logic underlying Prop. A.31 tothis case. Namely, Prop. 3.49 says that those co-slice objects that are H -injective between H -connected algebrasdo have a cofibrant replacement by a relative Sullivan algebra: A min B ∈ W (cid:47) (cid:47) AB coslice object (cid:52) (cid:52) (cid:89)(cid:57) ∈ RelSulAlg ⊂ Cof (cid:107) (cid:107) (198)Now, first to see that the derived adjunction restricts to the given subcategories: In one direction, it is clear that L ( φ op ) ! preserves H and H , as this functor is given by precomposition with the quasi-isomorphism φ . In theother direction: R ( φ op ) ∗ is given by pushout along φ of a cofibrant representative of the given coslice object, andby (198) we may take that cofibrant representative to be a relative Sullivan algebra. But then Prop. 3.44 impliesthat the pushout has the same cohomology.Finally, to see that this restriction of the derived adjunction is an equivalence of categories, hence that thederived unit (325) and derived counit (326) are isomorphisms on these subcategories. This follows just as in thealternative proof (336) of Prop. A.31, using for the fibrant objects ρ there the opposites of the good fibrations givenby (198), for which Prop. 3.44 guarantees the required properness condition. Lemma 3.108 (Pasting composition with relative Sullivan model of local coefficient bundle) . LetA (cid:47) (cid:47) local coefficient bundle A (cid:12) G ρ (cid:15) (cid:15) BG (199) be a local coefficient bundle (31) in Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def. 3.52), and let • PLdR (cid:0) A (cid:12) G (cid:1) (cid:79) (cid:79) Ω • PLdR ( ρ ) (cid:111) (cid:111) p min BGA (cid:12) G ∈ W CE (cid:0) l ( A (cid:12) G ) (cid:1) (cid:79) (cid:79) CE ( l ρ ) Ω • PLdR (cid:0) BG (cid:1) (cid:111) (cid:111) p min BG ∈ W CE (cid:0) l ( BG ) (cid:1) (200) be its minimal relative Sullivan model (142) , which exists by Prop. 3.70. Then the pasting precomposition with thesquare (200) is a natural isomorphism of hom-functors on the homotopy categories from Lemma 3.107: Ho (cid:16)(cid:0) DiffGradedCommAlgebras ≥ R (cid:1) Ω • PLdR ( BG ) / proj (cid:17) H − conn H − inj (cid:0) Ω • PLdR ( ρ ) , − (cid:1)(cid:0) L (cid:0) ( p min BG ) op (cid:1) ! (cid:1) op (cid:39) (cid:15) (cid:15) Ho (cid:16)(cid:0) DiffGradedCommAlgebras ≥ R (cid:1) CE ( l BG ) / proj (cid:17) H − conn H − inj (cid:0) Ω • PLdR ( ρ ) ◦ p min BG , − (cid:1) ( − ) ◦ p min BGA (cid:12) G (cid:39) (cid:15) (cid:15) Ho (cid:16)(cid:0) DiffGradedCommAlgebras ≥ R (cid:1) CE ( l BG ) / proj (cid:17) H − conn H − inj (cid:0) CE ( l ρ ) , − (cid:1) (201) Here the first step is the derived left co-base change along φ (Example A.18), while the second is composition withthe diagram 200 regarded as a morphism in the co-slice under CE ( l BG ) .Proof. First notice that Ω • PLdR ( ρ ) is indeed an injection on H , by the assumption that the fiber A is connected (asin the proof of Prop. 3.70). With that, the first step is an isomorphism by Lemma 3.107, while the second step isevidently an isomorphism, since the weak equivalence φ A (cid:12) G becomes an isomorphism in the homotopy category(and still so in the coslice homotopy category, by Example A.10).53 The (differential) non-abelian character map
We introduce the character map in non-abelian cohomology (Def. 4.2) and then discuss how it specializes to:§4.1 – the Chern-Dold character on generalized cohomology;§4.2 – the Chern-Weil homomorphism on degree-1 non-abelian cohomology;§4.3 – the Cheeger-Simons differential characters on degree-1 non-abelian cohomology.
Definition 4.1 (Rationalization in non-abelian cohomology) . For A ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def. 3.52)we write ( η R A ) ∗ : non-abeliancohomology H ( − ; A ) H ( − ; η R A ) rationalization (cid:47) (cid:47) non-abelianreal cohomology H (cid:0) − ; L R A (cid:1) (202)for the cohomology operation (Def. 2.17) from non-abelian A -cohomology (Def. 2.1) to non-abelian real coho-mology (Def. 3.72), which is induced (24) by the rationalization map η R A (Def. 3.55). Definition 4.2 (Non-abelian character map) . Let X , A ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def. 3.52) such that X admits the structure of a smooth manifold. Then we say that the non-abelian character map in non-abelian A -cohomology (Def. 2.1) is the cohomology operation (Def. 4.1) non-abeliancharacter map ch A : non-abeliancohomology H ( X ; A ) ( η R A ) ∗ rationalization (cid:47) (cid:47) non-abelianreal cohomology H (cid:0) X ; L R A (cid:1) (cid:39) non-abeliande Rham theorem (cid:47) (cid:47) non-abeliande Rham cohomology H dR ( X ; l A ) (203)from non-abelian A -cohomology (Def. 2.1) to non-abelian de Rham cohomology (Def. 3.84) with coefficients inthe rational Whitehead L ∞ -algebra l A of A (Prop 3.63), which is the composite of (i) the operation (202) of rationalization of coefficients (Def. 4.1), (ii) the equivalence (163) of the non-abelian de Rham theorem (Theorem 3.87). We prove (Theorem 4.8) that the non-abelian character map reproduces the Chern-Dold character on generalizedcohomology theories (recalled as Def. 4.6) and in particular the Chern character on topological K-theory (Example4.10).
Proposition 4.3 (Dold’s equivalence [Do65, Cor. 4][Hil71, Thm. 3.18][Ru98, §II.3.17]) . Let E be a general-ized cohomology theory (Example 2.13). Then its rationalization E R is equivalent to ordinary cohomology withcoefficients in the rationalized stable homotopy groups of E:E n R ( X ) do E (cid:39) (cid:47) (cid:47) (cid:76) k ∈ Z H n + k (cid:0) X ; π k ( E ) ⊗ R R (cid:1) . Remark 4.4 (Rational stable homotopy theory) . In modern stable homotopy theory, Dold’s equivalence (Prop.4.3) is a direct consequence of the fundamental theorem [SSh01, Thm. 5.1.6] that rational spectra are direct sumsof Eilenberg-MacLane spectra with coefficients in the rationalized stable homotopy groups [BMSS19, Prop. 2.17].But we may explicitly re-derive Dold’s equivalence using the unstable rational homotopy theory from §3:
Proposition 4.5 (Dold’s equivalence via non-abelian real cohomology) . Let E be a generalized cohomology theory(Example 2.13) and let n ∈ N such that the nth coefficient space (17) is of R -finite homotopy type (Def. 3.52)E n ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil . Then there is a natural equivalence between the non-abelian real cohomology (Def. 3.72) with coefficients in E n and ordinary cohomology with coefficients in the rationalized homotopy groups of E:H (cid:0) − ; L R E n (cid:1) (cid:39) (cid:77) k ∈ N H n + k (cid:0) − ; π k ( E ) ⊗ Z R (cid:1) . (204)54 roof. Since E n is an infinite-loop space, it is necessarily nilpotent (Example 3.54). We may assume withoutrestriction that it is also connected, for otherwise we apply the following argument to each connected component(Remark 3.53). Hence E n ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def. 3.52) and the discussion in §3.2 applies:Again since E n is a loop space (17), Prop. 3.74 gives H ( − ; L R E n ) (cid:39) ⊕ k ∈ N H k ( − ; π k ( E n ) ⊗ Z R ) . The claim followsfrom the definition of stable homotopy groups as π k − n ( E ) = π k ( E n ) for k , n ≥ Definition 4.6 (Chern-Dold character [Bu70][Hil71, p. 50]) . Let E be a generalized cohomology theory (Example2.13). The Chern-Dold character in E -cohomology theory is the cohomology operation to ordinary cohomologywhich is the composite of rationalization in E -cohomology with Dold’s equivalence (Prop. 4.3): Chern-Doldcharacter ch E : E • ( − ) rationalizationin E -cohomoloy (cid:47) (cid:47) (18) (cid:39) (cid:15) (cid:15) E • R ( − ) (cid:39) Dold’s equivalence do E (cid:47) (cid:47) (18) (cid:39) (cid:15) (cid:15) (cid:76) k H • + k (cid:0) − ; π k ( E ) ⊗ Z R (cid:1) H ( − ; E • ) ( η R E • ) ∗ (202) (cid:47) (cid:47) H ( − ; L R E • ) (cid:39) (204) (cid:51) (cid:51) . (205)Here the bottom part in (205) serves to make the nature of the top maps fully explicit, using Example 2.13, Def.4.1 and Prop. 4.5. Remark 4.7 (Rationalization in the Chern-Dold character) . That the first map in the Dold-Chern character (205)is the rationalization localization is stated somewhat indirectly in the original definition [Bu70] (the concept ofrationalization was fully formulated later in [BK72]). The role of rationalization in the Chern-Dold character ismade fully explicit in [LSW16, §2.1]. The same rationalization construction of the generalized Chern character,but without attribution to [Bu70] or [Do65], is considered in [HS05, §4.8] (see also [BN14, p. 17]).We now come to the main result in this section:
Theorem 4.8 (Non-abelian character subsumes Chern-Dold character) . Let E be a generalized cohomology theory(Example 2.13) and let n ∈ N such that the nth coefficient space (17) is of R -finite homotopy type (Def. 3.52). Letmoreover X be a smooth manifold of connected, nilpotent, R -finite homotopy type (Def. 3.52).Then the non-abelian character (Def. 4.2) coincides with the Chern-Dold character (Def. 4.6) on E-cohomologyin degree n, in that the following diagram commutes:H (cid:0) X ; E n (cid:1) ch En (cid:47) (cid:47) (cid:79) (cid:79) (18) (cid:39) H dR (cid:0) l E n (cid:1) (cid:39) (163) (204) (cid:15) (cid:15) E n ( X ) ch En (cid:47) (cid:47) (cid:76) k H n + k (cid:0) X ; π k ( E ) ⊗ Z R (cid:1) . (206) Here the equivalence on the left is from Example 2.13, while the equivalence on the right is the inverse non-abeliande Rham theorem (Theorem 3.87) composed with that from Prop. 4.5.Proof.
Since E n is an infinite-loop space, it is necessarily nilpotent (Example 3.54). We may assume withoutrestriction that it is also connected, for otherwise we apply the following argument to each connected component(Remark 3.53). Hence E n ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def. 3.52) and the discussion in §3.2 and §3.3 applies:The non-abelian de Rham isomorphism (163) in the definition (203) of the non-abelian character cancelsagainst its inverse on the right of (206). Commutativity of the remaining diagram H (cid:0) X ; E n (cid:1) ( η R En ) ∗ (cid:47) (cid:47) (cid:79) (cid:79) (18) (cid:39) H (cid:0) X ; L R E n (cid:1) (cid:39) (204) (cid:15) (cid:15) E n ( X ) ch En (cid:47) (cid:47) (cid:76) k H n + k (cid:0) X ; π k ( E ) ⊗ Z R (cid:1) is the very definition of the Chern-Dold character (Def. 4.6).55 xample 4.9 (de Rham homomorphism in ordinary cohomology) . On ordinary integral cohomology (Example2.2), the non-abelian character (Def. 4.2) reduces to extension of scalars from the integers to the real numbers,followed by the de Rham isomorphism, in that the following diagram commutes: H (cid:0) − ; B n + Z (cid:1) ch Bn + Z non-abelian characteron ordinary cohomology (cid:47) (cid:47) (cid:79) (cid:79) (9) (cid:39) H dR (cid:0) − ; l B n + Z (cid:1) (cid:39) (137) (160) (cid:15) (cid:15) H n + ( − ; Z ) extensionof scalars (cid:47) (cid:47) H n + ( − ; R ) (cid:39) ordinaryde Rham isomorphism (cid:47) (cid:47) H n + ( − ) Example 4.10 (Chern character on complex K-theory) . The spectrum (17) representing complex K-theory has 0thcomponent space KU (cid:39) Z × B U (19). Here the connected components B U, the classifying space of the infiniteunitary group (20), are clearly of finite R -type (since their real cohomology is the ring of universal Chern classes,e.g. [Ko96, Thm. 2.3.1]). Therefore, Theorem 4.8 applies and says that the non-abelian character map (Def. 4.2)for coefficients in Z × B U reduces to the Chern-Dold character on complex K-theory. This, in turn, is equivalent(by [Hil71, Thm. 5.8]) to the original Chern character ch on complex K-theory [Hi56, §12.1][BH58, §9.1][AH61,§1.10] (review in [Hil71, §V]): non-abelian cohomologywith Z × B U -coefficients H ( X ; Z × B U ) ch Z × B U non-abelian character mapwith Z × B U -coefficients (cid:47) (cid:47) non-abelian de Rham homologywith l ( Z × B U ) -coefficients H dR (cid:0) X ; l ( Z × B U ) (cid:1) KU ( X ) ch traditional Chern character (cid:47) (cid:47) (cid:76) k ∈ Z H k dR ( X ) ∇ (cid:2) tr exp ( F ∇ ) (cid:3) Example 4.11 (Pontrjagin character on real K-theory) . The
Pontrjagin character ph on real topological K-theory(see [GHV73, §9.4][IK99][Ig08][GS18b, §2.1]) is defined to be the compositeKSpin • ( − ) (cid:47) (cid:47) KSO • ( − ) (cid:47) (cid:47) (cid:102) ph • (cid:52) (cid:52) KO • ( − ) cplx (cid:47) (cid:47) ph • (cid:50) (cid:50) KU • ( − ) ch • (cid:47) (cid:47) (cid:76) k H • ( − ; R ) of the complexification map (on representing virtual vector bundles) with the Chern character on complex K-theory(Example 4.10). (i) By naturality of the complexification map and since the complex Chern character is a Chern-Dold character (by[Hil71, Thm. 5.8]), so is the Pontrjagin character, as well as its restriction (cid:101) ph to oriented real K-theory KSO andfurther to ph on KO-theory and to Spin K-theory, etc. (ii)
The connected components B O of the classifying space KO for real topological K-theory are of finite R -type(since the real cohomology is the ring of universal Pontrjagin classes). Therefore, Theorem 4.8 applies and saysthat the non-abelian Chern character (Def. 4.2) for coefficients in Z × B SO coincides with the Pontrjagin character (cid:101) ph in KSO-theory:
Pontrjagin characteron oriented real K-theory (cid:101) ph (cid:39) ch Z × B SO . (iii) By Remark 3.53, the construction extends to the Pontrjagin character ph on KO-theory. (iv)
The same applies to the further restriction of the Pontrjagin character to KSpin; see [LD91][Th62] for somesubtleties involved and [Sa08, §7] for interpretation and applications.
Example 4.12 (Chern-Dold character on Topological Modular Forms) . The connective ring spectrum tmf of topo-logical modular forms [Ho94, §9][Ho02, §4] (see [DFHH14]) is, essentially by design, such that under rational-ization it yields the graded ring of rational modular forms (e.g [DH11, p. 2]): topologicalmodular forms π • ( tmf ) ( − ) ⊗ Z R (cid:47) (cid:47) rationalmodular forms mf R • (cid:39) R (cid:2) deg = (cid:122)(cid:125)(cid:124)(cid:123) c , deg = (cid:122)(cid:125)(cid:124)(cid:123) c (cid:3) . (207)56t follows that the Chern-Dold character (Def. 4.6) on tmf takes values in real cohomology with coefficients inmodular forms tmf • ( − ) ch • tmf Chern-Dold characteron topological modular forms (cid:47) (cid:47) H • (cid:0) − ; mf R • (cid:1) . (208)(This is often considered over the rational numbers, sometimes over the complex numbers [BE13, Fig. 1]; we mayjust as well stay over the real numbers, by Remark 3.51, to retain contact to the de Rham theorem.)By Theorem 4.8 this is an instance of the non-abelian character map: Chern-Dold character ontopological nodular forms ch • tmf (cid:39) ch tmf • . Example 4.13 (The Hurewicz/Boardman homomorphism on topological modular forms) . The spectrum tmf (Ex-ample 4.12) carries the structure of a suitable ( E ∞ ) ring spectrum and hence receives an essentially unique homo-morphism of ring spectra from the sphere spectrum: Σ ∞ S = S e tmf (cid:47) (cid:47) tmf . This is also known as the
Hurewicz homomorphism or rather the
Boardman homomorphism (e.g. [Ad75, §II.7][Ko96,§4.3]) for tmf. The Boardman homomorphism on tmf happens to be a stable weak equivalence in degrees ≤
6, inthat it is an isomorphism on stable homotopy groups in these degrees [Ho02, Prop. 4.6][DFHH14, §13]: π s •≤ = π •≤ ( S ) π •≤ ( e tmf ) (cid:39) (cid:47) (cid:47) π •≤ ( tmf ) . Hence, in particular, when X is a manifold of dimension dim ( X ) ≤
9, the Boardman homomorphism identifiesthe stable Cohomotopy (Example 2.16) of X in degree 4 with tmf (cid:0) X (cid:1) (by Prop. A.37): stable4-Cohomotopy π s (cid:0) X (cid:1) = S ( X ) ch S (cid:41) (cid:41) Boardman homomorphism e (cid:39) (cid:47) (cid:47) tmf -cohomologyin degree 4 tmf ( X ) . ch tmf4 (cid:118) (cid:118) H ( X ) (209)In this situation, the character map from Example 4.12 extracts exactly the datum of a real 4-class. Remark 4.14 (Clarifying the role of tmf in string theory) . Since the famous computation of [Wi87] showed thatthe partition function of the heterotic string lands in modular forms, and since the theorem of [AHS01][AHR10]showed that, mathematically, this statement lifts through (what we call above) the tmf-Chern-Dold character (208),there have been proposals about a possible role of tmf-cohomology theory in controlling elusive aspects of stringtheory (see [KS05][Sa10][DH11][ST11][Sa14][GJF18][GPPV18][Sa19]). While good progress has been made, itmight be fair to say that the situation has remained inconclusive. But with the non-abelian generalization (Def. 4.2)of the Chern-Dold character in hand, we may ask for a non-abelian enhancement (Example 2.24) of tmf-theoryon string background spacetimes. By Example 4.13, this is, in degree 4, equivalent to asking for a non-abelianenhancement of stable Cohomotopy theory (Example 2.25). This exists canonically: given by actual Cohomotopytheory (Example 2.10). We consider the non-abelian character map on twisted 4-Cohomotopy in Example 5.23below. The concluding Prop. 5.24 shows that this does capture core aspects of non-perturbative string theory.
Example 4.15 (Chern-Dold character on integral Morava K-theory) . We highlight that a particularly interest-ing example of the Chern-Dold character, which is not widely known, is that on integral Morava K-theory,whose codomain in real cohomology has a rich coefficient system. Morava K-theories K ( n ) [JW75] (reviewedin [Wu89][Ru98, §IX.7]) form a sequence of spectra labeled by chromatic level n ∈ N and by a prime p (notation-ally left implicit). Their coefficient ring is pure torsion, and hence vanishes upon rationalization. However, thereis an integral version (cid:101) K ( n ) , highlighted in [KS03][Sa10][Buh11][SW15][GS17b], which has an integral p -adiccoefficient ring: (cid:101) K ( n ) ∗ = Z p [ v n , v − n ] , with deg ( v n ) = ( p n − ) . (210)57his theory more closely resembles complex K-theory than is the case for K ( n ) ; in fact, for n =
1, it coincides withthe p -completion of complex K-theory.Therefore, the Chern-Dold character (Def. 4.6) on integral Morava K-theory [GS17b, p. 53] is of the formch Mor : (cid:101) K ( n )( − ) (cid:47) (cid:47) H ∗ (cid:0) − ; Q p [ v n , v − n ] ⊗ Q R (cid:1) , (211)where we used (210) in (205) together with the fact that the rationalization of the p -adic integers is the rational(here: real, by Remark 3.51) p -adic numbers Z p ⊗ Z R (cid:39) Q p ⊗ Q R .Now Q p is not finite-dimensional over Q , whence Q p ⊗ R is not finite-dimensional over R , so that the classify-ing space for integral Morava K-theory is not of R -finite type (Def. 3.52). Therefore, our proof of the non-abeliande Rham theorem (Theorem 3.87), being based on the fundamental theorem of dgc-algebraic rational homotopytheory (Prop. 3.60), does not immediately apply to integral Morava K-theory coefficients; and hence the non-abelian character on integral Morava K-theory with de Rham codomain, in the form defined in Def. 4.2, is notestablished here. While this is a purely technical issue, as discussed in Remark 3.53, further discussion is beyondthe scope of the present article. We prove (Theorem 4.26) that the non-abelian character subsumes the Chern-Weil homomorphism (recalled asProp. 4.21, review in [Ch51, §III][KN63, §XII][CS74, §2][MS74, §C][FSSt10, §2.1]) in degree-1 non-abeliancohomology.
Chern-Weil theory.
For definiteness, we recall the statements of Chern-Weil theory that we need to prove Theorem4.26 below.
Remark 4.16 (Attributions in Chern-Weil theory) . (i)
What came to be known as the
Chern-Weil homomorphism (recalled as Def. 4.21 below) seems to be first publicly described by H. Cartan (in May 1950), in his prominent
S´eminaire [Ca50, §7], published as [Ca51]. Later that year at the ICM (in Aug.-Sep. 1950), Chern discussesthis construction in a talk [Ch50, (10)], including a brief reference to unpublished work by Weil (which remainedunpublished until appearance in Weil’s collected works [We49]) for the proof that the construction is independentof the choice of connection (which is stated with an announcement of a proof in [Ca50, §7]). (ii)
The new result of Chern’s talk was the observation [Ch50, (15)] – later called the fundamental theorem in[Ch51, §III.6], recalled as Prop. 4.23 below – that this differential-geometric construction coincides with thetopological construction of real characteristic classes (Example 2.21). This crucially uses the identification [Ch50,(11)] of the real cohomology of classifying space BG with invariant polynomials, later expanded on by Bott [Bo73,p. 239]. (Various subsequent authors, e.g. [Fr02, (1.14)], suggest to prove Chern’s equation (15) by making senseof a connection on the universal G -bundle (which is possible though notoriously subtle, e.g. [Mo79]); but theproof in [Ch50] simply observes that for any given domain manifold the classifying space for G -bundles may betruncated to a finite cell complex (Prop. A.37), thus carrying a finite dimensional smooth G -bundle with ordinaryconnection. This argument was later worked out in [NR61][NR63][Sc80]). (iii) It is this fundamental theorem [Ch50, (15)][Ch51, §III.6] which allows to identify the Chern-Weil homomor-phism as an instance of the non-abelian character, in Theorem 4.26 below.
Notation 4.17 (Principal bundles with connection) . For G ∈ LieGroups X ∈ SmoothManifolds, we write G Connections ( X ) / ∼ (cid:47) (cid:47) (cid:47) (cid:47) G Bundles ( X ) / ∼ (212)for the forgetful map from the set of isomorphism classes of G -bundles equipped with connections to those of G -bundles without connection, over X .The function (212) is surjective and admits sections, corresponding to a choice of the class of a principalconnection on any class of G -principal bundles. Note, parenthetically, that the classical Chern character ch itself can be extended to cohomology theories with values in graded Q -algebras; see, e.g., [Ma06]. efinition 4.18 (Invariant polynomials [We49][Ca50, §7]) . For g ∈ LieAlgebras R , fin , we writeinv • ( g ) : = Sym (cid:0) b g ∗ (cid:1) G ∈ GradedCommAlgebras ≥ R for the graded sub-algebra (71) on those elements in the symmetric algebra (74) of the linear dual of g shifted up(Def. 3.7) into degree 2, which are invariant under the adjoint action of G on g ∗ . Definition 4.19 (Characteristic forms [Ca50, §7][Ch50, (10)]) . Let G be a finite-dimensional Lie group with Liealgebra g , and let P p (cid:47) (cid:47) X be G -principal bundle with connection ∇ (Def. 4.17). Then for ω ∈ inv n ( g ) an invariantpolynomial (Def. 4.18), its evaluation on the curvature 2-form F ∇ ∈ Ω ( P ) ⊗ g of the connection yields a differentialform ω ( F ∇ ) ∈ Ω n dR ( X ) p ∗ (cid:47) (cid:47) Ω n dR ( P ) which, by the second condition on an Ehresmann connection, is basic , namely in the image of the pullback opera-tion along the bundle projection p , as shown. Regarded as a differential form on X , this is called the characteristicform corresponding to ω . Lemma 4.20 (Characteristic de Rham classes of characteristic forms [We49][Ch50, p. 401][Ch51, §III.4]) . Theclass in de Rham cohomology (cid:2) ω ( F ∇ ) (cid:3) ∈ H n dR ( X ) of a characteristic form in Def. 4.19 is independent of the choice of connection ∇ and depends only on theisomorphism class of the principal bundle P. Definition 4.21 (Chern-Weil homomorphism [Ca50, §7][Ch50, (10)]) . Let G be a finite-dimensional Lie group,with classifying space denoted BG . The Chern-Weil homomorphism is the composite map
Chern-Weilhomomorphism cw G : G Bundles ( X ) / ∼ (cid:47) (cid:47) G Connections ( X ) / ∼ (cid:47) (cid:47) Hom (cid:0) inv • ( g ) , H • dR ( X ) (cid:1) principal bundle [ P ] (cid:31) (cid:47) (cid:47) with connection [ P , ∇ ] (cid:31) (cid:47) (cid:47) (cid:0) invariantpolynomial ω de Rham class ofcharacteristic form (cid:2) ω ( F ∇ ) (cid:3) (cid:1) , (213)where the first map is any section of (212), given by choosing any connection on a given principal bundle; andthe second map is the construction of characteristic forms according to Def. 4.19. (The Hom on the right is thatin GradedCommAlgebras ≥ R .) By Lemma 4.20 the second map is well-defined (and its composition with the firstturns out to be independent of the choices made, by Prop. 4.23 below).That this construction is useful, in that it produces interesting real characteristic classes of G -principal bundles(Example 2.21), is the following statement: Proposition 4.22 (Abstract Chern-Weil homomorphism [Ch50, (11)][Ch51, §III.5][Bo73, p. 239]) . Let G bea finite-dimensional, compact Lie group, with Lie algebra denoted g . Then the real cohomology algebra of itsclassifying space BG is isomorphic to the algebra of invariant polynomials (Def. 4.18): inv • ( g ) (cid:39) H • ( BG ; R ) ∈ GradedCommAlgebras ≥ R . (214)We can also obtain the following: Proposition 4.23 (Fundamental theorem of Chern-Weil theory [Ch50, (15)][Ch51, §III.6] (Rem. 4.16)) . Let Gbe a finite-dimensional compact Lie group. Then the Chern-Weil homomorphism (Def. 4.21) coincides with theoperation of pullback of universal characteristic classes along the classifying maps of G-bundles (Example 2.21),in that the following diagram commutes:H ( X ; BG ) c c ∗ ( − ) pullback ofuniversal characteristic classesalong classifying map (26) (cid:47) (cid:47) (cid:79) (cid:79) (11) (cid:39) Hom (cid:0) H • ( BG ; R ) , H • ( X ; R ) (cid:1) (cid:39) (214) (cid:15) (cid:15) G Bundles ( X ) / ∼ cw G Chern-Weil homomorphism (213) (cid:47) (cid:47)
Hom (cid:0) inv • ( g ) , H • dR ( X ) (cid:1) (215) Here the isomorphism on the left is from Example 2.3, while that from the right is from Prop. 4.22 and using thede Rham theorem. hern-Weil homomorphism as a special case of the non-abelian character.Lemma 4.24 (Sullivan model of classifying space) . Let G be a finite-dimensional, compact and simply-connectedLie group, with Lie algebra denoted g . Then the minimal Suillvan model (Def. 3.47) of its classifying space BG isthe graded algebra of invariant polynomials (Def. 4.18), regarded as a dgc-algebra with vanishing differential: (cid:0) inv ( g ) , d = (cid:1) (cid:39) CE ( l BG ) ∈ DiffGradedCommAlgebras ≥ R . (216) Proof.
According to [FOT08, Example 2.42], we haveCE ( l BG ) (cid:39) (cid:0) H • ( BG ; R ) , d = (cid:1) . (217)The composition of (217) with the isomorphism (214) from Prop. 4.22 yields the desired (216). Lemma 4.25 (Non-abelian de Rham cohomology with coefficients in a classifying space) . Let G be a finite-dimensional, compact and simply-connected Lie group, with Lie algebra denoted g . Then the non-abelian deRham cohomology (Def. 3.84) with coefficients in the rational Whitehead L ∞ -algebra l BG (Prop. 3.63) of theclassifying space is canonically identified with the codomain of the classical Chern-Weil construction (213) : nonabeliande Rham cohomology H dR (cid:0) X ; l BG (cid:1) (cid:39) traditional codomain ofChern-Weil construction Hom (cid:0) inv • ( g ) , H • dR ( X ) (cid:1) . (218) Proof.
Consider the following sequence of natural bijections: H dR (cid:0) X ; l BG (cid:1) : = DiffGradedCommAlgebras ≥ R (cid:0) CE (cid:0) l BG (cid:1) , Ω • dR ( X ) (cid:1) / ∼ (cid:39) DiffGradedCommAlgebras ≥ R (cid:16)(cid:0) inv • ( g ) , d = (cid:1) , Ω • dR ( X ) (cid:17) / ∼ (cid:39) GradedCommAlgebras ≥ R (cid:16) inv • ( g ) , Ω • dR ( X ) closed (cid:17) / ∼ (cid:39) GradedCommAlgebras ≥ R (cid:16) inv • ( g ) , (cid:0) Ω • dR ( X ) closed (cid:1) / ∼ (cid:17) (cid:39) GradedCommAlgebras ≥ R (cid:0) inv • ( g ) , H • dR ( X ) (cid:1) = : Hom (cid:0) inv • ( g ) , H • dR ( X ) (cid:1) . Here the first line is the definition (Def. 3.84). After that, the first step is Lemma 4.25. The second step unwindswhat it means to hom out of a dgc-algebra with vanishing differential (which is generator-wise as in Example 3.79),while the third and fourth steps unwind what this means for the coboundary relations (which is generator-wise asin Prop. 3.86). The last line just matches the result to the abbreviated notation used in (213).
Theorem 4.26 (Non-abelian character map subsumes Chern-Weil homomorphism) . Let G be a finite-dimensionalcompact, connected and simply-connected Lie group, with Lie algebra g . Let X ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def. 3.52) be equipped with the structure of a smooth manifold. Then the non-abelian character ch BG (Def 4.2)on non-abelian cohomology (Def. 2.1) of X with coefficients in BG coincides with the Chern-Weil homomorphism cw G (Def. 4.21) with coefficients in G, in that the following diagram (of cohomology sets) commutes:H ( X ; BG ) ch BG non-abelian character (cid:47) (cid:47) (cid:79) (cid:79) (11) (cid:39) H dR ( X ; l BG ) (cid:39) (218) (cid:15) (cid:15) G Bundles ( X ) / ∼ cw G Chern-Weil homomorphism (cid:47) (cid:47)
Hom (cid:0) inv • ( g ) , H • dR ( X ) (cid:1) (219) Here the isomorphism on the left is from Example 2.3, while that on the right is from Lemma 4.25.Proof.
First, notice that BG is simply connected (hence nilpotent), by the assumption that G is connected, and thatit is of finite rational type by Prop. 4.22. Hence, with Def. 3.52, BG ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil . (220)60ow, by Definition 4.2, the non-abelian character map on the top of (219)ch BG : H ( X ; BG ) ( η R BG ) (cid:47) (cid:47) H (cid:0) X ; L R BG (cid:1) (cid:39) (cid:47) (cid:47) H dR (cid:0) X ; L R BG (cid:1) sends a classifying map X c (cid:47) (cid:47) BG ∈ H ( X ; BG ) = Ho (cid:0) TopologicalSpaces Qu (cid:1) ( X , BG ) first to its composite with the rationalization map (Def. 3.55). By the fundamental theorem (Theorem 3.60 (i),using (220)), this is given by the derived adjunction unit D η BG of R exp (cid:97) Ω • PLdR (121): X c (cid:47) (cid:47) BG L R BG (cid:39) D η BG (cid:47) (cid:47) R exp ◦ Ω • PLdR ( BG ) ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1)(cid:0) X , L R BG (cid:1) = H (cid:0) X ; , L R BG (cid:1) . Moreover, by part (ii) of the fundamental theorem, the adjunct of the morphism D η BG ◦ c under (121) is Ω • PLdR ( X ) (cid:111) (cid:111) c ∗ Ω • PLdR ( BG ) ∈ Ho (cid:0)(cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj (cid:1) (using that Ω • PLdR ( D η R ) is an equivalence, by reflectivity of rationalization (113)). Hence it is the pullback op-eration of rational cocyles on BG along the classifying map c . Sending this further along the isomorphism to thebottom right in (219) (via Theorem 3.87 and Lemma 4.25) gives, by (164):ch BG : c Ω • dR ( X ) (cid:111) (cid:111) c ∗ Ω • PLdR ( BG ) (cid:111) (cid:111) (cid:39) inv • ( g ) ∈ Ho (cid:0)(cid:0) DiffGradedCommAlgebras ≥ R (cid:1) proj (cid:1) . (221)In conclusion, we have found that the commutativity of (219) is equivalent to the statement that the characteristicforms obtained by the Chern-Weil construction (213) represent the pullback (221) of the universal real character-istic classes on BG along the classifying map c of the underlying principal bundle (Example 2.21). This is the caseby the fundamental theorem of Chern-Weil theory, Prop. 4.23. Example 4.27 (de Rham representative of tangential Sp ( ) -twist) . For X a smooth 8-dimensional spin manifoldequipped with tangential Sp ( ) -structure τ (59), Theorem 4.26 says that there exists a smooth Sp ( ) -principalbundle on X equipped with an Ehresmann connection ∇ such that the rationalization (Def. 3.55) of the twist τ corresponds, under the non-abelian de Rham theorem (Theorem 3.87) to a flat l B Sp ( ) -valued differential formwhose components are the characteristic forms of the Sp ( ) -principal connection ∇ : H (cid:0) X ; B Sp ( ) (cid:1) ( η R BG ) ∗ −! H (cid:0) X ; L R B Sp ( ) (cid:1) (cid:39) H dR (cid:0) X ; l B Sp ( ) (cid:1) τ L R τ ! Ω • dR ( X ) (cid:111) (cid:111) τ dR R (cid:20) χ , p (cid:21)(cid:14)(cid:32) d p = d χ = (cid:33) = CE (cid:0) l B Sp ( ) (cid:1) p ( ∇ ) (cid:111) (cid:111) (cid:31) p χ ( ∇ ) (cid:111) (cid:111) (cid:31) χ Here on the right we are using [CV98, Thm . 8.1], see [FSS20, Lemma 2.12] to identify generating universalcharacteristic classes on B Sp ( ) : p is the first Pontrjagin class (of degree 4) and χ = (cid:16) p − (cid:0) p (cid:1) (cid:17) is theEuler 8-class, which on B Sp ( ) happens to be proportional to the I -polynomial (see [FSS19b, Prop. 3.7]). We show (Theorem 4.46) that the non-abelian character map induces secondary non-abelian cohomology opera-tions (Def. 4.42) which subsume the Cheeger-Simons homomorphism, recalled around (258) below, with valuesin ordinary differential cohomology, recalled around (245) below. We follow [FSSt10] [SSS12][Sch13] where theCheeger-Simons homomorphism, generalized to higher principal bundles, is called the ∞ -Chern-Weil homomor-phism .Underlying this is a differential enhancement of the non-abelian character map (Def. 4.32), and an inducednotion of differential non-abelian cohomology (Def. 4.33) on smooth ∞ -stacks (recalled as Def. A.44).61 he differential non-abelian character map. We introduce (in Def. 4.32 below) the differential refinement ofthe non-abelian character map; given as before by rationalization, but now followed not by a map to non-abelian deRham cohomology, but to its refinement by the full cocycle space of flat non-abelian differential forms (Def. 4.28below). It is this refinement of the codomain of the character map that allows it to be fibered over the smooth spaceof actual flat non-abelian differential forms (instead of just their non-abelian de Rham classes), thus producingdifferential non-abelian cohomology (Def. 4.33 below).
Definition 4.28 (Moduli ∞ -stack of flat L ∞ -algebra valued forms [Sch13, 4.4.14.2]) . Let A ∈ SimplicialSets be ofconnected, nilpotent, R -finite homotopy type (Def. 3.52). In view of the system of sets (Def. 3.77) X Ω dR (cid:0) X ; l A (cid:1) ∈ Setsof flat non-abelian differential forms with coefficient in the Whitehead L ∞ -algebra l A of A (Prop. 3.63), which arecontravariantly assigned to smooth manifolds X , we consider in Ho ( SmoothStacks ∞ ) (Def. A.44): (i) the smooth space of flat l A-valued differential forms Ω dR (cid:0) − ; l A (cid:1) flat : = (cid:18) R n (cid:16) ∆ [ k ] Ω dR (cid:0) R n ; l A (cid:1) flat (cid:17)(cid:19) , (222)regarded as a simplicially constant simplicial presheaf (355); (ii) the smooth ∞ -stack of flat l A-valued differential forms (Example 3.82) (cid:91) exp ( l A ) : = (cid:18) R n (cid:16) ∆ [ k ] Ω dR (cid:0) R n × ∆ k ; l A (cid:1) flat (cid:17)(cid:19) (223)which to any Cartesian space assigns the simplicial set that in degree k is the set of flat l A -valued differential formson the product manifold of the Cartesian space with the standard smooth k -simplex ∆ k ⊂ R k ; (iii) the canonical inclusion smooth space offlat l A -valued forms Ω ( − ; l A ) flat atlas (cid:47) (cid:47) smooth ∞ -stack offlat l A -valued forms (cid:91) exp ( l A ) (cid:18) R n (cid:16) ∆ [ k ] Ω dR (cid:0) R n ; l A (cid:1) flat (cid:17)(cid:19) (cid:31) (cid:127) (cid:47) (cid:47) (cid:18) R n (cid:16) ∆ [ k ] Ω dR (cid:0) R n × ∆ k ; l A (cid:1) flat (cid:17)(cid:19) (224)exhibiting Ω ( − ; l A ) (222) as the presheaf of 0-simplices in the simplicial presheaf (cid:91) exp ( l A ) (223) (more abstractly:this is the canonical atlas of the smooth moduli ∞ -stack, see [SS20b, Prop. 2.70]). Lemma 4.29 (Moduli ∞ -stack of flat forms is equivalent to discrete rational ∞ -stack) . For A ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def. 3.52), the evident inclusion (by inclusion of polynomial forms intosmooth differential forms followed by pullback along pr ∆ k ) Disc (cid:0) L R A (cid:1) (cid:39) Disc ◦ R exp ◦ CE (cid:0) l A (cid:1) ∈ W (cid:47) (cid:47) (cid:91) exp (cid:0) l A (cid:1)(cid:18) R n (cid:16) ∆ [ k ] Ω PLdR (cid:0) ∆ k ; l A (cid:1) flat (cid:17)(cid:19) (cid:31) (cid:127) (cid:47) (cid:47) (cid:18) R n (cid:16) ∆ [ k ] Ω dR (cid:0) R n × ∆ k ; l A (cid:1) flat (cid:17)(cid:19) (225) of the image under Disc (359) of the dg-algebraic model (122) for the rationalization of A (Def. 3.55) given bythe fundamental theorem (Prop. 3.60), into the moduli ∞ -stack of flat l A-valued differential forms (Def. 4.28) is anequivalence in Ho ( SmoothStacks ∞ ) (Def. A.44).Proof. By Prop. 3.62, the inclusion is for each R n a weak equivalence (130) in SimplicialSets Qu (Example A.8),hence is a weak equivalence already in the global projective model structure on simplicial presheaves, and hencealso in the local projective model structure. (Example A.43).62 emma 4.30 (Moduli ∞ -stack of closed differential forms is shifted de Rham complex) . For n ∈ N , we have an equivalence in Ho ( SmoothStacks ∞ ) (Def. A.44) from themoduli ∞ -stack (cid:91) exp (cid:0) b n R (cid:1) of flat differen-tial forms (Def. 4.28) with values in the lineLie ( n + ) -algebra b n R (Example 3.27) tothe image under the Dold-Kan construction(Def. A.53) of the smooth de Rham complex Ω • dR ( − ) (Example 3.23), naturally regardedas a presheaf on CartesianSpaces (352) withvalues in connective chain complexes (Exam-ple A.48) (i.e., with de Rham differential low-ering the chain degree) shifted up in degree byn and then homologically truncated in degree0, as shown on the right. (cid:91) exp (cid:0) b n R (cid:1) (cid:39) (cid:47) (cid:47) DK ... Ω ( − ) d Ω ( − ) d ... d Ω n + ( − ) clsd ∈ Ho ( SmoothStacks ∞ ) Proof.
First observe, with Example 3.79, that the simplicial presheaf (cid:91) exp (cid:0) b n R (cid:1) ( − ) = (cid:16) ∆ [ k ] Ω n + (cid:0) ( − ) × ∆ k (cid:1) clsd (cid:17) (226)naturally carries the structure of a presheaf of simplicial abelian groups, given by addition of differtial forms.Therefore, by the Dold-Kan Quillen equivalence (Prop A.52), it is sufficient to prove that we have a quasi-isomorphism of presheaves of chain complexes from the corresponding normalized chain complex (360) of (226)to the shifted and truncated de Rham complex itself: N (cid:16) ∆ [ k ] Ω n + (cid:0) ( − ) × ∆ k (cid:1) clsd (cid:17) (cid:39) (cid:82) ∆ • (cid:47) (cid:47) (cid:16) · · · ! ! ! Ω ( − ) d ! Ω ( − ) d ! · · · d ! Ω n + ( − ) clsd (cid:17) . (227)We claim that such is given by fiber integration of differential forms over the simplices ∆ k :First, to see that fiber integration does constitute a chain map, we compute for ω ∈ Ω • dR (cid:0) ( − ) × ∆ k (cid:1) clsd on theleft of (227): (cid:90) ∆ k ∂ ω = ( − ) k (cid:90) ∂ ∆ k ω = d (cid:90) ∆ k ω , (228)where the first step is the definition of the differential in the normalized chain complex (360) and the second stepis the fiberwise Stokes formula (158).Finally, to see that (cid:82) ∆ • is a quasi-isomorphism, notice that the chain homology groups on both sides are H k ( − ) = (cid:26) R | k = n + | otherwiseover each Cartesian space: For the left hand side this follows via the weak equivalence (130) from the fundamentaltheorem (Prop. 3.60) via Example 3.67, while for the right hand side this follows from the Poincar´e lemma.Hence it is sufficient to see that fiber integration over ∆ n + is an isomorphism on the ( n + ) st chain homologygroups. But a generator of this group on the left is clearly given by the pullback pr ∗ ∆ n + ω of any ω ∈ Ω n + ( ∆ n + ) of unit weight and supported in the interior of the simplex. That this is sent under (cid:82) ∆ n + to a generator ± ∈ R (cid:39) Ω ( − ) clsd on the right follows by the projection formula (159). Remark 4.31 (Moduli of closed forms via stable Dold-Kan correspondence) . Expressed in terms of the stableDold-Kan construction DK st (Prop. A.55) via the derived stabilization adjunction (Example A.41), Lemma 4.30says, equivalently, that: (cid:91) exp (cid:0) b n R (cid:1) (cid:39) R Ω ∞ (cid:16) DK st (cid:0) Ω • dR ( − ) ⊗ Z b n + R (cid:1)(cid:17) ∈ Ho ( SmoothStacks ∞ ) , (229)where now Ω • dR ( − ) ∈ PSh (cid:0)
CartesianSpaces , ChainComplexes Z (cid:1) is in non-positive degrees, with Ω ( − ) in degree0, and where b n + R (Def. 3.7) is concentrated on R in degree n + efinition 4.32 (Differential non-abelian character map [FSS15b, §4]) . Given A ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def. 3.52), the differential non-abelian character map in A -cohomology theory, to be denoted ch A , is the mor-phism in Ho ( SmoothStacks ∞ ) (356) from Disc ( A ) (359) to the moduli ∞ -stack of flat l A -valued forms (cid:91) exp ( l A ) (223) given by the composite coefficient space asgeometrically discretemoduli ∞ -stack Disc ( A ) Disc ( η PLdR A ) (cid:47) (cid:47) Disc ( D η PLdR A ) rationalization (122) (cid:50) (cid:50) differential non-abelian character map ch A (cid:44) (cid:44) Disc ◦ exp ◦ Ω • PLdR ( A ) (133) Disc ◦ exp ( p min ) (cid:47) (cid:47) Disc ◦ R exp ◦ CE ( l A ) ∈ W (225) (cid:47) (cid:47) moduli ∞ -stack offlat l A -valueddifferential forms (cid:91) exp ( l A ) (230)of (a) the image under Disc (359) of the derived adjunction unit D η PLdR A (325) of the PS de Rham adjunction (128),specifically with (co-)fibrant replacement p min being the minimal Sullivan model replacement (111); (recalling thatexp is a contravariant functor),with (b) the weak equivalence from Lemma 4.29. Differential non-abelian cohomology.Definition 4.33 (Differential non-abelian cohomology [FSS15b, §4]) . For A ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def. 3.52) we say that: (i) the moduli ∞ -stack of Ω A-connections is the object A diff ∈ Ho ( SmoothStacks ∞ ) in the homotopy categoryof smooth ∞ -stacks (Def. A.44), which is given by the homotopy pullback (Def. A.23) of the smooth space offlat non-abelian differential forms Ω dR ( − ; l A ) flat (224) along the differential non-abelian character map ch A (Def.4.32): moduli ∞ -stackof Ω A -connections A diff c A universal characteristic classin non-abelian A -cohomology (cid:15) (cid:15) F A l A -valuedcurvature forms (cid:47) (cid:47) (hpb) smooth space offlat l A -valued forms Ω dR ( − ; l A ) flat atlas (cid:15) (cid:15) Disc ( A ) ch A differential non-abeliancharacter map (cid:47) (cid:47) (cid:91) exp ( l A ) moduli ∞ -stack offlat l A -valued forms ∈ Ho ( SmoothStacks ∞ ) ; (231) (ii) the differential non-abelian cohomology of a smooth ∞ -stack X ∈ Ho ( SmoothStacks ∞ ) (356) with coefficientsin A is the structured non-abelian cohomology (Remark 2.27) with coefficients in the moduli ∞ -stack A diff of Ω A -connections (231), hence the hom-set in the homotopy category of ∞ -stacks (Def. A.44) from X to A diff (cid:98) H (cid:0) X ; A (cid:1) : = H (cid:0) X ; A diff (cid:1) : = Ho ( SmoothStacks ∞ ) (cid:0) X , A diff (cid:1) . (232) (iii) We call the non-abelian cohomology operations induced from the maps in (231) as follows (see (4)): (a) characteristic class : (cid:98) H (cid:0) X ; A (cid:1) ( c A ) ∗ (cid:47) (cid:47) H (cid:0) Shp ( X ) ; A (cid:1) (Def. 2.1) (233) (b) curvature : (cid:98) H (cid:0) X ; A (cid:1) ( F A ) ∗ (cid:47) (cid:47) Ω dR (cid:0) X ; l A (cid:1) flat (Def. 3.77) (234) (c) differential character : (cid:98) H (cid:0) X ; A (cid:1) ( ch A ◦ c A ) ∗ (cid:47) (cid:47) H dR (cid:0) X ; l A (cid:1) (Def. 3.84) (235)In differential enhancement of Example 2.13, we have the following.64 ifferential generalized cohomology.Example 4.34 (Differential generalized cohomology) . Let E • be a generalized cohomology theory (Example 2.13)with representing spectrum E (17) which is connective and whose component spaces E n are of finite R -type, sothat their connected components are, by Example 3.54, in Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def. 3.52). (i) Then differential non-abelian cohomology, in the sense of Def. 4.33, with coefficients in the component spaces E • , coincides with canonical differential generalized E -cohomology in the traditional sense of [HS05, §4.1][Bun12,Def. 4.53][BG13, §2.2][BNV13, §4.4]: generalizeddifferential cohomology (cid:98) E n ( − ) (cid:39) (cid:98) H ( − ; E n ) . (236) (ii) Here “canonical”, in the sense of [Bun12, Def. 4.46], refers to choosing the curvature differential form coef-ficients to be π • ( E ) ⊗ R (instead of some chain complex quasi-isomorphic to this). By Example 3.69, this choicecorresponds in our Def. 4.33 to the minimality (Def. 3.47) of the minimal Sullivan model CE ( l E n ) for E n (Prop.3.63) that controls the flat L ∞ -algebra valued differential forms Ω dR ( − ; l E n ) flat (Def. 3.77) in the top right of (250). (iii) Hence for canonical/minimal curvature coefficients, we have from Example 3.69, Lemma 4.30 and Remark229 that (cid:91) exp (cid:0) l E n (cid:1) (cid:39) R Ω ∞ (cid:16) DK st (cid:0) Ω • dR ( − ) ⊗ Z π • ( E n ) (cid:1)(cid:17) ∈ Ho ( SmoothStacks ∞ ) (237)and Ω dR (cid:0) − ; l E n (cid:1) flat (cid:39) R Ω ∞ (cid:16) DK st (cid:0) Ω • dR ( − ) ⊗ Z π • ( E n ) (cid:1) ≤ (cid:17) ∈ Ho ( SmoothStacks ∞ ) . (238) (iv) With this, the equivalence 236 follows by observing that the defining homotopy pullback diagram (231) fordifferential non-abelian cohomology with coefficients in A : = E n (351) is the image under R Ω ∞ (350) of thedefining homotopy pullback diagram for canonical differential E -cohomology according to [HS05, (4.12)] [Bun12,Def. 4.51][BNV13, (24)], and using that right adjoints preserve homotopy pullbacks: ( E ) diff c E (cid:15) (cid:15) F E (cid:47) (cid:47) (hpb) Ω dR ( − ; l E ) flatatlas (cid:15) (cid:15) Disc ( E ) ch E (cid:47) (cid:47) (cid:91) exp ( l E ) moduli ∞ -stackof Ω E -connections (cid:39) R Ω ∞ Diff ( E , can ) (cid:15) (cid:15) (cid:47) (cid:47) (hpb) (cid:0) Ω • dR ( − ) ⊗ Z π • ( E ) (cid:1) ≤ (cid:15) (cid:15) Disc ( E ) H R ∧ ( − ) (cid:47) (cid:47) Ω • dR ( − ) ⊗ Z π • ( E ) “differential function spectrum”of differential generalized E -cohomology (239)The same applies to ( E n ) diff , by replacing E with L Σ n E (350) on the right of (239). Remark 4.35 (The canonical atlas for the moduli stack of connections) . The operation ( − ) ≤ in (238) is the naivetruncation functor on the category of chain complexesChainComplexes Z ( − ) ≤ (cid:47) (cid:47) ChainComplexes ≤ Z (cid:0) · · · ∂ −! V ∂ −! V ∂ − −! V − ∂ − −! V − ! · · · (cid:1) (cid:0) V ∂ − −! V − ∂ − −! V − ! · · · (cid:1) . In contrast to the homological truncation involved in Ω ∞ (367), this naive truncation is not homotopy-invariant anddoes not have a derived functor. Instead, as seen from (238) and (224), once regarded in differential non-abeliancohomology, this operation serves to construct the canonical atlas [SS20b, Prop. 2.70] of the moduli ∞ -stack offlat l E n -valued differential forms.Via the defining homotopy pullback (231), (239) this becomes hallmark of differential cohomology: Differ-ential cohomology is the universal solution to lifting the values of the character map from cohomology classes tocochain representatives, namely to curvature forms. 65n differential enhancement of Example 2.14 and Example 4.10 we have: Example 4.36 (Differential complex K-theory) . With the coefficient space A : = KU = Z × B U (19) for topo-logical complex K-theory (Example 2.14), the corresponding differential non-abelian cohomology theory (Def.4.33) is, by Example 4.34, differential K-theory, whose diagram (4) of cohomology operations is of this form (see[HS05][BS09][BS12][GS17b]) (cid:98) H (cid:0) X ; KU (cid:1) (cid:39) (cid:99) KU ( X ) F KU0 (cid:47) (cid:47) c KU0 (cid:15) (cid:15) (cid:110) (cid:8) F k ∈ Ω k dR ( X ) (cid:9) k ∈ N (cid:12)(cid:12) d F k = (cid:111) (cid:15) (cid:15) KU ( X ) ch (cid:47) (cid:47) (cid:76) k ∈ N H k dR (cid:0) X (cid:1) , (240)where the bottom map is the ordinary Chern character from Example 4.10, and the curvature differential forms areidentified as in Example 3.94. Examples of differential non-abelian cohomology.
In differential enhancement of Example 2.3, we have:
Proposition 4.37 (Differential non-abelian cohomology of principal connections) . Let G be a compact Lie groupwith classifying space BG (12) . Then there is a natural map over manifolds X , shown dashed in (241) , from equiv-alence classes of G-principal connections (Notation 4.17) to differential non-abelian cohomology with coefficientsin BG (Def. 4.33) which covers the classification of G-principal bundles by plain non-abelian cohomology withcoefficients in BG (Example 2.3), in that the following diagram commutes:G
Connections ( X ) / ∼ (cid:47) (cid:47) forgetconnection (cid:15) (cid:15) differentialnon-abelian cohomology (cid:98) H ( X ; BG ) c BG (cid:15) (cid:15) G Bundles ( X ) / ∼ (cid:39) (cid:47) (cid:47) H ( X ; BG ) non-abelian cohomology (241) Proof.
By Lemma 4.25, the differential form coefficient in the given case is Ω dR ( − ; l BG ) flat (cid:39) Hom R (cid:16) inv • ( g ) , Ω • dR ( − ) clsd (cid:17) . Therefore, with Example 3.67, we find that (cid:16) ∆ [ k ] Hom R (cid:0) inv • ( g ) , Ω • dR ( ∆ k ) clsd (cid:1)(cid:17) (cid:39) ∏ k K (cid:0) inv n ( g ) , n (cid:1) ∈ Ho (cid:0) SimplicialSets Qu (cid:1) is a product of Eilenberg-MacLane spaces (10) for real coefficient groups spanned by the invariant polynomials,and so the defining homotopy pullback (231) is here of this form: BG diff (cid:47) (cid:47) (cid:15) (cid:15) (hpb) Hom R (cid:0) inv • ( g ) , Ω • dR ( − ) clsd (cid:1) (cid:15) (cid:15) Disc ( BG ) ( c k ) k ∈ N (cid:47) (cid:47) Disc (cid:16) ∏ k ∈ N K (cid:0) inv n ( g ) , n (cid:1)(cid:17) , where the bottom map classifies the real characteristic classes of BG via Example 2.2. It follows (by ExampleA.26) that maps into BG diff are equivalence classes of triples (cid:98) H ( X ; BG ) (cid:39) (cid:0) f , φ , ( α k ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( α k ) (cid:47) (cid:47) f (cid:15) (cid:15) Hom R (cid:0) inv • ( g ) , Ω • dR ( − ) clsd (cid:1) (cid:15) (cid:15) BG (cid:47) (cid:47) Disc (cid:16) ∏ k ∈ N K (cid:0) inv n ( g ) , n (cid:1)(cid:17) φ (cid:114) (cid:122) (242)consisting of (a) a classifying map f for a G -principal bundle (Example 2.3), (b) a set of closed differential forms α labeled by the invariant polynomials, and (c) a set of coboundaries φ in real cohomology between these differentialforms and the pullbacks f ∗ c k . 66ow, given a G -connection ∇ on a G -principal bundle f ∗ EG over X , we obtain such a triple by (a) taking f tobe the classifying map of the underlying G -principal bundle, (b) taking α k : = ω k ( F ∇ ) to be the characteristic forms(Def. 4.19) of the connection, and (c) taking φ to be given by the relative Chern-Simons forms [CS74] between thegiven connection and the pullback along f of the universal connection (see Remark 4.16). This construction is aninvariant of the isomorphism class of the connection (see [HS05, p. 28]) and hence defines the desired map (241): G Connections ( X ) / ∼ (cid:47) (cid:47) (cid:98) H ( X ; BG ) (cid:2) f ∗ EG , ∇ (cid:3) (cid:2) f , (cid:0) cs k ( ∇ , f ∗ ∇ univ ) (cid:1) , (cid:0) ω k ( F ∇ ) (cid:1)(cid:3) (243)In differential enhancement of Example 2.10, we have: Example 4.38 (Differential Cohomotopy [FSS15b]) . The canonical differential enhancement of (unstable) Coho-motopy theory (Example 2.10) in degree n is differential non-abelian cohomology (Def. 4.33) with coefficients in S n : differentialCohomotopy (cid:98) π n ( − ) : = (cid:98) H (cid:0) − ; S n (cid:1) . (i) By Example 3.81, a cocycle (cid:98) C ∈ (cid:98) π ( X ) in differential 4-Cohomotopy has as curvature (231) a pair consistingof a differential 4-form G and a differential 7-form G , satisfying the Cohomotopical Bianchi identity shown here: differential4-Cohomotopy (cid:98) π ( X ) cohomotopical curvature F S (cid:47) (cid:47) Ω (cid:0) X ; l S (cid:1) flat (cid:98) C cohomotopicallycharge-quantized C -field (cid:40) G ( (cid:98) C ) , G ( (cid:98) C ) ∈ Ω • dR ( X ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d G ( (cid:98) C ) = − G ( (cid:98) C ) ∧ G ( (cid:98) C ) d G ( (cid:98) C ) = (cid:41) . (244)Such differential form data is exactly what characterizes the flux densities of the C -field in 11-dimensional su-pergravity (up to the self-duality constraint G = (cid:63) G ). By comparison with Dirac’s charge quantization (2), wethus see that a natural candidate for charge quantization of the supergravity C -field is (nonabelian/unstable) 4-Cohomotopy theory π [Sa13, §2.5][FSS16a, §2][BMSS19, §3] (review in [FSS19a, §7]) or rather: differential4-Cohomotopy theory (cid:98) π [FSS15b, p. 9][GS20]. (ii) The consequence of this Cohomotopical charge quantization is readily seen from the Hurewicz operation onCohomotopy theory (Example 2.26): The de Rham class of the 4-flux density is constrained to be integral, henceto be in the image of the de Rham homomorphism (Example 4.9) and its cup square is forced to vanish (cid:2) G ( (cid:98) C ) (cid:3) ∈ H (cid:0) X ; Z (cid:1) (cid:47) (cid:47) H (cid:0) X (cid:1) , (cid:2) G ( (cid:98) C ) (cid:3) ∪ (cid:2) G ( (cid:98) C ) (cid:3) = . This leads to interpretation via Massey products [KS05], with corresponding differential refinement in [GS17a]. (iii)
Passing from 11-dimensional supergravity to M-theory, the curvature data in (244) is expected to be sub-jected to more refined topological constraints, forcing the class of G to be integral up to a fractional shift bythe first Pontrjagin class of the tangent bundle, and deforming its cup square to a quadratic function with non-trivial “background charge” ([FSS19b, Table 1][GS20]). We see, in Prop. 5.24 below, that these more subtleM-theoretic constraints on the C -field flux densities are imposed by charge quantization in – hence lifting throughthe non-abelian character map of – the corresponding twisted non-abelian cohomology theory, namely: J-twisted
Differential ordinary cohomology.
The ordinary differential cohomology (cid:98) H • ( X ) [SiSu08] of a smooth manifold X combines ordinary integral cohomology classes (Example 2.2) with closed differential forms that represent thesame class in real cohomology, in that it makes a diagram of the following form commute: ordinarydifferential cohomology (cid:98) H • ( X ) underlyingintegral class (cid:15) (cid:15) curvature (cid:47) (cid:47) Ω • dR ( X ) clsd viade Rham theorem (cid:15) (cid:15) H • ( X ; Z ) rationalization (cid:47) (cid:47) H • ( X ; R ) (245)67n fact, differential cohomology is universal with this property, but not at the coarse level of cohomology sets shownabove (where the universal property is shallow) but at the fine level of of complexes of sheaves of coefficients (i.e.of moduli ∞ -stacks), as made precise in Prop. 4.40 below.In degree 2, ordinary differential cohomology classifies ordinary U ( ) -principal bundles (equivalently: com-plex line bundles) with connection [Bry93, §II], and the curvature map in (245) assigns their traditional curvature2-form. In degree 3 ordinary differential cohomology classifies bundle gerbes with connection [Mu96][SWa07]with their curvature 3-form. In general degree it classifies higher bundle gerbes with connection [Ga97], or equiv-alently higher U ( ) -principal bundles with connection [FSS12b, 2.6].One construction of ordinary differential cohomology over smooth manifolds is given in [CS85, §1], nowknown as Cheeger-Simons characters . An earlier construction over schemes, now known as
Deligne cohomology (Example 4.39), due independently to [De71, §2.2][MM74, §3.1.7][AM77, §III.1] and brought to seminal applica-tion in [Bei85] (review in [EV88]) is readily adapted to smooth manifolds [Bry93, §I.5][Ga97]. The advantage ofDeligne cohomology over Cheeger-Simons characters is that is immediately generalizes from smooth manifolds tosmooth ∞ -stacks, [FSSt10, §3.2.3][FSS12b, §2.5], such as to orbifolds [SS20a] and to moduli ∞ -stacks of higherprincipal connections where it yields higher Chern-Simons functionals [SSS12][FSS12a][FSS13a][FSS15a], aswell as allowing for twists in a systematic manner [GS18c][GS19b].In differential enhancement of Example 2.12, we have: Example 4.39 (Ordinary differential cohomology on smooth ∞ -stacks [FSSt10, §3.2.3][FSS12b, §2.5]) . Let n ∈ N . (i) The smooth
Deligne-Beilinson complex in degree n + n + n + • : = (cid:16) · · · (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) Z (cid:31) (cid:127) (cid:47) (cid:47) Ω ( − ) d (cid:47) (cid:47) Ω ( − ) d (cid:47) (cid:47) · · · d (cid:47) (cid:47) Ω n dR ( − ) (cid:1) . (246) (ii) The de Rham differential in degree 0 gives a morphism of presheaves of complexesDB n + • ( , , ··· , , d ) (cid:47) (cid:47) Ω n + ( − ) clsd (247)from the Deligne-Beilinson complex (246) to the presheaf of closed ( n + ) -forms, regarded as a presheaf of chaincomplexes regarded in degree 0. (iii) Ordinary differential cohomology is stacky non-abelian cohomology (Remark 2.27) with coefficients in theDeligne-Beilinson complex (246) regarded as a smooth ∞ -stack (Def. A.44) under the ∞ -stackified Dold-Kanconstruction from Example A.53 (hence sheaf hypercohomology with coefficients in the Deligne complex): ordinarydifferential cohomology (cid:98) H n + (cid:0) X (cid:1) : = Ho ( SmoothStacks ∞ ) (cid:16) X , Dold-Kancorrespondence L loc ◦ DK (cid:0) Deligne-Beilinsoncomplex DB n + • (cid:1)(cid:17) . (248) (iv) The curvature map on ordinary differential cohomology is the cohomology operation induced by (247): ordinarydifferential cohomology (cid:98) H n + (cid:0) X (cid:1) F curvature (cid:47) (cid:47) Ω n + (cid:0) X (cid:1) clsd Ho ( SmoothStacks ∞ ) (cid:16) X , L loc ◦ DK (cid:0) DB n + • (cid:1)(cid:17) Ho ( SmoothStacks ∞ )( X , L loc ◦ DK ( d )) (cid:47) (cid:47) Ho ( SmoothStacks ∞ ) (cid:16) X , L loc ◦ DK (cid:0) Ω n + ( − ) clsd (cid:1)(cid:17) (249) Proposition 4.40 (Differential non-abelian cohomology subsumes differential ordinary cohomology [FSSt10,Prop. 3.2.26]) . Let n ∈ N and consider A = B n U ( ) (cid:39) K ( Z , n + ) (Example 2.12). Then: (i) Differential non-abelian A-cohomology (Def. 4.33) coincides with ordinary differential cohomology (Def. 4.39): ordinarydifferential cohomology (cid:98) H n + (cid:0) X (cid:1) (cid:39) (cid:98) H (cid:0) X ; B n U ( ) (cid:1) . (250) (ii) The abstract curvature map in differential A-cohomology (231) reproduces the ordinary curvature map (249) . roof. First we use the Dold-Kan correspondence (Prop. A.50) to obtain a convenient presentation of the differ-ential character: (a)
Since the Dold-Kan construction DK (Def. A.53) realizes homotopy groups from homology groups (362),and since Eilenberg-MacLane spaces are characterized by their homotopy groups (10), we have the vertical iden-tifications on the left of the following diagram:
Disc (cid:0) B n + Z (cid:1) ch Bn U ( ) (cid:44) (cid:44) η R Bn + Z (cid:47) (cid:47) Disc (cid:0) B n + R (cid:1) (cid:39) (cid:47) (cid:47) (cid:91) exp ( b n R ) (cid:39) (cid:82) ∆ • (cid:15) (cid:15) DK Z ... (cid:31) (cid:127) (cid:47) (cid:47) DK R ... (cid:31) (cid:127) (cid:47) (cid:47) DK Ω ( − ) d Ω ( − ) d ... d Ω n + ( − ) clsd (251)Under this identification, it is clear that the rationalization map η R B n + Z (Def. 3.55) is presented by the canonicalinclusion of the integers into the real numbers, as on the bottom left of (251).Moreover, the right vertical equivalence in (251) is that from Lemma 4.30. (b) Since the differential character (230) in the present case evidently comes from a morphism of (presheavesof) simplicial abelian groups, with group structure given by addition of ordinary differential forms (Example 3.79),we may, using the Dold-Kan correspondence (Prop. A.50), analyze the remainder of the diagram on normalizedchain complexes N ( − ) (361).Using this, it follows by inspection of the bottom map in (230) that the bottom right square in (251) commutes,with the bottom morphism on the right being the canonical inclusion of (presheaves of) chain complexes.Now to use this presentation for identifying the resulting homotopy fiber product: (i) Since the DK-construction (Def. A.53), applied objectwise over CartesianSpaces, is a right Quillen functor intothe global model structure from Example A.43, and since ∞ -stackification preserves homotopy pullbacks (LemmaA.46), it is now sufficient to show, by definition (248), that the homotopy pullback (Def. A.23) along the bottommap in (251), formed in presheaves of chain complexes is the Deligne-Beilinson complex (246). For this, by(329) it is sufficient to find a fibration replacement of the bottom map in (251) whose ordinary fiber product with Ω n + ( − ) clsd is the Deligne-Beilinson complex. This is the case for the following factorization: Z i Ω ( − ) d Ω ( − ) d ... d Ω n − ( − ) d Ω n dR ( − ) i (cid:15) (cid:15) d (cid:47) (cid:47) (pb) ... Ω n + ( − ) clsd i (cid:15) (cid:15) Z ... n ( n , n ) ∈ W (cid:47) (cid:47) Z ⊕ Ω ( − ) i ↙ − id d Ω ( − ) ⊕ Ω ( − ) d ↙ + id d Ω ( − ) ⊕ Ω ( − ) d ↙ − id d ... ... ... d ↙ d Ω n − ( − ) ⊕ Ω n dR ( − ) d ↙ Ω n dR ( − ) pr pr ...pr d ∈ Fib (cid:47) (cid:47) Ω ( − ) d Ω ( − ) d Ω ( − ) d ... d Ω n dR ( − ) d Ω n + ( − ) clsd (252)69ere the total bottom morphims is the total bottom morphism from (251), factored as a weak equivalence (quasi-isomorhism) followed by a fibration (positve degreewise surjection). The ordinary pullback of the fibration isshown, and represents the homotopy pullback, since all chain complexes are projectively fibrant. (ii) Finally, the top morphism in (252), thus being the abstract curvature map (231) is seen to coincide with thecurvature map (247) on the Deligne complex.
Secondary non-abelian cohomology operations.
We define secondary non-abelian cohomology operations (Def.4.42 below) which generalize the classical notion of secondary characteristic classes (Theorem 4.46, see Remark4.47 for the terminology) to higher non-abelian cohomology. To formulate the concept in this generality, we needa technical condition (Def. 4.41) which happens to be trivially satisfied in the classical case (Lemma 4.44 below):
Definition 4.41 (Absolute minimal model) . For A , A ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def. 3.52) we say thatan absolute minimal model for a morphism A c (cid:47) (cid:47) A in SimplicialSets is a morphism l A c (cid:47) (cid:47) l A betweenthe respective Whitehead L ∞ -algebras (Prop. 3.63) which makes the square on the left and hence the square on thefar right of the following diagram commute: Ω • dRPL ( A ) (cid:111) (cid:111) p min A (cid:79) (cid:79) CE (cid:0) l A (cid:1) (cid:79) (cid:79) c Ω • dRPL ( A ) (cid:111) (cid:111) p min A CE (cid:0) l A (cid:1) , ∈ DiffGradedCommAlgebras ≥ R A c (cid:15) (cid:15) η PLdR A (cid:47) (cid:47) D η PLdR A (cid:44) (cid:44) exp ◦ Ω • PLdR ( A ) exp ◦ Ω • PLdR ( c ) (cid:15) (cid:15) exp ( p min A ) (cid:47) (cid:47) exp ◦ CE ( l A ) exp ◦ CE ( c ) (cid:15) (cid:15) A η PLdR A (cid:47) (cid:47) D η PLdR A (cid:50) (cid:50) exp ◦ Ω • PLdR ( A ) exp ( p min A ) (cid:47) (cid:47) exp ◦ CE ( l A ) , ∈ SimplicialSets (253)hence a morphism that yields a transformation between exactly those derived adjunction units D η PLdR (325) ofthe PL-de Rham adjunction (119) that are given by minimal fibrant replacement. In this case, the commutingdiagram (253) evidently extends to a strict transformation between the differential non-abelian characters (230) onthe A i (Def. 4.32), in that the following diagram of simplicial presheaves (Def. 353) commutes:Disc ( A ) Disc ( c ) (cid:15) (cid:15) ch A (cid:47) (cid:47) (cid:91) exp ( l A ) (cid:91) exp ( c ) (cid:15) (cid:15) Disc ( A ) ch A (cid:47) (cid:47) (cid:91) exp ( l A ) ∈ PSh (cid:0)
CartesianSpaces , SimplicialSets (cid:1) . (254)In differential enhacement of Def. 2.17 we have: Definition 4.42 (Secondary non-abelian cohomology operation) . Let A c (cid:47) (cid:47) A in SimplicialSets, with inducedcohomology operation (Def. 2.17) H ( − ; A ) c ∗ (cid:47) (cid:47) H ( − ; A ) , have an absolute minimal model c (Def 4.41). Then the corresponding secondary non-abelian cohomology opera-tion is the structured cohomology operation (Remark 2.27) (cid:98) H ( − ; A ) ( c diff ) ∗ secondarynon-abelian character (cid:47) (cid:47) (cid:98) H ( − ; A ) (255)on differential non-abelian cohomology (Def. 4.33) which is induced (29) by the dashed morphism c diff in thefollowing diagram, which in turn is induced from c and c (254) by the universal property of the defining homotopypullback operation (230): Notice that the existence of morphisms c making this diagram commute is not guaranteed; it is only the existence of the relative minimal morphism l A ( c ) from Prop. 3.70 which is guaranteed to make the square (142) commute. econdary/differentialcohomology operation ( A ) diff c diff (cid:47) (cid:47) c A (cid:15) (cid:15) F A (cid:40) (cid:40) ( A ) diff c A (cid:15) (cid:15) F A (cid:40) (cid:40) Ω dR (cid:0) − ; l A (cid:1) flat (cid:15) (cid:15) c ∗ (cid:47) (cid:47) Ω dR (cid:0) − ; l A (cid:1) flat (cid:15) (cid:15) plain/primarycohomology operation Disc ( A ) Disc ( c ) (cid:47) (cid:47) transformation ofdifferential characters ch A (cid:40) (cid:40) Disc ( A ) ch A (cid:40) (cid:40) (cid:91) exp ( l A ) c ∗ (cid:47) (cid:47) (cid:91) exp ( l A ) . (256)The left and right squares are the homotopy pullback squares defining differential non-abelian cohomology (Def.4.33) while the bottom square is the transformation of differential non-abelian characters (Def. 4.32) from (254).In differential enhancement of Examples 2.26, 4.13 we have: Example 4.43 (Secondary non-abelian Hurewicz/Boardman homomorphism to differential K-theory) . Considerthe map S β (cid:47) (cid:47) B U ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) from the 4-sphere to the classifying space of the infinite unitary group (20) which classifies a generator in π (cid:0) B U (cid:1) (cid:39) Z . By Example 3.68 and Examples 3.69, 3.94 the corresponding Whitehead L ∞ -algebras (Prop. 3.63) are as shownhere: CE (cid:0) l S (cid:1) (cid:111) (cid:111) CE (cid:0) l B U (cid:1) (cid:39) ⊗ k ∈ N CE (cid:0) l K ( Z , k ) (cid:1) R (cid:20) ω , ω (cid:21)(cid:14)(cid:32) d ω = − ω ∧ ω d ω = (cid:33) (cid:111) (cid:111) ω | k = | else (cid:111) (cid:91) f k (cid:63) (cid:95) R ... f , f , (cid:14) ... d f = d f = (257)The morphism shown in (257) evidently restricts to the relative rational Whitehead L ∞ -algebra inclusion (Prop.3.70) on the factor K ( R , ) ⊂ L R B U and is zero elsewhere, hence fits into the required diagram (253) exhibiting itas an absolute minimal model (Def. 4.41) for β (by the commuting diagram in Prop. 3.50). Cheeger-Simons homomorphism.
Where the construction of the Chern-Weil homomorphism (Def. 4.21) invokesconnections on principal bundles without actually being sensitive to this choice (by Prop. 4.23), the
Cheeger-Simons homomorphism [CS85, §2][HS05, §3.3] (based on [CS74]) is a refinement of the Chern-Weil homomor-phism, now taking values in differential ordinary cohomology (Example 4.39), that does detect connection data(hence “differential” data): G Connections ( X ) / ∼ forgetconnection (cid:15) (cid:15) cs G Cheeger-Simonshomomorphism (cid:47) (cid:47)
Hom Z (cid:16) H • ( BG ; Z ) , differentialcohomology (cid:98) H • ( X ) (cid:17) curvature map (cid:15) (cid:15) G Bundles ( X ) / ∼ cw G Chern-Weilhomomorphism (cid:47) (cid:47)
Hom R (cid:16) inv • ( g ) , H • dR ( X ) de Rhamcohomology (cid:17) (258)We discuss how the general notion of secondary non-abelian cohomology operations (Def. 4.42) specializes onordinary principal bundles to the Cheeger-Simons homomorphism, and hence generalizes it to higher non-abeliancohomology: 71 emma 4.44 (Characteristic classes of G -principal bundles have absolute minimal models) . Let G be a connectedcompact Lie group with classifying space BG (12) . For n ∈ N , let [ c ] ∈ H n + ( BG ; Z ) be a universal integralcharacteristic class (Example 2.4). Then every representative classifying map BG c (cid:47) (cid:47) B n + Z has an absoluteminimal model in the sense of Def. 4.41.Proof. By Lemma 4.24, the minimal Sullivan model for BG has vanishing differential, while the minimal Sullivanmodel of B n + Z is a tensor factor (by Example 3.67), whose inclusion is already the relative minimal Sullivanmodel l B n + Z ( c ) (Prop. 3.70) of c . Therefore, settingCE ( c ) : = CE (cid:0) l B n + Z ( c ) (cid:1) : R [ c ] (cid:14) ( d c = ) (cid:31) (cid:127) (cid:47) (cid:47) inv • ( g ) (259)gives the required morphism of minimal models that makes makes the square (253) commute, by (142).In differential enhancement of Example 2.18 we have: Definition 4.45 (Secondary characteristic classes of differential non-abelian G -cohomology) . Let G be a connectedcompact Lie group with classifying space BG (12). By Lemma 4.44), the construction of secondary characteristicclasses (Def. 4.42, on differential non-abelian G -cohomology (Example 4.37) exists generally, and yields a Z -linear map of the form H (cid:0) BG ; B • Z (cid:1) ( − ) diff (cid:47) (cid:47) (cid:98) H (cid:0) BG diff ; B • Z (cid:1) = H (cid:0) BG diff ; B • Z diff (cid:1) , where on the right we have the ordinary differential non-abelian cohomology (Prop. 4.40) of the moduli ∞ -stack BG diff (231). Combined with the composition operation in Ho ( SmoothStacks ∞ ) (A.44) this gives a map (cid:98) H (cid:0) X ; BG (cid:1) × H (cid:0) BG ; B • Z (cid:1) id × ( − ) diff (cid:47) (cid:47) H (cid:0) X ; BG diff (cid:1) × H (cid:0) BG diff ; B • Z diff (cid:1) ◦ (cid:47) (cid:47) H (cid:0) X ; B • Z diff (cid:1) = (cid:98) H (cid:0) X ; B • Z (cid:1) which is Z -linear in its second argument, and whose hom-adjunct is (cid:98) H ( X ; BG ) ∇ ( c c diff ( ∇ )) (cid:47) (cid:47) Hom Z (cid:0) H ( BG ; B • Z ) , (cid:98) H ( X ; B • Z ) (cid:1) . (260) Theorem 4.46 (Secondary non-abelian cohomology operations subsume Cheeger-Simons homomorphism) . LetG be a connected compact Lie group, with classifying space denoted BG (12) . Then the canonical construction (260) of secondary characteristic classes on differential non-abelian G-cohomology (Def. 4.45) coincides with theCheeger-Simons homomorphim (258) , in that the following diagram commutes:G
Connections ( X ) (241) (cid:15) (cid:15) cs G Cheeger-Simonshomomorphism (cid:47) (cid:47)
Hom Z (cid:16) H • ( BG ; Z ) , differentialordinarycohomology (cid:98) H • ( X ) (cid:17) (cid:79) (cid:79) (cid:39) (250) (cid:98) H ( X ; BG ) differential non-abeliancohomology ∇ ( c c diff ( ∇ )) secondarynon-abelian cohomology operations (cid:47) (cid:47) Hom Z (cid:16) H ( BG ; B • Z ) , (cid:98) H ( X ; B • Z ) (cid:17) , (261) where on the left we have the map from G-connections to differential non-abelian G-cohomology from Prop. 4.37,and on the right the identification of ordinary differential cohomology from Prop. 4.40.Proof. Let c ∈ H (cid:0) BG ; B • Z (cid:1) be a characteristic class, and let ( f ∗ EG , ∇ ) be a G -principal bundle equipped with a G -connection. By Prop. 4.37, its image in differential non-abelian cohomology is given by the first map in thefollowing diagram G Connections ( X ) / ∼ (cid:47) (cid:47) (cid:98) H ( X ; BG ) ( c diff ) ∗ (cid:47) (cid:47) (cid:98) H (cid:0) X ; B n + Z (cid:1) (cid:39) (cid:47) (cid:47) (cid:98) H n + ( X ) (cid:2) f ∗ EG , ∇ (cid:3) (cid:2) f , (cid:0) cs k ( ∇ , f ∗ ∇ univ ) (cid:1) (cid:0) ω k ( F ∇ ) (cid:1)(cid:3) (cid:2) f ∗ c , cs c ( ∇ , f ∗ ∇ univ ) , c ( F ∇ ) (cid:3) (262)Here the triple of data are the three components (Example A.26) of a map into the defining homotopy pullbackof differential non-abelian cohomology (242). Therefore, the secondary operation induced by the transformation(256) of these homotopy pullbacks, which in the present case is of this form:72 G diff c diff secondarycharacteristic class (cid:47) (cid:47) (cid:38) (cid:38) c BG (cid:15) (cid:15) B n + Z diff (cid:40) (cid:40) c Bn + Z (cid:15) (cid:15) Ω dR ( − ; l BG ) flat (cid:15) (cid:15) (cid:47) (cid:47) Ω dR ( − ; l B n + Z ) flat (cid:15) (cid:15) BG c characteristic class (cid:47) (cid:47) ch BG (cid:39) (cid:39) B n + Z ch Bn + Z (cid:40) (cid:40) (cid:91) exp ( l BG ) c ∗ (cid:47) (cid:47) (cid:91) exp ( l B n + Z ) , (263)acts (a) on the first component in the triple by postcomposition with c , hence as f f ∗ c : = c ◦ f and (b) on the other two components by composition with c , which by (259) corresponds to projecting out theChern-Simons form and characteristic form corresponding to c , respectively. This is shown as the second map in(262). Hence we are reduced to showing that the total map in (262) gives the Cheeger-Simons homomorphism.This statement is the content of [HS05, §3.3]. Remark 4.47 (Secondary characteristic classes of G -connections) . The traditional reason for referring to theCheeger-Simons homomorphism (261) as producing secondary invariants is that Cheeger-Simons classes cs G ( P , ∇ ) ∈ (cid:98) H ( X ) may be non-trivial even if the underlying characteristic class cw G ( P ) (the “primary” class) vanishes. In thiscase the cs G ( P , ∇ ) are also called Chern-Simons invariants . (i) This happens, in particular, when the G -connection ∇ is flat, F ( ∇ ) = (ii) In fact, the proof of Theorem 261, via the triples (242) of homotopy data, shows that, in this case, cs G ( P , ∇ ) measures how (or “why”) cw G ( P ) vanishes, namely by which class of homotopies. (iii) Here we may understand secondary classes more abstractly, and explicitly related to the non-abelian charactermap: Where a (primary) non-abelian cohomology operation, according to Def. 2.17, is induced by a morphism ofcoefficient spaces (24), a secondary non-abelian cohomology operation, according to Def. 4.42, is induced (255)by a morphism of non-abelian character maps (254) – hence by a morphism of morphisms – on these coefficientspaces. (iv)
Note that classical secondary cohomology operations themselves admit differential refinements. For instance,for the case of Massey products as secondary operations for the cup product [GS17a]. While these can also fit intoour context on general grounds, we will not demonstrate that explicitly here.73
The twisted (differential) non-abelian character map
We introduce the character map in twisted non-abelian cohomology (Def. 5.4) and then discuss how it specializesto: §5.1 – the twisted Chern character on (higher) K-theory;§5.3 – the twisted character on Cohomotopy theory.
Rationalization in twisted non-abelian cohomology.
In generalization of Def. 4.1 we now define rationalizationof local coefficient bundles (31). This operation is transparent in the language of ∞ -category theory, where it simplyamounts to forming the pasting composite with the homotopy-coherent naturality square of the rationalization unit η R : X τ (cid:32) (cid:32) τ -twisted cocycle withlocal coefficients ρ c (cid:47) (cid:47) A (cid:12) G ρ (cid:124) (cid:124) BG (cid:39) (cid:113) (cid:121) rationalization X τ (cid:31) (cid:31) τ -twisted cocycle with rationalized local coefficients L R ρ c (cid:47) (cid:47) A (cid:12) G ρ (cid:124) (cid:124) η R A (cid:12) G (cid:47) (cid:47) L R (cid:0) A (cid:12) G (cid:1) L R ρ (cid:2) (cid:2) BG η R BG (cid:35) (cid:35) L R BG (cid:39) (cid:112) (cid:120) (cid:39) (cid:116) (cid:124) (264)Slightly less directly but equivalently, this operation is the composite of (a) “base change” along η R BG from theslice over BG to the slice over L R BG , (b) followed by the composition with the naturality square, now regarded asa morphism in the slice over L R BG : X τ (cid:32) (cid:32) τ -twisted cocycle withlocal coefficients ρ c (cid:47) (cid:47) A (cid:12) G ρ (cid:124) (cid:124) BG (cid:39) (cid:113) (cid:121) base change X τ (cid:34) (cid:34) c (cid:47) (cid:47) A (cid:12) G ρ (cid:122) (cid:122) BG η R BG (cid:15) (cid:15) L R BG (cid:39) (cid:111) (cid:119) compositionin slice X τ (cid:34) (cid:34) τ -twisted cocycle withlocal coefficients ρ c (cid:47) (cid:47) A (cid:12) G ρ (cid:122) (cid:122) η R A (cid:12) G (cid:47) (cid:47) L R ( A (cid:12) G ) L R ρ (cid:122) (cid:122) BG η R BG (cid:15) (cid:15) L R BG (cid:39) (cid:111) (cid:119) (cid:39) (cid:115) (cid:123) It is in this second form that the operation lends itself to formulation in model category theory (Def. 5.2 below).For that we just need to produce a rectified (strictly commuting) incarnation of the η R -naturality square: Lemma 5.1 (Rectified rationalization unit on coefficient bundle) . LetA (cid:47) (cid:47) local coefficient bundle A (cid:12) G ρ (cid:15) (cid:15) BG (265) be a local coefficient bundle (31) in Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def. 3.52), and let Ω • PLdR (cid:0) A (cid:12) G (cid:1) (cid:79) (cid:79) Ω • PLdR ( ρ ) (cid:111) (cid:111) p min BGA (cid:12) G ∈ W CE (cid:0) l BG ( A (cid:12) G ) (cid:1) (cid:79) (cid:79) CE ( l p ) Ω • PLdR (cid:0) BG (cid:1) (cid:111) (cid:111) p min BG ∈ W CE (cid:0) l ( BG ) (cid:1) (266) be its minimal relative Sullivan model (142) , which exists by Prop. 3.70. Then the composite of the image of (266)74 nder exp with the Ω • PLdR (cid:97) exp -adjunction unit (from Prop. 3.59): D η PLdR ρ : = A (cid:12) G η PLdR A (cid:12) G (cid:47) (cid:47) ρ (cid:15) (cid:15) D η PLdR A (cid:12) G (cid:39) η R A (cid:12) G (cid:44) (cid:44) exp ◦ Ω • PLdR (cid:0) A (cid:12) G (cid:1) exp ◦ Ω • PLdR ( ρ ) (cid:15) (cid:15) exp (cid:0) p min BG A (cid:12) G (cid:1) (cid:47) (cid:47) exp ◦ CE (cid:0) l BG ( A (cid:12) G ) (cid:1) exp ◦ CE ( l p ) (cid:15) (cid:15) BG η PLdR BG (cid:47) (cid:47) D η PLdR BG (cid:39) η R BG (cid:50) (cid:50) exp ◦ Ω • PLdR (cid:0) BG (cid:1) exp (cid:0) p min BG (cid:1) (cid:47) (cid:47) exp ◦ CE (cid:0) l ( BG ) (cid:1) (267) is, after passage (317) to the classical homotopy category (Example A.33), equivalent to the naturality square ofthe rationalization unit on ρ (112) : D η PLdR ρ (cid:39) η R ρ . Proof.
By Prop. 3.43 the right part of (267) is the image under exp of a fibrant replacement morphism. By(325) this identifies the diagram as the naturality square of the derived PLdR adjunction unit, and by (122) in thefundamental theorem (Prop. 3.60) this implies the claim.
Definition 5.2 (Rationalization in twisted non-abelian cohomology) . Given a local coefficient bundle ρ and its rec-tified rationalization unit D η PLdR ρ as in Lemma 5.1 we say that rationalization in twisted non-abelian cohomologywith local coefficients ρ (Def. 2.29) is the twisted non-abelian cohomology operation (Def. 2.41) (cid:0) η R ρ (cid:1) ∗ : H τ ( X ; A ) (cid:0) D η PLdR ρ ◦ ( − ) (cid:1) ◦ L (cid:0) η R BG (cid:1) ! (cid:47) (cid:47) H L R τ (cid:0) X ; L R A (cid:1) (268)given by the composite of (a) derived left base change L ( η R BG ) ! (Example A.18) along the rationalization unit (112) on the classifying spaceof twists, (b) composition with the rectified rationalization unit (267) on the coefficient bundle, regarded as a morphism inthe homotopy category (317) of the slice model category (Example A.10) of SimplicialSets Qu (Example A.8) overexp ◦ CE ( l BG )) . Remark 5.3 (Commutativity of rationalization over twisting) . The existence of the transformation (267) from alocal coefficient bundle to its rationalization, inducing the cohomology operation (268) from any twisted cohomol-ogy theory to its twisted rational cohomology, may be thought of as exhibiting commutativity of rationalizationover twisting. For twisted KO-theory this is discussed in [GS19d, Prop. 4].
Twisted non-abelian character map.
In generalization of Def. 4.2 we set:
Definition 5.4 (Twisted non-abelian character map) . Let X ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def. 3.52) equippedwith the structure of a smooth manifold, and A (cid:47) (cid:47) local coefficient bundle A (cid:12) G ρ (cid:15) (cid:15) BG (269)be a local coefficient bundle (31) in Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def. 3.52). Then the twisted non-abeliancharacter map in twisted non-abelian cohomology is the twisted cohomology operation twistednon-abeliancharacter map ch ρ : twistednon-abeliancohomology H τ ( X ; A ) ( η R ρ ) ∗ rationalization (cid:47) (cid:47) twistednon-abelianreal cohomology H L R τ (cid:0) X ; L R A (cid:1) (cid:39) twistednon-abeliande Rham theorem (cid:47) (cid:47) twistednon-abeliande Rham cohomology H τ dR dR ( X ; l A ) (270)from twisted non-abelian A -cohomology (Def. 2.29) to twisted non-abelian de Rham cohomology (Def. 3.98) withlocal coefficients in the rational relative Whitehead L ∞ -algebra l ρ of ρ (Prop. 3.75) which is the composite of (i) the operation (268) of rationalization of local coefficients (Def. 5.2), (ii) the equivalence (191) of the twisted non-abelian de Rham theorem (Theorem 3.104).75 .1 Twisted Chern character on higher K-theory We discuss how the twisted non-abelian character map reproduces the the twisted Chern character in twisted topo-logical K-theory [BCMMS02, §6.3][MaSt03][AS06, §7] – see also [TX06][MaSt06, §6][FrHT08, §2][BGNT08][GT10, §4][Ka12, §8.3][GS19a, §3.2][GS19c] – (Prop. 5.5) and in twisted iterated K-theory [LSW16, §2.2] (Prop.5.10).
Character maps on higher-twisted ordinary K-theories.
In twisted enhancement of Example 4.10, we have:
Proposition 5.5 (Twisted Chern character in twisted topological K-theory) . Consider twisted complex topologicalK-theory KU τ ( − ) (Example 2.36), for degree-3 twists given (via Example 2.11) by τ ∈ H (cid:0) − ; B U ( ) (cid:1) (cid:39) H ( − ; , Z ) , and regarded, via (48) , as twisted non-abelian cohomology with local coefficients in Z × B U (cid:12) B U ( ) (47) . Thenthe twisted non-abelian character map (Def. 5.4) ch τ Z × B U is equivalent to the traditional twisted Chern character ch τ on twisted K-theory with values in H -twisted de Rham cohomology (Def. 3.99): twisted non-abeliancharacter map ch τ Z × B U (cid:39) twistedChern character ch τ . Proof.
That the codomain of the twisted non-abelian character map, in this case, is indeed H -twisted de Rham co-homology is the content of Prop. 3.100. With this, and due to the twisted non-abelian de Rham theorem (Theorem3.104), it is sufficient to see that the general rationalization map of local non-abelian coefficients from Def. 5.2reproduces the rationalization map underlying the twisted Chern character. This is manifest from comparing therationalization operation (264), that is made formally precise by Def. 5.2, to the description of the twisted Cherncharacter as given in [FrHT08, (2.8)-(2.9)]. Remark 5.6 (Twisted Pontrjagin character in twisted KO-theory) . Similarly, an analogous statement holds for thetwisted Pontrjagin character (as in Example 4.11) on twisted real K-theory [GS19d, Prop. 2].There are also higher twists of ordinary K-theory, specifically twists of K-theory by Cohomotopy (Example2.10):
Example 5.7 (Chern character on higher cohomotopy-twisted K-theory) . For any k ∈ N , there is a non-trivial localcoefficient bundle (31) for twists of topological K-theory (Example 2.14) over the ( k + ) -sphere [MMS20, §2.1]:KU (cid:47) (cid:47) Y k + ρ Y k + (cid:15) (cid:15) S k + . (271) (i) By Def. 2.29, this defines a twisted version of topological K-theory (Example 2.14) with the twist τ = λ inCohomotopy theory (Example 2.10): cohomotopically-twisted topological K-theory K λ ( X ) : = H λ ( X ; KU ) , twist in Cohomotopy [ λ ] ∈ π k + ( X ) . (272)Direct comparison shows that this recovers the higher twisted K-theory of [MMS20, Def. 2.5]. (ii) Moreover, by Def. 5.4, we obtain the character map on cohomotopically twisted topological K-theory (272),and by Theorems 3.87, 3.98 and Examples 3.67, 3.68 we see that it lands in λ dR = H k + -twisted de Rham coho-mology (Example 3.95, Prop. 3.102): K λ ( X ) : = H λ (cid:0) X ; KU (cid:1) ch λ KU0 (cid:47) (cid:47) H λ dR dR (cid:0) X ; l KU (cid:1) (cid:39) H • + H k + ( X ) . (273) (iii) A map of this form has been defined in [MMS20, §2.2] by direct construction on form representatives. How-ever, [MMS20, Thm. 4.19] implies that this component construction coincides with the rationalization map (Def.5.2) on the local coefficient bundle (271), up to application of the de Rham theorem. Therefore, the twisted charac-ter map (273) obtained as a special case of Def. 5.4, reproduces the MMS-Character on higher (cohomotopically)twisted K-theory, from [MMS20, §2.2]. 76 emark 5.8 (Charge quantization of spherical T-duality in M-theory) . For k =
3, the character map (273) on7-Cohomotopy-twisted K-theory is a candidate for charge quantization (2) of the super-rational M-theory fieldsparticipating in 3-spherical T-duality over 11-dimensional super-spacetime, as derived in [FSS18, Prop. 4.17, Rem.4.18] (review in [SS18, (8), (19)]). However, the 7-Cohomotopy-twisted K-theory character has some spuriousfields of 2-periodic degree in its image, which are not seen in the physics application, where the field degreesare 6-periodic [Sa09, §3][FSS18, (65)]. Another candidate for charge-quantization of the super-rational M-theoryfields participating in 3-spherical T-duality, possibly more accurately reflecting the physics, is the character mapon twisted higher K-theory [LSW16], which we turn to next (Lemma 5.10).
Character map on twisted higher K-theory.Lemma 5.9 (Higher twisted de Rham coefficients inside rational twisted iterated K-theory) . There is a non-trivialtwisted cohomology operation (Def. 2.41) from (a) twisted non-abelian de Rham cohomology (Def. 3.98) withcoefficients in the relative rational Whitehead L ∞ -algebra (Prop. 3.70) of the coefficient bundle (53) of twistediterated K-theory (Example 2.39) to (b) higher twisted de Rham cohomology (Def. 3.101) regarded as twistednon-abelian de Rham cohomology via Prop. 3.102):H τ dR dR (cid:16) − ; l K ◦ r − ( ku ) (cid:17) φ ∗ (cid:47) (cid:47) H τ dR dR (cid:16) − ; (cid:76) k ∈ N b rk R (cid:17) , (274) given, under the twisted non-abelian de Rham theorem (Theorem 3.104) by the LSW-character from [LSW16, §2.2]applied to rational coefficients. Proposition 5.10 (Twisted Chern character in twisted iterated K-theory) . For r ∈ N , r ≥ , consider twisted iteratedK-theory (cid:0) K ◦ r − ( ku ) (cid:1) τ (Example 2.39), for degree- ( r + ) twists given (via Example 2.12) by τ ∈ H (cid:0) − ; B r U ( ) (cid:1) (cid:39) H r + ( − ; , Z ) , and regarded, via Example 2.39, as twisted non-abelian cohomology with local coefficients in (cid:0) K ◦ r − ( ku ) (cid:1) . Thenthe twisted non-abelian character map (Def. 5.4) ch τ K ◦ r − ( ku ) composed with the projection operation (274) ontohigher twisted de Rham cohomology, (Def. 3.101) from Lemma 5.9, is equivalent to the LSW character map ch r − [LSW16, Def. 2.20] restricted along the connective inclusion twistedLSW character ch τ r − (cid:39) φ ∗ projection ontohigher twistedde Rham cohomology ◦ twisted non-abeliancharacter map ch τ K ◦ r − ( ku ) . Proof.
After unwinding the definitions, the statement reduces to the commutativity of the square diagram in[LSW16, p. 15]: The top morphism there is the plain rationalization map (Def. 5.2), the right vertical mor-phism is φ ∗ from Lemma 5.9 before passing from real to de Rham cohomology, the left morphism is restriction tothe connective part and the bottom morphism is the LSW character. We introduce twisted differential non-abelian cohomology (Def. 5.13 below) and discuss how the correspondingtwisted differential non-abelian character subsumes existing constructions on twisted differential K-theory (Exam-ples 5.19 Example 5.22 below).
Twisted differential non-abelian cohomology.
From the perspective of structured non-abelian cohomology (Re-mark 2.27) that we have developed, it is now evident how to canonically combine (a) twisted non-abelian cohomology (Def. 2.29) with (b) differential non-abelian cohomology (Def. 4.33) to get twisted differential non-abelian cohomology: 77 efinition 5.11 (Differential non-abelian local coefficient bundles) . Let A (cid:47) (cid:47) local coefficient bundle A (cid:12) G ρ (cid:15) (cid:15) BG be a local coefficient bundle (31) in Ho (cid:0) TopologicalSpaces Qu (cid:1) fin R ≥ , nil (Def. 3.52). (i) By Lemma 3.71, Lemma 5.1, and using that exp preserves fibrations (Prop. 3.62), this induces a homotopyfibration (Def. A.22) in Ho ( SmoothStacks ∞ ) (Def. A.44) of differential non-abelian character maps (Def. 4.32) ofthis form:Disc ( A ) ch A differential non-abelian character mapwith coefficients in fiber space (cid:47) (cid:47) hofib ( Disc ( ρ )) (cid:39) (cid:39) (cid:91) exp ( l A ) (cid:111) (cid:111) atlashofib (( l ρ ) ∗ ) (cid:40) (cid:40) Ω dR ( − ; l A ) flat hofib (( l ρ ) ∗ ) (cid:42) (cid:42) Disc (cid:0) A (cid:12) G (cid:1) ch BGA (cid:12) G twisted differential non-abelian character map (cid:47) (cid:47) Disc ( ρ ) (cid:15) (cid:15) (cid:91) exp (cid:0) l ( A (cid:12) G ) (cid:1) (cid:111) (cid:111) atlas ( l ρ ) ∗ (cid:15) (cid:15) Ω dR (cid:0) − ; l BG ( A (cid:12) G ) (cid:1) flat ( l ρ ) ∗ (cid:15) (cid:15) Disc ( BG ) ch BG differential non-abelian character mapwith coefficients in space of twists (cid:47) (cid:47) (cid:91) exp ( l BG ) (cid:111) (cid:111) atlas Ω dR ( − ; l BG ) flat (275) (ii) Here the twisted differential non-abelian character map ch BGA (cid:12) G is defined just as in Def. 4.32, but with coeffi-cients the relative Whitehead L ∞ -algebra l BG ( A (cid:12) G ) (Prop. 3.70), as opposed to the absolute Whitehead L ∞ -algebra l ( A (cid:12) G ) (Prop. 3.63). Remark 5.12 (Differential local coefficient bundles) . Since homotopy limits commute over each other, passage tothe homotopy fiber products (Def. A.23) formed from the horizontal stages of (275) yields a homotopy fibrationof moduli ∞ -stacks of ∞ -connections (231) of this form: Ω dR (cid:0) − ; l BG ( A (cid:12) BG ) (cid:1) flat atlas (cid:42) (cid:42) ( l ρ ) ∗ (cid:15) (cid:15) A diff moduli ∞ -stack of Ω A -connections hofib ( ρ diff ) (cid:47) (cid:47) (cid:0) A (cid:12) G (cid:1) diff BG ρ diff differential non-abelianlocal coefficient bundle (cid:15) (cid:15) c BGA (cid:12) G (cid:41) (cid:41) F BGA (cid:12) G (cid:50) (cid:50) (cid:91) exp (cid:0) l BG ( A (cid:12) G ) (cid:1) ( l ρ ) ∗ (cid:15) (cid:15) Disc (cid:0) A (cid:12) G (cid:1) ch BGA (cid:12) G (cid:49) (cid:49) Disc ( ρ ) (cid:15) (cid:15) Ω dR (cid:0) − ; l BG (cid:1) flat atlas (cid:42) (cid:42) BG diff moduli ∞ -stack of G -connections F BG (cid:50) (cid:50) c BG (cid:41) (cid:41) (cid:91) exp (cid:0) l BG (cid:1) Disc (cid:0) BG (cid:1) ch BG (cid:49) (cid:49) (276) Definition 5.13 (Twisted differential non-abelian cohomology) . Given a differential non-abelian local coefficientbundle ρ diff (276) according to Def. 5.11, we say that: (i) A differential twist on a X ∈ Ho ( SmoothStacks ∞ ) (Def. A.44) is a cocycle τ diff in differential non-abeliancohomology with coefficients in BG (Def. 4.33) (cid:2) τ diff (cid:3) ∈ (cid:98) H (cid:0) X ; BG (cid:1) . (277) (ii) The τ diff -twisted differential non-abelian cohomology with local coefficients in ρ diff is the structured (Remark2.27) τ diff twisted non-abelian cohomology (Def. 2.29) with coefficients in ρ diff , hence the hom-set in the ho-motopy category (Def. A.14) of the slice model structure (Def. A.10) of the local projective model structureSmoothStacks ∞ on simplicial presheaves over CartesianSpaces (Example A.43) from τ diff (277) to ρ diff (276):78 wisted differentialnon-abelian cohomology (cid:98) H τ diff (cid:0) X ; A (cid:1) : = Ho (cid:0) SmoothStacks / BG diff ∞ (cid:1) ( τ diff , ρ diff ) = X differential cocycle c diff (cid:47) (cid:47) τ diff differentialtwist (cid:35) (cid:35) ( A (cid:12) G ) diff BG ρ diff differential localcoefficients (cid:121) (cid:121) BG diff (cid:39) (cid:111) (cid:119) (cid:14) homotopyrelative BG diff (278) (iii) The twisted non-abelian cohomology operations induced from the maps in (276) we call (see (4)): (a) characteristic class : (cid:98) H τ diff (cid:0) X ; A (cid:1) c τ A : = (cid:0) c BGA (cid:12) G (cid:1) ∗ (cid:47) (cid:47) H τ (cid:0) Shp ( X ) ; A (cid:1) (Def. 2.29) (279) (b) curvature: (cid:98) H τ diff (cid:0) X ; A (cid:1) F τ dR A : = (cid:0) F BGA (cid:12) G (cid:1) ∗ (cid:47) (cid:47) Ω τ dR dR (cid:0) X ; l A (cid:1) flat (Def. 3.92) (280) (c) differential character: (cid:98) H τ diff (cid:0) X ; A (cid:1) ch τ A : = (cid:0) ch BGA (cid:12) G ◦ c BGA (cid:12) G (cid:1) ∗ (cid:47) (cid:47) H τ dR dR (cid:0) X ; l A (cid:1) (Def. 3.98) (281) Twisted differential non-abelian cohomology as non-abelian ∞ -sheaf hypercohomology. While the formula-tion of twisted differential non-abelian cohomology as hom-sets in a slice of SmoothStacks ∞ (Def. 5.13) is naturaland useful, we indicate how this is equivalently incarnated as a non-abelian sheaf hypercohomology over X . Thisserves to make the connection to existing literature (in Example 5.18 below), but is not otherwise needed for thedevelopment here. We shall be brief, referring to [SS20b] for some technical background that is beyond the scopeof our presentation here. Proposition 5.14 ( ´Etale ∞ -topos over ∞ -stack [SS20b, Prop. 3.33, Rem. 3.34]) . For X ∈ Ho ( SmoothStacks ∞ ) (Def. A.44) let Ho (cid:0) ´Et X (cid:1) (cid:31) (cid:127) L i X (cid:47) (cid:47) Ho (cid:0) SmoothStacks / X ∞ (cid:1) be the full subcategory of the homotopy category (Def. A.14) of the slice model structure over X (Example A.10)of the local projective model structure on simplicial presheaves (Example A.43) on those E ! X which are localdiffeomorphisms ([SS20b, Def. 3.26]). (i) The inclusion L i X is a left-exact homotopy co-reflection, in that it preserves finite homotopy limits and has aderived right adjoint R LcclCnstnt (sending ∞ -bundles to their ∞ -sheaves of ∞ -sections). (ii) There is a global section functor R Γ X from Ho (cid:0) ´Et X (cid:1) to Ho (cid:0) TopologicalSpaces Qu (cid:1) (Example A.33) whichalso admits a left exact left adjoint: ∞ -bundles over X Ho (cid:0) SmoothStacks / X ∞ (cid:1) (cid:111) (cid:111) L i X (cid:63) (cid:95) R LcclConstnt ∞ -sheaf of local sections ⊥ (cid:47) (cid:47) ∞ -sheaves over X Ho (cid:0) ´Et X (cid:1) (cid:111) (cid:111) ∆ X R Γ X global sections ⊥ (cid:47) (cid:47) Ho (cid:0) TopologicalSpaces Qu (cid:1) . (282) Definition 5.15 (Non-abelian ∞ -sheaf hypercohomology over ∞ -stacks) . Given X ∈ Ho ( SmoothStacks ∞ ) (Def.A.44) and A ∈ Ho (cid:0) ´Et X (cid:1) (Prop. 5.14) we say that the set of connected components of the derived global sections (282) of A over X H (cid:0) X , A (cid:1) : = π (cid:0) R Γ X ( A ) (cid:1) is the non-abelian ∞ -sheaf hypercohomology of X with coefficients in A . Lemma 5.16 (Twisted differential non-abelian cohomology as non-abelian ∞ -sheaf hyper-cohomology) . Given adifferential twist τ diff (277) on some X ∈ Ho ( SmoothStacks ∞ ) (356) consider the objectA τ diff : = R LcllCnstnt X (cid:0) R τ ∗ diff ( A (cid:12) G ) diff (cid:1) ∈ Ho (cid:0) ´Et X (cid:1) (283) in the ´etale ∞ -topos over X Prop. 5.14. The non-abelian ∞ -sheaf hypercohomology (Def. 5.15) of A τ diff over X coincides with the τ diff -twisted differential non-abelian cohomology of X (Def. 5.13): non-abelian ∞ -sheaf hypercohomology π R Γ X (cid:0) A τ diff (cid:1) (cid:39) twisted differentialnon-abelian cohomology (cid:98) H τ diff (cid:0) X , A (cid:1) . (284)79 roof. As in [SS20b, Remark 3.34].It is useful to decompose this construction of twisted differential cohomology via ∞ -sheaf hypercohomologyagain as a homotopy pullback of corresponding ∞ -sheaves representing plain twisted cohomology and plain twisteddifferential forms: Remark 5.17 (Homotopy pullback of ∞ -sheaves representing twisted differential cohomology) . Given a differen-tial twist τ diff (277) on some X ∈ Ho ( SmoothStacks ∞ ) (356) with components ( τ , τ dR , L R τ ) (Example A.26), (i) Consider the pullback stacks over X in the following diagram R τ ∗ Disc (cid:0) A (cid:12) G (cid:1) (cid:15) (cid:15) (cid:47) (cid:47) (cid:32) (cid:32) Disc (cid:0) A (cid:12) G (cid:1) (cid:32) (cid:32) ρ (cid:15) (cid:15) R ( L R τ ) ∗ (cid:91) exp (cid:0) l BG ( A (cid:12) G ) (cid:1) (cid:30) (cid:30) (cid:15) (cid:15) (cid:47) (cid:47) (cid:91) exp (cid:0) l BG ( A (cid:12) G ) (cid:1) (cid:30) (cid:30) (cid:15) (cid:15) X τ (cid:47) (cid:47) Disc ( BG ) (cid:29) (cid:29) R τ ∗ dR Ω dR (cid:0) − ; l BG ( A (cid:12) G ) (cid:1) flat (cid:47) (cid:47) (cid:15) (cid:15) Ω dR (cid:0) − ; l BG ( A (cid:12) G ) (cid:1) flat (cid:15) (cid:15) X L R τ (cid:47) (cid:47) (cid:91) exp (cid:0) l BG (cid:1) (cid:33) (cid:33) X τ dR (cid:47) (cid:47) Ω dR ( − ; l BG ) Here the right hand side is (275) and all front-facing squares are homotopy pullbacks (Def. A.23). (ii)
By commutativity of homotopy limits over each other, these form a homotopy pullback square as on the rightof the following diagram, which gives, under the derived right adjoint R LcllCnstnt (282) a homotopy pullbackdiagram of ∞ -sheaves of sections as shown on the left: A τ diff (cid:15) (cid:15) (cid:47) (cid:47) (hpb) Ω (cid:0) − ; l A (cid:1) flat τ dR (cid:15) (cid:15) A τ (cid:47) (cid:47) (cid:91) exp (cid:0) l A (cid:1) L R τ : = R LcllCnstst R τ ∗ diff ( A (cid:12) G ) diff (cid:15) (cid:15) (cid:47) (cid:47) (hpb) R τ ∗ dR Ω dR (cid:0) − ; l BG ( A (cid:12) G ) (cid:1) (cid:15) (cid:15) R τ ∗ ( A (cid:12) G ) (cid:47) (cid:47) R ( L R τ ) ∗ (cid:91) exp (cid:0) l ( A (cid:12) G ) (cid:1) ∈ Ho (cid:0) ´Et X (cid:1) . (285)Here the top left item A τ diff from (283) is the ∞ -sheaf whose global sections give the τ diff -twisted differentialcohomology, by Lemma 5.16.In differential enhancement of Prop. 2.38 and in twisted enhancement of Example 4.34, we have: Example 5.18 (Twisted differential generalized cohomology) . Let X = X be a smooth manifold (Example A.45)and R be a suitable ring spectrum, and let E (hpb) (cid:47) (cid:47) R τ ∗ ρ R (cid:15) (cid:15) ( R ) (cid:12) GL R ( ) ρ R (cid:15) (cid:15) X τ (cid:47) (cid:47) B GL R ( ) be a twist for twisted generalized R -cohomology over X (51), as in Lemma 2.38. (i) Then the corresponding homotopy pullback diagram (285), which exhibits, by Lemma 5.16, twisted differentialnon-abelian cohomology (Def. 5.13) with coefficients in E as ∞ -sheaf hypercohomology (Def. 5.15), is the imageunder R Ω ∞ X of the homotopy pullback diagram of sheaves of spectra considered in [BN14, Def. 4.11], shown onthe right below, for canonical/minimal differential refinement as in Example 4.34:80 τ diff (cid:15) (cid:15) (cid:47) (cid:47) (hpb) Ω (cid:0) − ; l R (cid:1) flat τ dR (cid:15) (cid:15) R τ (cid:47) (cid:47) (cid:91) exp (cid:0) l R (cid:1) L R τ (cid:39) R Ω X ∞ Diff (cid:0) E (cid:1) (hpb) (cid:15) (cid:15) (cid:47) (cid:47) H M ≤ (cid:15) (cid:15) Disc ( E ) (cid:47) (cid:47) H M This is the twisted/parametrized analog of the relation (239). (ii)
Accordingly, the twisted differential generalized R -cohomology according to [BN14, Def. 4.13] is subsumedby twisted differential non-abelian cohomology, via Lemma 5.16.In differential enhancement of Prop. 5.5 and in twisted generalization of Example 4.36 we have: Example 5.19 (Twisted Chern character in twisted differential K-theory) . Consider again the local coefficientbundle KU (cid:47) (cid:47) KU (cid:12) B U ( ) ρ (cid:15) (cid:15) B U ( ) for complex topological K-theory (Example 2.36). By Example 5.18, the twisted differential non-abelian coho-mology theory (Def. 5.13) induced from these local coefficients is twisted differential K-theory, as discussed in[CMW09] for torsion twists (review in [BS12, §7]). By the diagram (4) of cohomology operations on twisteddifferential cohomology, one may regard the corresponding twisted curvature map (280) (cid:98) K τ diff (cid:0) X (cid:1) (cid:0) F τ dRKU0 (cid:1) ∗ (cid:47) (cid:47) Ω τ dR dR (cid:0) X ; l KU (cid:1) flat (with values in flat τ dR (cid:39) H -twisted differential forms, by Example 3.94) as an incarnation of the Chern charactermap on twisted differential K-theory. This is the perspective taken in [CMW09, p. 2][Pa18] for torsion twists, andin [BN14, p. 6] for general twists.However, in the spirit of the Cheeger-Simons homomorphism (4.3), any lift of a cohomology operation (here:rationalization) to differential cohomology should be enhanced all the way to a secondary cohomology operation(Def. 4.42, now to be generalized to a twisted secondary cohomology operation, Def. 5.21 below) whose codomainis itself a (twisted) differential cohomology theory. The twisted Chern character enhanced to a secondary coho-mology operation this way is Example 5.22 below, following the perspective taken in [GS19a, §3.2][GS19c, §2.3]. Secondary twisted non-abelian cohomology operations.
We introduce the twisted generalization of secondarynon-abelian cohomology operations (Def. 5.21 below). This requires the following twisted analog of the technicalcondition in Def. 4.41:
Definition 5.20 (Twisted absolute minimal model) . For A (cid:12) G ρ (cid:15) (cid:15) c t (cid:47) (cid:47) A (cid:12) G ρ (cid:15) (cid:15) BG c b (cid:47) (cid:47) BG ∈ SimplicialSetsa transformation (55) between local coefficient bundles (31), and for c b an absolute minimal model (Def. 4.41) ofthe map c b between spaces of twists, hence with induced transformation (254)Disc (cid:0) BG (cid:1) Disc ( c b ) (cid:47) (cid:47) ch BG (cid:35) (cid:35) Disc (cid:0) BG (cid:1) ch BG (cid:35) (cid:35) (cid:91) exp (cid:0) l BG (cid:1) ( c b ) ∗ (cid:47) (cid:47) (cid:91) exp (cid:0) l BG (cid:1) twistedabsolute minimal model is a lift of c b to a morphism l BG ( A (cid:12) G ) c t (cid:47) (cid:47) l BG ( A (cid:12) G ) (286)between the relative rational Whitehead L ∞ -algebras of the local coefficient bundles (Prop. 3.70) which (i) yields a transformationDisc (cid:0) A (cid:12) G (cid:1) Disc ( c t ) (cid:47) (cid:47) ch BG A (cid:12) G (cid:37) (cid:37) Disc (cid:0) A (cid:12) G (cid:1) ch BG A (cid:12) G (cid:37) (cid:37) (cid:91) exp (cid:0) l BG ( A (cid:12) G ) (cid:1) ( c t ) ∗ (cid:47) (cid:47) (cid:91) exp (cid:0) l BG ( A (cid:12) G ) (cid:1) of the twisted differential characters (275) (thus being an “absolute minimal model for c t relative to c b ”), (ii) compatible with the transformation of the differential characters on the twisting space, in that the followingcube commutes: Disc (cid:0) A (cid:12) G (cid:1) Disc ( c t ) (cid:47) (cid:47) ρ (cid:15) (cid:15) ch BG A (cid:12) G (cid:32) (cid:32) Disc (cid:0) A (cid:12) G (cid:1) ρ (cid:15) (cid:15) ch BG A (cid:12) G (cid:33) (cid:33) (cid:91) exp (cid:0) l BG ( A (cid:12) G ) (cid:1) l BG ( A (cid:12) G ) (cid:15) (cid:15) (cid:96) (cid:96) atlas ( c t ) ∗ (cid:47) (cid:47) (cid:91) exp (cid:0) l BG ( A (cid:12) G ) (cid:1) (cid:96) (cid:96) atlas ( l BG p ) ∗ (cid:15) (cid:15) Disc (cid:0) BG (cid:1) Disc ( c b ) (cid:47) (cid:47) ch BG (cid:30) (cid:30) Disc (cid:0) BG (cid:1) ch BG (cid:30) (cid:30) Ω (cid:0) − ; l BG ( A (cid:12) G ) (cid:1) flat ( c t ) ∗ (cid:47) (cid:47) ( l BG p ) ∗ (cid:15) (cid:15) Ω (cid:0) − ; l BG ( A (cid:12) G ) (cid:1) flat ( l BG p ) ∗ (cid:15) (cid:15) (cid:91) exp (cid:0) l BG (cid:1) (cid:97) (cid:97) atlas ( c b ) ∗ (cid:47) (cid:47) (cid:91) exp (cid:0) l BG (cid:1) (cid:97) (cid:97) atlas Ω (cid:0) − ; l BG (cid:1) flat ( c b ) ∗ (cid:47) (cid:47) Ω (cid:0) − ; l BG (cid:1) flat (287)At the level of dgc-algebras, the condition that c t (286) is a twisted absolute minimal model for the transfor-mation of local coefficient bundles means equivalently that it makes the following cube commute: Ω • PLdR (cid:0) A (cid:12) G (cid:1) (cid:111) (cid:111) Ω • PLdR ( c t ) (cid:102) (cid:102) p min BG A (cid:12) G (cid:79) (cid:79) Ω • PLdR ( ρ ) Ω • PLdR (cid:0) A (cid:12) G (cid:1) (cid:102) (cid:102) p min BG A (cid:12) G (cid:79) (cid:79) Ω • PLdR ( ρ ) CE (cid:0) l BG ( A (cid:12) G ) (cid:1) (cid:111) (cid:111) CE ( c t ) CE ( l ρ ) (cid:15) (cid:15) CE (cid:0) l BG ( A (cid:12) G ) (cid:1) CE ( l ρ ) (cid:15) (cid:15) Ω • PLdR ( BG ) (cid:103) (cid:103) p min BG (cid:111) (cid:111) Ω • PLdR ( c t ) Ω • PLdR ( BG ) (cid:103) (cid:103) p min BG CE ( l BG ) (cid:111) (cid:111) c b CE ( l BG ) (288)In differential enhancement of Def. 2.41 and in twisted generalization of Def. 4.42, we set:82 efinition 5.21 (Twisted secondary non-abelian cohomology operations) . Let A (cid:12) G ρ (cid:15) (cid:15) c t (cid:47) (cid:47) A (cid:12) G ρ (cid:15) (cid:15) BG c b (cid:47) (cid:47) BG ∈ SimplicialSetsbe a transformation (55) between local coefficient bundles (31), together with an absolute minimal model c b (Def.4.41) for the base map, and a compatible twisted absolute minimal model c t (Def. 5.20) for the total map. Thenforming stage-wise homotopy pullbacks (Def. A.23) in the required commuting cube (287) yields a transformationof corresponding differential coefficient bundles (276): ( A (cid:12) G ) diff ( c t ) diff (cid:47) (cid:47) ( ρ ) diff (cid:15) (cid:15) ( A (cid:12) G ) diff ( ρ ) diff (cid:15) (cid:15) ( BG ) diff ( c b ) diff (cid:47) (cid:47) ( BG ) diff ∈ PSh (cid:0)
CartesianSpaces , SimplicialSets (cid:1) . (289)This yields, in turn, a natural transformation of twisted differential non-abelian cohomology sets (Def. 5.13),hence a twisted secondary non-abelian cohomology operation , by pasting composition, hence by right derivedbase change (Example A.18) along ( ρ ) diff followed by composition with ( c t ) diff regarded as a morphism in theslice (Example A.10) over ( BG ) diff : (cid:98) H τ diff (cid:0) X ; A (cid:1) (( c t ) diff ◦ ( − )) ◦ (( ρ ) diff ) ∗ (cid:47) (cid:47) (cid:98) H ( c b ) diff ◦ τ diff (cid:0) X ; A (cid:1) . In differential enhancement of Prop. 5.5, we have:
Example 5.22 (Twisted differential character on twisted differential K-theory) . Consider the rationalization (Def.3.55) over the actual rational numbers (see Remark 3.51) of the local coefficient bundle (47) for degree-3 twistedcomplex topological K-theory (Example 2.36). (i)
This is captured by the diagramKU (cid:12) B U ( ) ρ (cid:15) (cid:15) η Q KU0 (cid:12) B U ( ) (cid:47) (cid:47) L Q (cid:0) KU (cid:12) B U ( ) (cid:1) L R ρ (cid:15) (cid:15) B U ( ) η Q B ( ) (cid:47) (cid:47) L Q (cid:0) B U ( ) (cid:1) (290)regarded as a transformation of local coefficient bundles from twisted K-theory to twisted even-periodic rationalcohomology: L Q KU (cid:39) Ω ∞ (cid:16) (cid:77) k Σ k H Q (cid:124) (cid:123)(cid:122) (cid:125) = : H per Q (cid:17) . (ii) Since rationalization is idempotent (113), which here means that L R ◦ L Q (cid:39) L R , in this situation an absoluteminimal model (Def. 4.41) of the base map c b = η R B U ( ) and a twisted absolute minimal model (Def. 5.20) of thetotal map c t = η R K (cid:12) B U ( ) exist and are given, respectively, simply by the identity morphisms c b : = id l B U ( ) and c t : = id l B ( ) ( K (cid:12) B U ( )) . (iii) Therefore, the induced twisted secondary cohomology operation Def. 5.21 exists, and is for each differentialtwist τ diff a transformation (cid:98) K τ diff (cid:0) X (cid:1) ch τ diffdiff : = (cid:16) η R K (cid:12) B U ( ) (cid:17) diff (cid:47) (cid:47) (cid:92) H per Q L Q τ diff (cid:0) X (cid:1) (291)from twisted differential K-theory to twisted differential periodic rational cohomology theory.83 iv) This is the twisted differential Chern character map on twisted differential complex K-theory as conceived in[GS19a, §3.2][GS19c, Prop. 4]. The analogous statement holds for the twisted differential Pontrjagin character (asin Example 4.11) on twisted differential real K-theory [GS19d, Thm. 12]. (v)
Notice that this construction is close to but more structured than the plain curvature map on twisted differentialK-theory (Example 5.19): If we considered the transformation of local coefficients as in (290) but for rationaliza-tion L R over the real numbers (Remark 3.51), then the induced twisted secondary cohomology operation wouldbe equivalent to the twisted curvature map. Instead, (291) refines the plain curvature map to a twisted secondaryoperation that retains information about rational periods.84 .3 Twisted character on twisted differential Cohomotopy We discuss here (Example 5.23 below) the twisted non-abelian character map on J-twisted Cohomotopy (Example2.40) in degree 4, and on Twistorial Cohomotopy (Example 2.44). We highlight the induced charge quantization(Prop. 5.24 below) and comment on the relevance to high energy physics (Remark 5.29).These twisted non-abelian cohomotopical character maps have been introduced and analyzed in [FSS19b] and[FSS20]. The general theory of non-abelian characters developed here shows how these cohomotopical charactersare cousins both of generalized abelian characters such as the Chern character on twisted higher K-theory (§5.1),notably of the character on topological modular forms (by Example 4.13, and Remark 4.14) as well as of non-abelian characters such as the Chern-Weil homomorphism (§4.2) and the Cheeger-Simons homomorphism (§4.3).
Cohomotopical character maps.Example 5.23 (Character map on J-twisted Cohomotopy and on Twistorial Cohomotopy) . Let X be an 8-dimensionalsmooth spin manifold equipped with tangential Sp ( ) -structure τ (59). We have the twisted non-abelian charactermaps (Def. 5.4) on J-twisted Cohomotopy (Example 2.40) in degree 4, and on Twistorial Cohomotopy (Example2.44): C P (cid:12) Sp ( ) Borel-equivariantizedtwistor fibration t H (cid:12) Sp ( ) (cid:15) (cid:15) TwistorialCohomotopy T τ ( X ) : = H τ (cid:0) X ; C P (cid:1) character map onTwistorial Cohomotopy ch τ C P (cid:47) (cid:47) cohomology operationalong twistor fibration ( t H ) ∗ (cid:15) (cid:15) H τ dR dR (cid:0) X ; l C P (cid:1) ( l t H ) ∗ (cid:15) (cid:15) = H , F , G , G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d H = G − p ( ∇ ) − F ∧ F , d F = , d G = − (cid:0) G − p ( ∇ ) (cid:1) ∧ (cid:0) G + p ( ∇ ) (cid:1) − χ ( ∇ ) , d G = (cid:14) ∼ H F G G ! ! ! ! G G (cid:15) (cid:15) S (cid:12) Sp ( ) π τ (cid:0) X (cid:1) J-twisted4-Cohomotopy : = H τ (cid:0) X ; S (cid:1) ch τ S character map inJ-twisted Cohomotopy (cid:47) (cid:47) H τ dR dR (cid:0) X ; l S (cid:1) = (cid:40) G , G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d G = − (cid:0) G − p ( ∇ ) (cid:1) ∧ (cid:0) G + p ( ∇ ) (cid:1) − χ ( ∇ ) , d G = , (cid:41)(cid:14) ∼ Here: (i)
The twisted non-abelian de Rham cohomology targets on the right are as shown, by Example 3.96. (Inparticular the twisted curvature forms in the first line are relative to l S .) (ii) The vertical twisted non-abelian cohomology operation (Def. 2.41) on the left is induced from the Borel-equivariantized twistor fibration (61), and that on the right from its associated morphism of rational White-head L ∞ -algebras (Prop. 3.70). Proposition 5.24 (Charge-quantization in J-twisted Cohomotopy [FSS19b, Prop. 3.13][FSS20, Cor. 3.11]) . Con-sider the twisted non-abelian character maps (Def. 5.4) in J-twisted Cohomotopy and in Twistorial Cohomotopyfrom Example 5.23. (i)
A necessary condition for a flat Sp ( ) -twisted l S -valued differential form datum ( G , G ) to lift through theJ-twisted cohomotopical character map (i.e. to be in its image) is that the de Rham class of G , when shifted bythe fourth fraction of the Pontrjagin form, is in the image, under the de Rham homomorphism (Example 4.9), of anintegral class: (cid:2) G − p ( ∇ ) (cid:3) ∈ H ( X ; Z ) (cid:47) (cid:47) H ( X ) . (292) (ii) A necessary condition for a flat Sp ( ) -twisted l C P -valued differential form datum ( G , G , F , H ) to liftthrough the character map in Twistorial Cohomotopy is that the de Rham class of G shifted by the fourth fractionof the Pontrjagin form is in the image, under the de Rham homomorphism (Example 4.9), of an integral class, andas such equal to the [ F ] cup-square: (cid:2) G − p ( ∇ ) (cid:3) = (cid:2) F ∧ F (cid:3) ∈ H ( X ; Z ) (cid:47) (cid:47) H ( X ) . (293)85 wisted differential Cohomotopy theory.Definition 5.25 (Differential twists for twistorial Cohomotopy) . Let X be an 8-dimensional smooth spin manifoldequipped with tangential Sp ( ) -structure τ (59). By (41) in Example 2.33, by (57) in Example 2.42, and by (11)in Example 2.3, we have [ τ ] ∈ H τ fr (cid:0) X ; O ( n ) / Sp ( ) (cid:1) ( Bi ) ∗ (cid:47) (cid:47) H (cid:0) X ; B Sp ( ) (cid:1) (cid:39) Sp ( ) Bundles ( X ) / ∼ . (294)This gives, in particular, the class of a smooth principal Sp ( ) -bundle P ! X to which the tangent bundle T X isassociated. With (212), we may choose an Sp ( ) -connection ∇ on P , and, by Prop. 4.37, this connection has aclass [ τ diff ] in differential non-abelian cohomology (Def. 4.33) with coefficient in BG : H (cid:0) X ; B Sp ( ) (cid:1) (cid:39) Sp ( ) Bundles ( X ) / ∼ (cid:111) (cid:111) (cid:111) (cid:111) Sp ( ) Connections ( X ) / ∼ (cid:47) (cid:47) (cid:98) H (cid:0) X ; B Sp ( ) (cid:1) [ τ ] (cid:111) (cid:111) (cid:47) (cid:47) [ P ] (cid:111) (cid:111) (cid:31) [ ∇ ] (cid:31) (cid:47) (cid:47) [ τ diff ] . Any such τ diff serves as a differential twist (277) for twistorial Cohomotopy in the following.In twisted generalization of Example 4.38, we have: Example 5.26 (Differential twistorial Cohomotopy) . Let X be a spin 8-manifold equipped with tangential Sp ( ) -structure τ (59), and with a corresponding differential twist τ diff (Def. 5.25). (i) Consider the local coefficient bundle (54), S ! S (cid:12) Sp ( ) J C P −! B Sp ( ) , for J-twisted 4-Cohomotopy (Exam-ple 2.40) pulled back along B Sp ( ) (cid:39) −! B Spin ( ) ! B O ( ) . This induces, via Def. 5.13, a twisted differentialnon-abelian cohomology theory (cid:99) T τ diff ( − ) , which we call J-twisted differential 4-Cohomotopy , whose value onmanifolds X = X × R k sits in a cohomology operation diagram (4) of this form: differentialJ-twisted4-Cohomotopy (cid:98) π τ ( X ) (cid:15) (cid:15) F τ S J-twistedcohomotopicalcurvature (cid:47) (cid:47)
J-twisted cohomotopical Bianchi identities (Example 3.96) (cid:40) G , G ∈ Ω • dR ( X ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d G = − (cid:0) G − p ( ∇ ) (cid:1) ∧ (cid:0) G + p ( ∇ ) (cid:1) − χ ( ∇ ) , d G = (cid:41) (cid:15) (cid:15) π τ ( X ) J-twisted4-Cohomotopy (Example 2.40) ch τ S character mapon J-twisted Cohomotopy (Example 5.23) (cid:47) (cid:47) H τ dR (cid:0) X ; l S (cid:1) . J-twistedde Rham cohomology (Def 3.98) (295) (ii)
Consider the local coefficient bundle (61) C P ! C P (cid:12) Sp ( ) J C P −! B Sp ( ) for twistorial Cohomotopy (Def.2.44). This induces, via Def. 5.13, a twisted differential non-abelian cohomology theory (cid:99) T τ diff ( − ) , which wecall differential twistorial Cohomotopy , whose value on manifolds X = X × R k sits in a cohomology operationdiagram (4) of this form: differentialtwistorialCohomotopy (cid:99) T τ diff ( X ) (cid:15) (cid:15) F τ dR C P twistorialcurvature (cid:47) (cid:47) twistorial Bianchi identities (Example 3.96) H , F , G , G ∈ Ω • dR ( X ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d H = G − p ( ∇ ) − F ∧ F , d F = , d G = − (cid:0) G − p ( ∇ ) (cid:1) ∧ (cid:0) G + p ( ∇ ) (cid:1) − χ ( ∇ ) , d G = (cid:15) (cid:15) T τ ( X ) twistorialCohomotopy (Example 2.44) ch τ C P character mapon twistorial Cohomotopy (Example 5.23) (cid:47) (cid:47) H τ dR dR (cid:0) X ; l C P (cid:1) . twistorialde Rham cohomology (Def 3.98) (296)86 roposition 5.27 (Twisted secondary cohomology operation induced by twistor fibration) . The defining twistednon-abelian cohomology operation (62) from twistorial Cohomotopy (Example 2.44) to J-twisted 4-Cohomotopy(Example 2.40), induced by the Sp ( ) -equivariatized twistor fibration t H (cid:12) Sp ( ) (61) refines to a twisted secondarycohomology operation (Def. 5.21) from differential twistorial Cohomotopy to differential J-twisted Cohomotopy(Example 5.26): differentialtwistorialCohomotopy (cid:99) T τ diff (cid:0) X (cid:1) twisted secondarycohomology operation (( t H (cid:12) Sp ( )) diff ) ∗ along Sp ( ) -equivariantizedtwistor fibration (cid:15) (cid:15) c τ C P (cid:47) (cid:47) T τ (cid:0) X (cid:1) ( t H (cid:12) Sp ( )) ∗ twisted primarycohomology operation (cid:15) (cid:15) differentialJ-twisted4-Cohomotopy (cid:98) π τ (cid:0) X (cid:1) c τ S (cid:47) (cid:47) π τ (cid:0) X (cid:1) Proof.
By Def. 5.21, we need to show that we have a twisted absolute minimal model (Def. 5.20) for the Sp ( ) -equivariantized twistor fibration (61). By (288) this means that we can find a morphism l B Sp ( ) ( C P (cid:12) Sp ( )) t H (cid:12) l Sp ( ) (cid:47) (cid:47) l B Sp ( ) S (cid:12) Sp ( )) (297)between the relative Whitehead L ∞ -algebras (Prop. 3.70) of the two local coefficient bundles, which makes thefollowing cube of transformations of derived PL-de Rham adjunction units commute: exp ◦ Ω • PLdR (cid:0) C P (cid:12) Sp ( ) (cid:1) exp ◦ Ω • PLdR ( J C P ) (cid:15) (cid:15) exp ◦ Ω • PLdR ( t H (cid:12) Sp ( )) (cid:47) (cid:47) exp (cid:18) p minSp ( ) C P (cid:12) Sp ( ) (cid:19) (cid:28) (cid:28) exp ◦ Ω • PLdR (cid:0) S (cid:12) Sp ( ) (cid:1) (cid:15) (cid:15) exp (cid:18) p min B Sp ( ) S (cid:12) Sp ( ) (cid:19) (cid:28) (cid:28) exp ◦ CE (cid:16) l B Sp ( ) (cid:0) C P (cid:12) Sp ( ) (cid:1)(cid:17) (cid:15) (cid:15) exp ◦ CE ( t H (cid:12) l Sp ( )) (cid:47) (cid:47) exp ◦ CE (cid:16) l (cid:0) S (cid:12) Sp ( ) (cid:1)(cid:17) (cid:15) (cid:15) exp ◦ Ω • PLdR (cid:0) B Sp ( ) (cid:1) exp (cid:0) p min B Sp ( ) (cid:1) (cid:28) (cid:28) exp ◦ Ω • PLdR (cid:0) B Sp ( ) (cid:1) exp ( p min B Sp ( ) ) (cid:28) (cid:28) exp ◦ CE (cid:0) l B Sp ( ) (cid:1) exp ◦ CE (cid:0) l B Sp ( ) (cid:1) But, from Example 3.96, we see that the total object of the relative Whitehead L ∞ -algebra of C P (cid:12) Sp ( ) relativeto l B Sp ( ) coincides with that relative to l B Sp ( ) S (cid:12) Sp ( ) . Therefore, we may take the twisted absolute minimalmodel (297) to be equal to the relative Whitehead L ∞ -algebra projection (the top arrow in Example 3.96). Thismakes the required top front square commute by the commuting triangle in Prop. 3.49. Remark 5.28 (Lifting against the twisted differential twistor fibration) . In terms of differential moduli ∞ -stacks (275), the result of Prop.5.27 with Example 5.26 says that lifting a twisted differential Coho-motopy cocycle (cid:98) C with 4-flux density G against the twisted differ-ential refinement (289) of the equivariantized twistor fibration (61)to a differential twistorial Cohomotopy cocycle ( (cid:98) C , (cid:98) C , (cid:98) C ) involves,on twisted curvature forms (280) the appearance of a 2-flux density F and of a 3-form H such that dH = G − p ( ∇ ) − F ∧ F . R , × X ( (cid:98) C , (cid:98) B , (cid:98) A ) (cid:47) (cid:47) (cid:127) (cid:95) (cid:15) (cid:15) (cid:0) C P (cid:12) Sp ( ) (cid:1) diff ( t H (cid:12) Sp ( )) diff twisted differentialtwistor fibration (cid:15) (cid:15) R , × X (cid:98) C C-field (cid:47) (cid:47) (cid:0) S (cid:12) Sp ( ) (cid:1) diff emark 5.29 (M-theory fields and Hypothesis H) . In summary, we have found: (i)
A cocyle (cid:98) C in J-twisted differential 4-Cohomotopy (Example 5.26) has as curvature/character forms (280): (a) a closed 4-form G , hence a 4-flux density, (b) a non-closed 7-form G ,satisfying the following Bianchi identities (Example 5.23) and integrality conditions (Prop. 5.24): differentialJ-twisted 4-Cohomotopy (cid:98) π τ ( X ) curvature (non-abelian character form representative) (cid:47) (cid:47) flat twisted cohomotopicaldifferential forms Ω τ dR dR (cid:0) X ; l S (cid:1) flat (cid:0) (cid:98) C (cid:1) G , G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) shifted C-field flux quantization (cid:2) G − p ( ω ) (cid:3) ∈ H ( X ; Z ) d G = d G = − (cid:0) G − p ( ω ) (cid:1) ∧ (cid:0) G + p ( ω ) (cid:1) − I ( ω ) C-field tadpole cancellation & M5 Hopf WZ term level quantization (298) (ii)
Lifting this cocycle through the twisted differential twistor fibration (Prop. 5.27) to a cocycle (cid:0) (cid:98) C , (cid:98) B , (cid:98) A (cid:1) indifferential twistorial Cohomotopy (Example 5.26) involves (Remark 5.28) adjoining to the 4-flux density G : (c) a closed 2-form curvature F , hence a 2-flux density, (d) a non-closed 3-form H ,such that these curvature/character forms satisfy the following Bianchi identities (Example 5.23) and integralityconditions (Prop. 5.24): differentialtwistorial Cohomotopy (cid:99) T τ diff ( X ) curvature (non-abelian character form representative) (cid:47) (cid:47) flat twistorialdifferential forms Ω τ dR dR (cid:0) X ; l C P (cid:1) flat (cid:0) (cid:98) C , (cid:98) B , (cid:98) A (cid:1) H , G , F , G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d H = G − p ( ω ) − F ∧ F , Hoˇrava-Witten Green-Schwarz mechanism (cid:2) G − p ( ω ) (cid:3) = (cid:2) F ∧ F (cid:3) ∈ H ( X ; Z ) d G = , d F = , d G = − (cid:0) G − p ( ω ) (cid:1) ∧ (cid:0) G + p ( ω ) (cid:1) − I ( ω ) C-field tadpole cancellation & M5 Hopf WZ term level quantization (299) (iii)
With these cohomotopical curvature/character forms interpreted as flux densities, this is the Bianchi identitiesand charge quantization expected in M-theory on the supergravity C-field ( (cid:98) C ), the heterotic B-field ( (cid:98) B ) and theheterotic S (cid:0) U ( ) (cid:1) ⊂ E gauge field ( (cid:98) A ) , with the following prominent features: (a) The charge quantization:(1) (cid:2) G − p ( ∇ ) (cid:3) ∈ H ( X ; Z ) is expected for the C-field in the M-theory bulk(see [FSS19b, Table 1] for pointers); (2) (cid:2) G − p ( ∇ ) (cid:3) = [ F ∧ F ] ∈ H ( X ; Z ) is expected on heterotic boundaries(see [FSS20, §1] for pointers). (300) (b) The quadratic functions:(1) G ( G − p ( ω )) ∧ ( G + p ( ω ))+ I ( ω ) constitute the Hopf Wess-Zumino term(see [FSS19c][SS20a]) (2) F F ∧ F constitute the 2nd Chern classof an S (cid:0) U ( ) (cid:1) ⊂ E bundle (see [FSS20, (7)]). (301) (iv) These are necessary, not yet sufficient constraints on cohomotopical lifts. The full constraints may be computedby a Postnikov tower analysis [GS20] and turn out to coincide with a list of further expected conditions in M-theory(see [FSS19b, Table 1]).All this suggests the
Hypothesis H [Sa13][FSS19b][FSS19c][SS19a][SS19b][BMSS19][SS20a][FSS20] that theelusive cohomology theory which controls M-theory in analogy to how K-theory controls string theory is: twisteddifferential non-abelian Cohomotopy theory. 88 ohomotopical character into K-theory.
We may regard the secondary non-abelian Hurewicz/Boardman ho-momorphism (Example 4.43) from differential 4-Cohomotopy (Example 4.38) to differential K-theory (Example4.36), as a non-abelian but K-theory valued character, lifting the target of the cohomotopical character (Example5.23) from rational cohomology to K-theory (compare [BSS19, Fig. 1]): differentialCohomotopy (cid:98) τ (cid:0) X (cid:1) β differential non-abelianBoardman homomorphism (Example 4.43) (cid:47) (cid:47) differentialK-theory (cid:99) KU (cid:0) X (cid:1) ch diff differentialChern character (Example (5.22)) (cid:47) (cid:47) differentialrational cohomology (cid:92) H per Q (cid:0) X (cid:1) (cid:111) (cid:111) charge-quantizationin M-theory (cid:111) (cid:111) charge-quantizationin string theory (302) (i) Lifting through this differential Boardman homomorphism induces secondary charge quantization conditionson
K-theory, analogous to (2) but invisible even in generalized cohomology, instead now coming from non-abeliancohomology theory. (ii)
In the plain version (302) (i.e. disregarding twisting and equivariant enhancement) the effect of β on curva-ture forms (234) is (by Example 4.43) to forget the quadratic function (301) on G and to inject what remains asthe 4-form curvature component F in differential K-theory: differentialCohomotopy (cid:98) τ (cid:0) X (cid:1)(cid:0) F S (cid:1) ∗ (cid:15) (cid:15) β secondary non-abelianBoardman homomorphism (cid:47) (cid:47) curvatures/flux densities differentialK-theory (cid:99) KU (cid:0) X (cid:1)(cid:0) F KU0 (cid:1) ∗ (cid:15) (cid:15) (cid:40) G , G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d G = − G ∧ G d G = (cid:41) G F G (cid:47) (cid:47) (cid:110)(cid:0) F k (cid:1) | d F k = (cid:111) . (303)This is a ‘cohomotopical enhancement’ of the reduction in [DMW02] of E bundles in M-theory to the K-theoryof type IIA string theory, now characterized by higher Postnikov stages of the Boardman homomorphism [GS20].The remaining RR-flux components in { F k } are also found in the cohomotopical character, through cohomo-logical double dimensional reduction formulated in parametrized homotopy theory: this is discussed in detail in[BMSS19]. (iii) The twisted generalization of the non-abelian Boardman homomorphism in (302) and (303) is more sub-tle, since the degree-3 twist of K-theory does not arise from the J-twist of Cohomotopy, but arises, togetherwith the further RR-flux components, from S -equivariantization/double dimensional reduction of Cohomotopy[FSS16a][BMSS19], reproducing the reduction of E bundles from M-theory to type IIA in [MaSa04]. Outlook: Equivariant enhancement.
The cohomotopical character into K-theory (302) is particularly interestingafter lifting further to equivariant non-abelian cohomology theory, where charge-quantizing/lifting of RR-fieldsin equivariant K-theory through the Boardman homomorphism on the left of (302) encodes pertinent “tadpolecancellation” conditions [SS19a][BSS19]. The character theory presented here lifts to the required equivari-ant differential non-abelian cohomology on orbi-orientifolds by combining it with the techniques developed in[HSS18][SS20b]. We shall discuss the resulting character map in equivariant (twisted differential) non-abeliancohomology in a followup article. 89
Model category theory
For ease of reference and to highlight some less widely used aspects needed in the main text, we record basicsof homotopy theory via model category theory [Qu67] (review in [Ho99][Hir02][Lu09, A.2]) and of homotopytopos theory [Re10] via model categories of simplicial presheaves [Bro73][Ja87][Du01] (review in [Du98][Lu09,§A.3.3][Ja15]).
Topology.
By TopologicalSpaces ∈ Categorieswe denote a convenient [St67] (in particular: cartesian closed) category of topological spaces such as compactly-generated spaces [Stri09] or ∆ -generated spaces [Du03], equivalently numerically-generated spaces [SYH10], orD-topological spaces [SS20a, Prop. 2.4]. Categories.
Let C be a category. (i) For X , A ∈ C a pair of objects, we write C ( X , A ) : = Hom C ( X , A ) for the set of morphisms from X to A . (ii) For C , D two categories, we denote a pair of adjoint functors between them by D (cid:111) (cid:111) LR ⊥ (cid:47) (cid:47) C ⇔ D ( L ( − ) , − ) (cid:39) C ( − , R ( − )) . (304) (iii) A Cartesian square in C we indicate by pullback notation f ∗ ( − ) and/or by the symbol “ (pb) ”: f ∗ A (cid:15) (cid:15) (cid:15) (cid:15) (pb) f ∗ p (cid:47) (cid:47) A p (cid:15) (cid:15) B f (cid:47) (cid:47) B . (305) (iv) Dually, a co-Cartesian square in C we indicate by pushout notation f ∗ ( − ) and/or by the symbol “ (po) ”: X q (cid:47) (cid:47) f (cid:15) (cid:15) (po) A q ∗ f (cid:15) (cid:15) Y (cid:47) (cid:47) f ∗ A . (306) Model categories.Definition A.1 (Weak equivalences) . A category with weak equivalences is a category C equipped with a sub-classW ⊂ Mor ( C ) of its morphisms, to be called the class of weak equivalences , such that (i) W contains the class of isomorphisms; (ii)
W satisfies the cancellation property (“2-out-of-3”): if in any commuting triangle in C Y g (cid:39) (cid:39) X f (cid:55) (cid:55) g ◦ f (cid:47) (cid:47) Z (307)two morphisms are in W, then so is the third. Definition A.2 (Weak factorization system) . A weak factorization system in a category C is a pair of sub-classesof morphisms Proj , Inj ⊂ Mor ( C ) such that (i) every morphisms X f (cid:47) (cid:47) Y in C may be factored through a morphism in Proj followed by one in Inj: f : X ∈ Proj (cid:47) (cid:47) Z ∈ Inj (cid:47) (cid:47) Y (308) (ii) For every commuting square in C with left morphism in Proj and right morphism in Inj, there exists a lift,namely a dashed morphism X (cid:47) (cid:47) ∈ Proj (cid:15) (cid:15) A ∈ Inj (cid:15) (cid:15) Y ∃ (cid:55) (cid:55) (cid:47) (cid:47) B (309)making the resulting triangles commute. (iii) Given Inj (resp. Proj), the class Proj (resp. Inj) is the largest class for which (309) holds.90 efinition A.3 (Model category [Jo08, Def. E.1.2][Rie09]) . A model category is a category C that has all smalllimits and colimits, equipped with three sub-classes of its class of morphisms, to be denotedW – weak equivalences Fib – fibrations
Cof – cofibrations such that (i)
The class W makes C a category with weak equivalences (Def. A.1); (ii) The pairs (cid:0)
Fib , Cof ∩ W (cid:1) and (cid:0) Fib ∩ W , Cof (cid:1) are weak factorization systems (Def. A.2).
Remark A.4 (Minimal assumptions) . By item (iii) in Def. A.2 a model category structure is specified already bythe classes W and Fib, or alternatively by the classes W and Cof. Moreover, it follows from Def. A.3 that also theclass W is stable under retracts [Jo08, Prop. E.1.3][Rie09, Lemma 2.4]: Given a commuting diagram in the modelcategory C of the form on the left here X (cid:47) (cid:47) f (cid:15) (cid:15) Y ∈ W (cid:15) (cid:15) (cid:47) (cid:47) X f (cid:15) (cid:15) A (cid:47) (cid:47) B (cid:47) (cid:47) A ⇒ f ∈ W (310)with the middle morphism a weak equivalence, then also f is a weak equivalence. Definition A.5 (Proper model category) . A model category C Def. A.3 is called (i) right proper , if pullback along fibrations preserves weak equivalences: X (cid:15) (cid:15) p ∗ f (cid:47) (cid:47) (pb) A p ∈ Fib (cid:15) (cid:15) Y f ∈ W (cid:47) (cid:47) B ⇒ p ∗ f ∈ W (311) (ii) left proper , if pushout along cofibrations preserves weak equivalences, hence if the opposite model category(Example A.9) is right proper.
Notation A.6 (Fibrant and cofibrant objects) . Let C be a model category (Def. A.3) (i) We write ∗ ∈ C for the terminal object and ∅ ∈ C for the initial object. (ii) An object X ∈ C is called: (a) fibrant if the unique morphism to the terminal object is a fibration, X ∈ Fib (cid:47) (cid:47) ∗ ; (b) cofibrant if the unique morphism from the initial object is a cofibration, ∅ ∈ Cof (cid:47) (cid:47) X .We write C fib , C cof , C coffib ⊂ C for the full subcategories on, respectively, fibrant objects, or cofibrant objects orobjects that are both fibrant and cofibrant. (iii) Given an object X ∈ C(a) A fibrant replacement is a factorization (308) of the terminal morphism as X j X ∈ Cof ∩ W (cid:47) (cid:47) PX q X ∈ Fib (cid:47) (cid:47) ∗ . (312) (b) A cofibrant replacement is a factorization (308) of the initial morphism as ∅ i X ∈ Cof (cid:47) (cid:47) QX p X ∈ Fib ∩ W (cid:47) (cid:47) X . (313) Example A.7 (Classical model structure on topological spaces [Qu67, §II.3][Hir15]) . The category of TopologicalSpacescarries a model category structure whose (i)
W – weak equivalence are the weak homotopy equivalences; (ii)
Fib – fibrations are the Serre fibrations.We denote this model category by TopologicalSpaces Qu ∈ ModelCategories . xample A.8 (Classical model structure on simplicial sets [Qu67, §II.3][GJ99]) . The category of SimplicialSetscarries a model category structure whose (i)
W – weak equivalence are those whose geometric realization is a weak homotopy equivalence; (ii)
Cof – cofibrations are the monomorphisms (degreewise injections).We denote this model category by SimplicialSets Qu ∈ ModelCategories . Example A.9 (Opposite model category [Hir02, §7.1.8]) . If C is a model category (Def. A.3) then the oppositeunderlying category becomes a model category C op with the same weak equivalences (up to reversal) and withfibrations (resp. cofibrations) the cofibrations (resp. fibrations) of C , up to reversal. Example A.10 (Slice model structure [Hir02, §7.6.4]) . Let C be a model category (Def. A.3) (i) For X ∈ C any object, the slice category C / X , whose objects are morphisms to X and whose morphisms arecommuting triangles in C over X C / X (cid:0) a , b (cid:1) : = A a (cid:39) (cid:39) f (cid:47) (cid:47) B b (cid:119) (cid:119) X becomes itself a model category, whose weak equivalence, fibrations and cofibrations are those morphims whoseunderlying morphisms f are such in C . (ii) Dually there is the coslice model category C X / : = (cid:0) ( C op ) / X (cid:1) op , being the opposite model category (ExampleA.9) of the slice category of the opposite of C : C / X ( a , b ) : = XA (cid:119) (cid:119) a f (cid:47) (cid:47) B (cid:39) (cid:39) b Homotopy categories.Definition A.11 (Path space objects [Qu67, Def. I.4]) . Let C be a model category (Def. A.3), and A ∈ C fib bea fibrant object (Notation A.6). Then a path space object for A is a factorization of the diagonal morphism ∆ A through a weak equivalence followed by a fibration: A ∆ A (cid:51) (cid:51) ∈ W (cid:47) (cid:47) Paths ( A ) ( p , p ) ∈ Fib (cid:47) (cid:47) A × A . (314) Definition A.12 (Right homotopy) . Let C be a model category (Def. A.3), X ∈ C cof a cofibrant object, A ∈ C fib a fibrant object (Notation A.6) and let Paths ( A ) be a path space object for A (Def. A.11). Then a right homotopy between a pair of morphisms f , g ∈ C ( X , A ) , to be denoted φ : f ⇒ r g or X f (cid:37) (cid:37) g (cid:57) (cid:57) A φ (cid:11) (cid:19) is a morphism φ ∈ C (cid:0) X , Paths ( A ) (cid:1) which makes this diagram commute: AX φ (cid:47) (cid:47) f (cid:53) (cid:53) g (cid:41) (cid:41) Paths ( A ) p (cid:79) (cid:79) p (cid:15) (cid:15) A Proposition A.13 (Right homotopy classes) . Let C be a model category, X ∈ C cof and A ∈ C fib (Notation A.6).Then right homotopy (Def. A.12) is an equivalence relation ∼ r on the hom-set C ( X , A ) . We write C ( X , A ) / ∼ r ∈ Sets (315) for the corresponding set of right homotopy classes of morphisms from X to A. efinition A.14 (Homotopy category of a model category) . For C a model category (Def. A.3), (i) we write Ho ( C ) : = (cid:0) C coffib (cid:1) / ∼ r ∈ Categories (316)for the category whose objects are those objects of C that are both fibrant and cofibrant (Notation A.6), and whosemorphisms are the right homotopy classes of morphisms in C (Def. 315): X , A ∈ C coffib ⇒ Ho ( C )( X , A ) : = C ( X , A ) / ∼ r and composition of morphisms is induced from composition of representatives in C . (ii) Given a choice of fibrant replacement P and of cofibrant replacement Q for each object of C (Notation A.6) weobtain a functor C γ C (cid:47) (cid:47) Ho ( C ) , (317)which (a) sends any object X ∈ C to PQX and sends (b) any morphism X f (cid:47) (cid:47) A to the right homotopy class (315)of any lift (309) PQ f obtained from any lift
Q f in the following diagrams: ∅ (cid:47) (cid:47) i X (cid:15) (cid:15) QY p Y (cid:15) (cid:15) QX Q f (cid:54) (cid:54) f ◦ p x (cid:47) (cid:47) Y (cid:32) QX j QY ◦ Q f (cid:47) (cid:47) j QX (cid:15) (cid:15) PQY q QY (cid:15) (cid:15) PQX (cid:47) (cid:47)
PQ f (cid:53) (cid:53) ∗ Proposition A.15 (Homotopy category is localization) . Given a model category C (Def. A.3) the functor C γ C (cid:47) (cid:47) Ho (cid:0) C (cid:1) (317) from Def. A.14 exhibits the homotopy category as the localization of the model category at its class of weakequivalences: γ C sends all weak equivalences in C to isomorphisms, and is the universal functor with this property. The restriction to fibrant-and-cofibrant objects in Def. A.14 is convenient for defining composition of mor-phisms, but for computing hom-sets in the homotopy category it is sufficient that the domain object is cofibrant,and the codomain fibrant:
Proposition A.16 ([Qu67, §I.1 Cor. 7]) . Let C be a model category (Def. A.3). For X ∈ C cof a cofibrant objectand A ∈ C fib a fibrant object, any choice of fibrant replacement PX and cofibrant replacement QA (Notation A.6).induces a bijection between the set of right homotopy classes (Def. A.12) and the hom-set in the homotopy category(Def. A.14) between X and A: C ( X , A ) / ∼ r C ( j X , p A ) (cid:39) (cid:47) (cid:47) Ho ( C )( X , A ) . Quillen adjunctions.Definition A.17 (Quillen adjunction) . Let D , C be model categories (Def. A.3). Then a pair of adjoint functors ( L (cid:97) R ) (304) between their underlying categories is called a Quillen adjunction , to be denoted D (cid:111) (cid:111) LR ⊥ Qu (cid:47) (cid:47) C (318)if the following equivalent conditions hold:• L preserves Cof, and R preserves Fib;• L preserves Cof and Cof ∩ W;• R preserves Fib and Fib ∩ W. Example A.18 (Base change Quillen adjunction) . Let C be a model category (Def. A.3), B , B ∈ C fib a pair offibrant objects (Notation A.6) and B f (cid:47) (cid:47) B ∈ C (319)a morphism. Then we have a Quillen adjunction (Def. A.17) C / B (cid:111) (cid:111) f ! f ∗ ⊥ Qu (cid:47) (cid:47) C / B (i) The left adjoint functor f ! is given by postcomposition in C with f (319): f ∗ : X τ (cid:40) (cid:40) c (cid:47) (cid:47) A p (cid:118) (cid:118) B X τ (cid:40) (cid:40) c (cid:47) (cid:47) A p (cid:118) (cid:118) B f (cid:15) (cid:15) B (320) (ii) The right adjoint functor f ∗ is given by pullback (305) along f (319).That these functors indeed form an adjunction f ! (cid:97) f ∗ follows from the defining universal property of the pullback(305): C / B (cid:0) f ∗ τ , ρ (cid:1) (cid:39) C / B (cid:0) τ , f ∗ ρ (cid:1) X c (cid:47) (cid:47) τ (cid:35) (cid:35) A ρ (cid:15) (cid:15) B f (cid:42) (cid:42) B ↔ X (cid:101) c (cid:47) (cid:47) τ (cid:35) (cid:35) f ∗ A f ∗ ρ (cid:15) (cid:15) (cid:47) (cid:47) (pb) A ρ (cid:15) (cid:15) B f (cid:42) (cid:42) B (321)That this adjunction is a Quillen adjunction (Def. A.17) follows since f ! (320) evidently preserves each of W andCof (even Fib) separately, by Example A.10. Lemma A.19 (Ken Brown’s lemma [Ho99, Lemma 1.1.12][Bro73]) . Given a Quillen adjunction L (cid:97)
R (Def. A.17), (i) the right Quillen functor R preserves all weak equialences between fibrant objects. (ii) the left Quillen functor L preserves all weak equivalences between cofibrant objects.
Proposition A.20 (Derived functors) . Given a Quillen adjunction ( L (cid:97) Qu R ) (Def. A.17), there are adjoint functors L L (cid:97) R R (304) between the homotopy categories (Def. A.14) Ho ( D ) (cid:111) (cid:111) L L R R ⊥ (cid:47) (cid:47) Ho ( C ) (322) whose composites with the localization functors (317) make the following squares commute up to natural isomor-phism: D R (cid:47) (cid:47) γ D (cid:15) (cid:15) ⇓ (cid:39) C γ C (cid:15) (cid:15) Ho ( D ) R R (cid:47) (cid:47) Ho ( C ) D (cid:111) (cid:111) L γ D (cid:15) (cid:15) (cid:39) ⇓ C γ C (cid:15) (cid:15) Ho ( D ) (cid:111) (cid:111) L L Ho ( C ) . These are unique up to natural isomorphism, and are called the left and right derived functors of L and R, respec-tively.
Example A.21 (Derived functors via (co-)fibrant replacement) . It is convenient to leave the localization functors γ (317) notationally implicit, and understand objects of C as objects of Ho ( C ) , via γ . Then: (i) The value of a left derived functor L L (Prop. A.20) on an object c ∈ C is equivalently the value of L on acofibrant replacement Qc (313): L L ( c ) (cid:39) L ( Qc ) ∈ Ho ( D ) . (323) (ii) The value of a right derived functor R R (Prop. A.20) on an object d ∈ D is equivalently the value of R on acofibrant replacement Pd (312): R R ( d ) (cid:39) R ( Pd ) ∈ Ho ( C ) . (324) (iii) The derived unit D η of the derived adjunction (322), is, on any cofibrant object c ∈ C cof , given by D η c : c η c (cid:47) (cid:47) R (cid:0) L ( c ) (cid:1) R ( j L ( c ) ) (cid:47) (cid:47) R (cid:0) PL ( c ) (cid:1) ∈ Ho ( C ) (325)where L ( c ) j L ( c ) (cid:47) (cid:47) PL ( c ) is any fibrant replacement (312). (iv) The derived co-unit D ε of the derived adjunction (322), is, on any fibrant object d ∈ D fib , given by D ε d : L (cid:0) QR ( d ) (cid:1) L ( p R ( d ) ) (cid:47) (cid:47) L (cid:0) R ( d ) (cid:1) ε d (cid:47) (cid:47) d ∈ Ho ( D ) (326)where QR ( d ) p R ( d ) (cid:47) (cid:47) R ( d ) is any cofibrant replacement (313).94 omotopy fibers and homotopy pullback.Definition A.22 (Homotopy fiber) . Let C be a model category (Def. A.3). (i) For A p (cid:47) (cid:47) B a morphism in C with B ∈ C fib ⊂ C a fibrant object (Notation A.6), and for ∗ b (cid:47) (cid:47) B a morphismfrom the terminal object (a “point in B ”), the homotopy fiber of p over b is the image in the homotopy category(317) of the ordinary fiber over b , i.e. the pullback (305) along b in C , of any fibration (cid:101) p weakly equivalent to p :hofib b ( ρ ) (cid:47) (cid:47) A ρ (cid:15) (cid:15) B : = γ C fib b ( (cid:101) p ) (cid:47) (cid:47) (cid:15) (cid:15) (pb) (cid:101) A (cid:101) ρ ∈ Fib (cid:15) (cid:15) (cid:111) (cid:111) ∈ W A ρ (cid:119) (cid:119) ∗ b (cid:47) (cid:47) B ∈ Ho ( C ) . (327)This is well-defined in that hofib b ( p ) ∈ Ho ( C ) depends on the choice of fibration replacement (cid:101) p only up to iso-morphism in the homotopy category. (ii) Dually, homotopy co-fibers are homotopy fibers in the opposite model category (Def. A.9).More generally:
Definition A.23 (Homotopy pullback) . Given a model category C (Def. A.3) and a pair of coincident morphisms A ρ (cid:15) (cid:15) X τ (cid:47) (cid:47) B between fibrant objects, the homotopy pullback of ρ along τ (or homotopy fiber product of ρ with τ ) is the imageof ρ , regarded as an object in the homotopy category (Def. A.14) of the slice model category (Example A.10)under the right derived functor (Prop. A.20) of the right base change functor along τ (Example A.18):Ho (cid:0) C / B (cid:1) (cid:51) A ρ (cid:15) (cid:15) B homotopypullback R τ ∗ A R τ ∗ ρ (cid:15) (cid:15) X : = Ho (cid:0) C / X (cid:1) , (328)By (321) the derived adjunction counit (326) on (328) gives a commuting square in (317) the homotopy categoryof C R τ ∗ A (cid:47) (cid:47) R τ ∗ ρ (cid:15) (cid:15) (hpb) A ρ (cid:15) (cid:15) X τ (cid:47) (cid:47) B : = γ C τ ∗ (cid:101) A (cid:47) (cid:47) (cid:15) (cid:15) (pb) (cid:101) A (cid:101) ρ ∈ Fib (cid:15) (cid:15) (cid:111) (cid:111) ∈ W A ρ (cid:119) (cid:119) X τ (cid:47) (cid:47) B ∈ Ho ( C ) . (329)This square in the homotopy category, together with its pre-image pullback square in the model category, is the homotopy pullback square of ρ along τ . Example A.24 (Homotopy fiber is homotopy pullback to the point) . Homotopy fibers (Def. A.22) are the homo-topy pullbacks (Def. A.23) to the terminal object, by (324).
Lemma A.25 (Factorization lemma [Bro73, p. 431]) . Let C be a model category (Def. A.3) and A ρ (cid:47) (cid:47) B ∈ C fib amorphism between fibrant objects. Then for Paths ( B ) a path space object for B (Def. A.11) the vertical compositein the following diagram A ∈ W (cid:47) (cid:47) ρ (cid:34) (cid:34) p ∗ A (cid:15) (cid:15) ∈ W (cid:47) (cid:47) (pb) A ρ (cid:15) (cid:15) Paths ( B ) p (cid:47) (cid:47) p (cid:15) (cid:15) BB (330) is a fibration, and in fact a fibration resolution of ρ , in that it factors ρ through a weak equivalence. xample A.26 (Homotopy pullback via triples) . Given a model category C (Def. A.3) and a pair of coincidentmorphisms A ρ (cid:15) (cid:15) X τ (cid:47) (cid:47) B between fibrant objects, Lemma A.25 says that the corresponding homotopy pullback (Def. A.23) is computed bythe following diagram R τ ∗ A (cid:15) (cid:15) (cid:47) (cid:47) φ (cid:38) (cid:38) A ρ (cid:15) (cid:15) Paths ( B ) p (cid:15) (cid:15) p (cid:47) (cid:47) BX τ (cid:47) (cid:47) B = τ ∗ (cid:0) p ◦ p ∗ ρ (cid:1) (cid:15) (cid:15) (cid:47) (cid:47) (pb) p ∗ A (cid:47) (cid:47) (cid:15) (cid:15) (pb) A ρ (cid:15) (cid:15) Paths ( B ) p (cid:15) (cid:15) p (cid:47) (cid:47) BX τ (cid:47) (cid:47) B Here the right hand side exhibits the left hand side as a limit cone. This means that the homotopy pullback R τ ∗ A isuniversally characterized by the fact that morphisms into it are triples ( f , g , φ ) , consisting of a pair of morphisms f , g to A , X , respectively, and a right homotopy φ (Def. A.12) between their composites with ρ and τ , respectively: C (cid:0) − ; R τ ∗ A (cid:1) (cid:39) ( f , g , φ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f (cid:47) (cid:47) g (cid:15) (cid:15) A ρ (cid:15) (cid:15) X τ (cid:47) (cid:47) B φ (cid:114) (cid:122) (331) Quillen equivalences.Lemma A.27 (Conditions characterizing Quillen equivalences) . Given a Quillen adjunction L (cid:97) Qu R (Def. A.17),the following conditions are equivalent: • The left derived functor (Prop. A.20) is an equivalence of homotopy categories (Def. A.14) Ho ( D ) (cid:111) (cid:111) L L (cid:39) Ho ( C ) . • The right derived functor (Prop. A.20) is an equivalence of homotopy categories (Def. A.14) Ho ( D ) R R (cid:39) (cid:47) (cid:47) Ho ( C ) . • Both of the following two conditions hold: (i)
The derived adjunction unit D η (325) is a natural isomorphism, hence (325) is a weak equivalence in C ; (ii) The derived adjunction counit D ε (326) is a natural isomorphism, hence (326) is a weak equivalencein D . • For c ∈ C cof and d ∈ D fib , a morphism out of L ( c ) is a weak equivalence precisely if its adjunct into R ( d ) is:L ( c ) f ∈ W (cid:47) (cid:47) d ⇔ c (cid:101) f ∈ W (cid:47) (cid:47) R ( d ) . (332) Definition A.28 (Quillen equivalence) . If the equivalent conditions from Lemma A.27 are met, a Quillen adjunc-tion L (cid:97) Qu R (Def. A.17) is called a Quillen equivalence , which we denote as follows: D (cid:111) (cid:111) LR (cid:39) Qu (cid:47) (cid:47) C . Hence:
Proposition A.29 (Derived equivalence of homotopy categories) . The derived adjunction (Prop. A.20) of a Quillenequivalence (Def. A.28) is an adjoint equivalence of homotopy categories (Def. A.14): Ho ( D ) (cid:111) (cid:111) L L R R (cid:39) (cid:47) (cid:47) Ho ( C ) . (333)96 emma A.30 (Quillen equivalence when left adjoint creates weak equivalences [EI19, Lemma 3.3]) . Let L (cid:97) Qu Rbe a Quillen adjunction (Def. A.17) such that the left adjoint functor L creates weak equivalences, in that for allmorphisms f in C we have f ∈ W C ⇔ L ( f ) ∈ W D . (334) Then L (cid:97) Qu R is a Quillen equivalence (Def. A.28) precisely if the adjunction co-unit ε d is a weak equivalence onall fibrant objects d ∈ C fib .Proof. By Lemma A.27, it is sufficient to check that the (i) derived unit and (ii) derived counit of the adjunctionare weak equivalences precisely if the ordinary counit is a weak equivalence. (ii)
For the derived counit (326) D ε c : L (cid:0) QR ( d ) (cid:1) L ( p R ( d ) ) ∈ W (cid:47) (cid:47) L (cid:0) R ( d ) (cid:1) ε d (cid:47) (cid:47) d we have that p R ( d ) is a weak equivalence (313), and since L preserves this, by assumption, so is L (cid:0) p R ( d ) (cid:1) . Therefore D ε d is a weak equivalence precisely if ε d is, by 2-out-of-3 (307). (i) For the derived unit (325) c η c (cid:47) (cid:47) R (cid:0) L ( c ) (cid:1) R ( j L ( c ) ) (cid:47) (cid:47) R (cid:0) PL ( c ) (cid:1) consider the composite of its image under L with the adjunction counit, as shown in the middle row of the followingdiagram: L ( c ) L ( η c ) (cid:47) (cid:47) j L ( c ) ∈ W (cid:50) (cid:50) L ( D η c ) (cid:44) (cid:44) L ◦ R (cid:0) L ( c ) (cid:1) L ◦ R ( j L ( c ) ) (cid:47) (cid:47) L ◦ R (cid:0) PL ( c ) (cid:1) ε PL ( c ) (cid:47) (cid:47) PL ( c ) . By the formula for adjuncts, this composite equals the adjunct of the derived adjunction unit, hence j L ( c ) , asshown by the bottom morphism, which is a weak equivalence (312). Now, since L creates weak equivalences byassumption, L ( D η c ) is a weak equivalence precisely if D η c is a weak equivalence. Therefore it follows, again by2-out-of-3 (307), that this is the case precisely if the adjunction counit ε is a weak equivalence on the fibrant object PL ( c ) . Proposition A.31 (Base change along weak equivalence in right proper model category) . Let C be a right propermodel category (Def. A.5). Then its base change Quillen adjunction (Example A.18) along any weak equivalenceB f ∈ W (cid:47) (cid:47) B ∈ C is a Quillen equivalence (Def. A.28): C / B (cid:111) (cid:111) f ! f ∗ (cid:39) Qu (cid:47) (cid:47) C / B . Proof.
Observe that B (cid:47) (cid:47) B is the terminal object of C / B , so that the fibrant objects of C / B correspond tothe fibrations in C over B . Therefore, the condition (332) says that for f ! (cid:97) f ∗ to be a Quillen equivalence it issufficient that in (321) c is a weak equivalence precisely if (cid:101) c is, assuming that ρ is a fibration: X c (cid:47) (cid:47) τ (cid:36) (cid:36) A ρ ∈ Fib (cid:15) (cid:15) B f ∈ W (cid:43) (cid:43) B ↔ X (cid:101) c (cid:47) (cid:47) τ (cid:38) (cid:38) f ∗ A f ∗ ρ (cid:15) (cid:15) ρ ∗ f ∈ W (cid:47) (cid:47) (pb) A ρ ∈ Fib (cid:15) (cid:15) B f ∈ W (cid:43) (cid:43) B (335)But under this assumption, right-properness implies that ρ ∗ f is a weak equivalence (311), so that the statementfollows by 2-out-of-3 (307). Alternative Proof.
The conclusion also follows with Lemma A.30: The left adjoint functor L = f ! clearly createsweak equivalences (334) (by the nature of the slice model structure, Example A.10), so that Lemma A.30 asserts97hat we have a Quillen equivalence as soon as the ordinary adjunction counit is a weak equivalence on all fibrantobjects. By (321), the adjunction counit on a fibration ρ ∈ Fib is the dashed morphism ρ ∗ f in the followingdiagram on the right: f ∗ A id (cid:47) (cid:47) f ∗ ρ (cid:39) (cid:39) f ∗ A f ∗ ρ (cid:15) (cid:15) ρ ∗ f ∈ W (cid:47) (cid:47) (pb) A ρ ∈ Fib (cid:15) (cid:15) B f ∈ W (cid:43) (cid:43) B ↔ f ∗ A ρ ∗ f ∈ W (cid:47) (cid:47) f ∗ ρ (cid:37) (cid:37) A ρ ∈ Fib (cid:15) (cid:15) B f ∈ W (cid:43) (cid:43) B (336)And hence this is a weak equivalence, again by right-properness. Example A.32 (Quillen equivalence between topological spaces and simplicial sets [Qu67]) . Forming simplicialsets constitutes a Quillen equivalence (Def. A.28)TopologicalSpaces Qu (cid:111) (cid:111) geometric realization |−| Sing singular simplicial complex (cid:39) Qu (cid:47) (cid:47) SimplicialSets Qu (337)between the classical model structure on topological spaces (Example A.7) and the classical model structure onsimplicial sets (Example A.8). Example A.33 (Classical homotopy category) . By Prop. A.29 and the derived adjunction (Prop. A.20) of the |−| (cid:97)
Sing-adjunction (Example A.32) is an equivalence between the homotopy categories (Def. A.14) of theclassical model category of topological spaces (Example A.7) and the classical model category of simplicial sets(Example A.8): Ho (cid:0)
TopologicalSpaces Qu (cid:1) (cid:111) (cid:111) L |−| R Sing (cid:39) (cid:47) (cid:47) Ho (cid:0) SimplicialSets Qu (cid:1) . (338)Either of these is the classical homotopy category . We refer to its objects as homotopy types , to be distinguishedfrom the actual topological spaces or simplicial set that represent them. Cell complexes.Proposition A.34 (Skeleta and truncation [May67, §II.8][DK84, §1.2 (vi)] ) . For each n ∈ N there is a pair ofadjoint functors SimplicialSets (cid:111) (cid:111) skcosk ⊥ (cid:47) (cid:47) SimplicialSets , (339) where sk n ( S ) is the simplicial sub-set generated by the simplices in S of dimension ≤ n (hence including only alltheir degenerate higher simplices), and where cosk n ( S ) : [ k ] SimplicialSets (cid:0) sk n ( ∆ [ k ]) , S (cid:1) . Here cosk n + preserves fibrant objects of the classical model structure (Example A.8) and models n-truncation, inthat: π k (cid:12)(cid:12) cosk n + ( S ) (cid:12)(cid:12) = for k ≥ n + and there are natural fibrations S p n (cid:47) (cid:47) cosk n ( S ) such that π k (cid:12)(cid:12) S (cid:12)(cid:12) π k | p n |(cid:39) (cid:47) (cid:47) π k (cid:12)(cid:12) cosk n + ( S ) (cid:12)(cid:12) for k ≤ n . For A ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) we write A ( n ) : = (cid:12)(cid:12) cosk n + (cid:0) Sing ( A ) (cid:1)(cid:12)(cid:12) (340)We say that A is n-truncated if it is equivalent to its n -truncation: A is n -truncated ⇔ A (cid:39) A ( n ) . (341)for its n -truncation. 98 xample A.35 (Simplicial sets are weakly equivalent to singular simplicial sets of their realization) . For S ∈ SimplicialSets, the unit of the adjunction (337) is a weak equivalence: S η S ∈ W (cid:47) (cid:47) Sing ( | S | ) . (342)Notice that, a priori, the characterization of Quillen equivalences (Lemma A.27) only says, with Example A.32,that the derived adjunction unit, hence the composite S η S (cid:47) (cid:47) Sing ( | S | ) Sing ( | j | S | | ) (cid:47) (cid:47) Sing ( P | S | ) is a weak equivalence, where j | S | is a Kan fibrant replacement for | S | . But since all topological spaces are fibrant(Example A.7), the above simpler condition follows. Example A.36 (Homotopy types of manifolds via triangulations) . For X ∈ TopologicalSpaces equipped with thestucture of an n -manifold, there exists a triangulation of X , namely an n -skeletal simplicial set (Prop. A.34)Tr ( X ) ∈ SimplicialSets , sk n (cid:0) Tr ( X ) (cid:1) = Tr ( X ) (343)equipped with a homeomorphism to X out of its geometric realization (337) | Tr ( X ) | p homeo (cid:47) (cid:47) X , (344)Since the inclusion Tr ( X ) (cid:31) (cid:127) η Tr ( X ) ∈ W (cid:47) (cid:47) Sing (cid:0) | Tr ( X ) | (cid:1) Sing ( p ) ∈ Iso (cid:47) (cid:47)
Sing ( X ) , (345)is a weak equivalence (by Example A.35), the triangulation represents the homotopy type (338) of the manifold. Proposition A.37 (Homotopy classes of maps out of n -manifolds) . Let X ∈ TopologicalSpaces admit the stuctureof an n-manifold. Then for any A ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) (Example A.33) the homotopy classes of mapsX (cid:47) (cid:47) A are in natural bijection to the homotopy classes into the ( n − ) -truncation (340) of A: Ho (cid:0) TopologicalSpaces Qu (cid:1)(cid:0) X , A (cid:1) (cid:39) Ho (cid:0) TopologicalSpaces Qu (cid:1)(cid:0) X , A ( n − ) (cid:1) (346) Proof.
Consider the following sequence of natural isomorphismsHo (cid:0)
TopologicalSpaces Qu (cid:1)(cid:0) X , A (cid:1) (cid:39) Ho (cid:0) SimplicialSets Qu (cid:1)(cid:0) Sing ( X ) , Sing ( A ) (cid:1) (cid:39) Ho (cid:0) SimplicialSets Qu (cid:1)(cid:0) Tr ( X ) , Sing ( A ) (cid:1) (cid:39) Ho (cid:0) SimplicialSets Qu (cid:1)(cid:16) sk n (cid:0) Tr ( X ) (cid:1) , Sing ( A ) (cid:17) (cid:39) Ho (cid:0) SimplicialSets Qu (cid:1)(cid:16) Tr ( X ) , cosk n (cid:0) Sing ( A ) (cid:1)(cid:17) (cid:39) Ho (cid:0) TopologicalSpaces Qu (cid:1)(cid:16)(cid:12)(cid:12) Tr ( X ) (cid:12)(cid:12) , (cid:12)(cid:12) cosk n (cid:0) Sing ( A ) (cid:1)(cid:12)(cid:12)(cid:17) (cid:39) Ho (cid:0) TopologicalSpaces Qu (cid:1)(cid:0) X , A ( n − ) (cid:1) . Here the first step is (A.33), using, with Example A.21, that all topological spaces are fibrant and all simplicial setscofibrant. The second step uses (345). The third step is (343). Using that we do not need to fibrantly replace theskeleton in the domain, by Prop. A.16, the fourth step is the skeleta-adjunction (339). The fifth step is the reverseof the first step, with the same argument on (co-)fibrancy. The last step uses (344) in the first argument and (340)in the second. The composite of these isomorphisms is the desired (346).
Proposition A.38 (Postnikov tower [GJ99, Cor. 3.7]) . Let X ∈ Ho (cid:0) TopologicalSpaces Qu (cid:1) (Example A.33). If X isconnected, then its sequence of n-truncations (340) forms a system of fibrations with homotopy fibers (Def. A.22)the Eilenberg-MacLane spaces (10) of the homotopy group in the given degree: .. (cid:15) (cid:15) K ( π ( X ) , ) hfib ( p X ) (cid:47) (cid:47) X ( ) p X (cid:15) (cid:15) K ( π ( X ) , ) hfib ( p X ) (cid:47) (cid:47) X ( ) p X (cid:15) (cid:15) K ( π ( X ) , ) hfib ( p X ) (cid:47) (cid:47) X ( ) p X (cid:15) (cid:15) X ( ) . If X is not connected then this applies to each of its connected components.
Stable model categories.Example A.39 (Looping/suspension-adjunction) . On the category of pointed topological spaces, equipped withthe coslice model structure under the point (Example A.10) of the classical model structure (Example A.7), theoperation of forming based loop spaces Ω X : = Maps ∗ / ( S , X ) is the right adjoint in a Quillen adjunction (Def.A.17) TopologicalSpaces ∗ / Qu (cid:111) (cid:111) ΣΩ ⊥ Qu (cid:47) (cid:47) TopologicalSpaces ∗ / Qu (347)whose left adjoint is the reduced suspension operation Σ X : = S ∧ X : = (cid:0) S × X (cid:1) / (cid:0) S × {∗ X } (cid:116) {∗ S } × X (cid:1) . Example A.40 (Stable model category of sequential spectra [BF78][GJ99, §X.4]) . There exists a model category(Def. A.3) SequentialSpectra BF whose objects are sequences E : = (cid:110) E n ∈ TopologicalSpaces , Σ E n σ n (cid:47) (cid:47) E n + (cid:9) n ∈ N of topological spaces E n and continuous function σ n from their suspension Σ E n (Example A.39) to the next spacein the sequences; and whose morphisms E f (cid:47) (cid:47) F are sequences of component maps E n f n (cid:47) (cid:47) F n that commute withthe σ s. Moreover:W – weak equivalences are the morphisms that induce isomorphisms on all stable homotopy groups π • ( X ) : = lim −! n π • + k ( X k ) (where the colimit is formed using the σ ’s);Cof – cofibrations are those morphisms E f (cid:47) (cid:47) F such that the maps E f ∈ Cof (cid:47) (cid:47) F and ∀ n ∈ N E n + (cid:116) Σ E n Σ F n ( f n + , σ Fn ) ∈ Cof (cid:47) (cid:47) F n + are cofibrations in the classical model structure on topological spaces (Example A.7).Fib – Fibrant objects are the Ω -spectra , namely those sequences of spaces { E n } for which the Σ (cid:97) Ω -adjunct (347)of each σ n is a weak equivalence: (cid:110) E n ∈ TopologicalSpaces ∗ / Qu , E n ˜ σ n ∈ W (cid:47) (cid:47) Ω E n + (cid:111) n ∈ N (348) Example A.41 (Derived stabilization adjunction) . The suspension/looping Quillen adjunction on pointed spaces(Example A.39) extends to a commuting diagram of Quillen adjunctions (Def. A.17) to and on the stable modelcategory of spectra (Example A.40)TopologicalSpaces ∗ / Qu (cid:111) (cid:111) ΣΩ ⊥ Qu (cid:47) (cid:47) Σ ∞ (cid:97) Qu (cid:15) (cid:15) (cid:79) (cid:79) Ω ∞ TopologicalSpaces ∗ / Qu Σ ∞ (cid:97) Qu (cid:15) (cid:15) (cid:79) (cid:79) Ω ∞ SequentialSpectra BF (cid:111) (cid:111) ΣΩ ⊥ Qu (cid:47) (cid:47) SequentialSpectra BF . (349)100uch that the bottom adjunction is a Quillen equivalence (Def. A.28), hence such that under passage to derivedadjunctions (Prop. A.20)Ho (cid:16) TopologicalSpaces ∗ / Qu (cid:1) (cid:111) (cid:111) L Σ R Ω ⊥ (cid:47) (cid:47) L Σ ∞ (cid:97) (cid:15) (cid:15) (cid:79) (cid:79) R Ω ∞ Ho (cid:16) TopologicalSpaces ∗ / Qu (cid:1) L Σ ∞ (cid:97) (cid:15) (cid:15) (cid:79) (cid:79) R Ω ∞ Ho (cid:16) SequentialSpectra BF (cid:1) (cid:111) (cid:111) L Σ R Ω (cid:39) (cid:47) (cid:47) Ho (cid:16) SequentialSpectra BF (cid:1) (350)the bottom adjunction is an equivalence, thus exhibiting the homotopy category of spectra as being stable underlooping/suspension.We say that (i) Ho (cid:0) SequentialSpectra BF (cid:1) is the stable homotopy category of spectra; (ii) the vertical adjunction ( L Σ ∞ (cid:97) R Ω ∞ ) is the stabilization adjunction between homotopy types (338) and spectra. (iii) the images of Σ ∞ are the suspension spectra . (iv) For E ∈ Ho (cid:0) SequentialSpectra BF (cid:1) and n ∈ N we write (for brevity and in view of (348)) E n : = R Ω ∞ (cid:0) ( L Σ ) n E (cid:1) ∈ Ho (cid:0) TopologicalSpaces ∗ / Qu (cid:1) (351)for the homotopy type of the n th component space of the spectrum. Smooth ∞ -stacks.Definition A.42 (Simplicial presheaves over Cartesian spaces) . We write (i)
CartesianSpaces : = (cid:8) R n smooth (cid:47) (cid:47) R n (cid:9) n i ∈ N (352)for the category whose objects are the Cartesian spaces R n , for n ∈ N , and whose morphisms are the smooth functions between these (hence the full subcategory of SmoothManifolds on the Cartesian spaces). (ii) PSh (cid:0)
CartesianSpaces , SimplicalSets (cid:1) : = Functors (cid:0)
CartesianSpaces op , SimplicalSets (cid:1) (353)for the category of functors from the opposite of CartesianSpaces (352) to SimplicialSets,
Example A.43 (Model structure on simplicial presheaves over Cartesian spaces [Du98][Du01][FSSt10, §A]) . Thecategory of simplicial presheaves over Cartesian spaces (Prop. 353) carries the following model category structures(Def. A.3): (i)
The global projective model structurePSh (cid:0)
CartesianSpaces , SimplicalSets (cid:1) proj (354)whoseW – weak equivalences are the morphisms which over each R n are weak equivalence in SimplicialSets Qu (Exam-ple A.8),Fib – fibrations are the morphisms which over each R n are fibrations in SimplicialSets Qu (Example A.8), (ii) The local projective model structureSmoothStack ∞ : = PSh (cid:0)
CartesianSpaces , SimplicalSets (cid:1) projloc (355)whose:W – weak equivalences are the morphisms whose stalk at 0 ∈ R n is a weak equivalence in SimplicialSets Qu (Example A.8), for all n ∈ N ;Cof – cofibrations are the morphisms with the left lifting property (309) against the class of morphisms which overeach R n are in Fib ∩ W of SimplicialSets Qu . 101 efinition A.44 (Homotopy category of smooth ∞ -stacks) . We writeHo ( SmoothStacks ∞ ) : = Ho (cid:16) PSh (cid:0)
CartesianSpaces , SimplicalSets (cid:1) projloc (cid:17) (356)for the homotopy category (Def. A.14) of the local projective model category of simplicial presheaves overCartesianSpaces (Example A.43). We say that the objects of Ho ( SmoothStacks ∞ ) (356) are smooth ∞ -stacks .For exposition of smooth ∞ -stack theory see [FSS12b, §2][FSS13a][SS20a, §1]. In particular, notice: Example A.45 (Smooth manifolds as smooth ∞ -stacks) . For X ∈ SmoothManifolds it is incarnated as a smooth ∞ -stack (Def. A.44) by the rule X = (cid:16) R n (cid:0) ∆ [ k ] SmoothManifolds ( R n , X ) (cid:1)(cid:17) . (357)This construction constitutes to a full embeddingSmoothManifolds (cid:31) (cid:127) (cid:47) (cid:47) Ho ( SmoothStacks ∞ ) of smooth manifolds into smooth ∞ -stacks. Lemma A.46 ( ∞ -Stackification preserves finite homotopy limits) . The identity functors constitute a Quillen ad-junction (Def. A.17) between the local and the global projective model categories of Example A.43:
PSh (cid:0)
CartesianSpaces , SimplicalSets (cid:1) projloc (cid:111) (cid:111) idid ⊥ Qu (cid:47) (cid:47) PSh (cid:0)
CartesianSpaces , SimplicalSets (cid:1) proj . Moreover, this is such that the derived left adjoint functor (Prop. A.20)L loc : Ho (cid:16)
PSh (cid:0)
CartesianSpaces , SimplicalSets (cid:1) proj (cid:17) L id (cid:47) (cid:47) Ho ( SmoothStacks ∞ ) (358) (the ∞ -stackification functor) preserves homotopy pullbacks (Def. A.23). Proposition A.47 (Shape Quillen adjunction [Sch13, Prop. 4.4.8][SS20a, Example 3.18]) . We have a Quillenadjunction (Def. A.17)
PSh (cid:0)
CartesianSpaces , SimplicalSets (cid:1) projloc
Shp (cid:47) (cid:47) (cid:111) (cid:111)
Disc ⊥ Qu SimplicialSets Qu between the projective local model structure on simplicial presheaves over CartesianSpaces (Example A.43) andthe classical model structure on simplicial sets (Example A.8), hence a derived adjunction (Prop. A.20) betweenhomotopy category of ∞ -stacks (Def. A.44) and the classical homotopy category (Example A.33) Ho ( SmoothStacks ∞ ) L Shp (cid:47) (cid:47) (cid:111) (cid:111) R Disc ⊥ Qu Ho (cid:0) SimplicialSets Qu (cid:1) whose (underived) right adjoint sends a simplicial set to the presheaf which is constant on that simplicial set: Disc ( S ) : = const ( S ) : (cid:0) R n S (cid:1) . (359) Homological algebra.Example A.48 (Projective model structure on connective chain complexes [Qu67, §II.4 (5.)]) . The categoryChainComplexes ≥ Z of connective chain complexes of abelian groups (i.e. concentrated in non-negative degreeswith differential of degree -1) carries a model category structure (Def. A.3) whoseW – weak equivalences are the quasi-isomorphisms (those inducing isomorphisms on all chain homology groups)Fib – fibrations are the positive-degree wise surjectionsCof – cofibrations are the morphisms with degreewise injective kernels.We write (cid:0) ChainComplexes ≥ Z (cid:1) proj for this model category.More generally: Example A.49 (Projective model structure on presheaves of connective chain complexes [Ja03, p. 7]) . The cate-gory of presheaves of connective chain complexes over CartesianSpaces (352) carries the structure of a model cat-egory whose weak equivalences and fibrations are objectwise those of (cid:0)
ChainComplexes ≥ Z (cid:1) proj (Example A.48).We write PSh (cid:0) CartesianSpaces , ChainComplexes ≥ Z (cid:1) proj for this model category.102 roposition A.50 (Dold-Kan correspondence [Do58, Thm 1.9][Kan58][GJ99, §III.2][SSh03a, §2.1]) . Given A • ∈ SimplicialAbelianGroups , its normalized chain complex is the connective chain complex of abelian groups (Ex-ample A.48) which in degree n ∈ N is the quotient of A n by the degenerate cells and whose differential is thealternating sum of the face maps:N ( A ) • : = (cid:110) N ( A ) n : = A n / σ ( A n + ) , ∂ n : = n ∑ i = ( − ) i d i : N ( A ) n (cid:47) (cid:47) N ( A ) n − (cid:111) n ∈ N ∈ ChainComplexes ≥ Z . (360) (i) This construction constitutes an adjoint equivalence of categories
ChainComplexes ≥ Z (cid:111) (cid:111) N (cid:39) (cid:47) (cid:47) SimplicialAbelianGroups (361) (ii) such that simplicial homotopy groups of A ∈ SimplicialAbelianGroups ! SimplicialSet are identified withchain homology groups of the normalized chain complex ([GJ99, Cor. III.2.5]): π • ( A ) (cid:39) H • ( NA ) . (362) Example A.51 (Model structure on simplicial abelian groups [Qu69, §III.2][SSh03a, §4.1]) . The category ofSimplicialAbelianGroups carries a model category structure (Def. A.3) whoseW – weak equivalences are the morphisms which are weak equivaleces as morphisms in SimplicialSets Qu (Ex-ample A.8)Fib – fibrations are the morphisms which are fibrations as morphisms in SimplicialSets Qu (Example A.8)In other words, this is the transferred model structure along the free/forgetful adjunction, which thus becomes aQuillen adjunction (Def. A.17):SimplicialAbelianGroup proj (cid:111) (cid:111) Z [ − ] ⊥ Qu (cid:47) (cid:47) SimplicialSets Qu . (363) Proposition A.52 (Dold-Kan Quillen equivalence [SSh03a, §4.1][Ja03, Lemma 1.5]) . With respect to the projec-tive model structure on connective chain complexes (Example A.48) and the projective model structure on sim-plicial abelian groups (Example A.51) the Dold-Kan correspondence (Prop. A.50) is a Quillen equivalence (Def.A.28): (cid:0)
ChainComplexes ≥ Z (cid:1) proj (cid:111) (cid:111) N (cid:39) Qu (cid:47) (cid:47) SimplicialAbelianGroups proj , (364) where both functors preserve all three classes of morphims, Fib , Cof and W , separately. Example A.53 (Dold-Kan construction [FSSt10, §3.2.3][FSS12b, §2.4]) . i)
We write DK for the total right adjointin the composite of the free Quillen adjunction (363) and the Dold-Kan equivalence (364): (cid:0)
ChainComplexes ≥ Z (cid:1) proj (cid:111) (cid:111) N (cid:39) Qu (cid:47) (cid:47) DK (cid:50) (cid:50) SimplicialAbelianGroups proj (cid:111) (cid:111) Z [ − ] ⊥ Qu (cid:47) (cid:47) SimplicialSets Qu . (365) ii) This extends to a right Quillen functor on global projective model categories of presheaves (Example A.43,Example A.49). whose right derived functor (Prop. A.20) R DK composed with the ∞ -stackification functor (358)is thus of the formHo (cid:16) PSh (cid:0)
CartesianSpaces , ChainComplexes ≥ Z (cid:1) proj (cid:17) derivedDold-Kan construction R DK (cid:47) (cid:47) ∞ -stackifiedDold-Kan construction (cid:44) (cid:44) Ho (cid:16) PSh (cid:0)
CartesianSpaces , SimplicialSets (cid:1) proj (cid:17) L loc ∞ -stackification (cid:15) (cid:15) Ho ( SmoothStacks ∞ ) and preserves homotopy pullbacks (by Lemma A.46). 103 xample A.54 (Projective model structure on unbounded chain complexes [Ho99, Thm. 2.3.11]) . The categoryChainComplexes Z of unbounded chain complexes of abelian groups carries a model category structure (Def. A.3)whose:W – weak equivalences are the quasi-isomorphisms;Fib – fibrations are the degreewise surjections.We write (cid:0) ChainComplexes Z (cid:1) proj for this model category. Proposition A.55 (Stable Dold-Kan construction) . The Dold-Kan construction (Def. A.53) lifts along the stabi-lization adjunction (Example A.41) from connective to unbounded chain complexes (Example A.54), such as tomake the following diagram commute: Ho (cid:16)(cid:0) ChainComplexes ≥ Z (cid:1) proj (cid:17) Dold-Kan correspondence R DK (cid:44) (cid:44) (cid:39) (cid:47) (cid:47) (cid:95)(cid:127) (cid:97) (cid:15) (cid:15) (cid:79) (cid:79) R Ω ∞ Ho (cid:0) SimplicialAbelianGroups proj (cid:1) (cid:47) (cid:47) Ho (cid:0) SimplicialSets Qu (cid:1) L Σ ∞ (cid:97) (cid:15) (cid:15) (cid:79) (cid:79) R Ω ∞ Ho (cid:16)(cid:0) ChainComplexes Z (cid:1) proj (cid:17) R DK st stable Dold-Kan construction (cid:50) (cid:50) (cid:39) (cid:47) (cid:47) Ho (cid:0) ( H Z ) ModuleSpectra (cid:1) (cid:47) (cid:47) Ho (cid:0) SequentialSpectra BF (cid:1) . (366) Here the right adjoint on chain complexes is the homological truncation from below: R Ω ∞ (cid:16) · · · ∂ −! V ∂ −! V ∂ −! V ∂ − −! V − ∂ − −! · · · (cid:17) = (cid:16) · · · ∂ −! V ∂ −! V ∂ −! ker ( ∂ − ) (cid:17) . (367) Proof. (i)
It is clear from inspection that the assignment (367) is right adjoint to the inclusion of connective chaincomplexes, so that we have a pair of adjoint functors (cid:0)
ChainComplexes Z (cid:1) proj (cid:111) (cid:111) (cid:63) (cid:95) Ω ∞ ⊥ Qu (cid:47) (cid:47) (cid:0) ChainComplexes ≥ Z (cid:1) proj . (368)Moreover, it is immediate that this is a Quillen adjunction (Def. A.17) between the projective model structure onconnective chain complexes (Example A.48) and that on unbounded chain complexes (Example A.54): Ω ∞ clearlypreserves fibrations (using that those between connective chain complexes need to be surjective only in positivedegrees!) and clearly preserves all weak equivalences. Finally, since all chain complexes in the projective modelstructure are fibrant, we have that with Ω ∞ also R Ω ∞ is given by (367), via Example A.21. (ii) A Quillen adjunction of the form (cid:0)
ChainComplexes Z (cid:1) proj (cid:111) (cid:111) H ⊥ (cid:39) (cid:47) (cid:47) DK st (cid:50) (cid:50) ( H Z ) ModuleSpectra (cid:111) (cid:111) ⊥ Qu (cid:47) (cid:47) SequentialSpectra BF (369)is established in [SSh01, §B.1], where (a) the first step is a Quillen equivalence (Def. A.28) between the projective model structure on unbounded chaincomplexes (Example A.54) and a model category of module spectra over the Eilenberg-MacLane spectrum H Z [SSh01, §B.1.11]; (b) the second step is a Quillen adjunction [SSh01, p. 37, item ii)] to the Bousfield-Friedlander model structure(Example A.40) whose right adjoint assigns underlying sequential spectra; such that (c) the composite right adjoint DK st (369) further composed with Ω ∞ on spectra (349) equals the composite of Ω ∞ on chain complexes (368) with the unstable Dold-Kan construction (365): Ω ∞ ◦ DK st (cid:39) DK ◦ Ω ∞ (by immediate inspection of the construction in [SSh01, p. 38-39]). (iii) By uniqueness of adjoints, this implies that the Quillen adjunction of the stable Dold-Kan construction (369)is intertwined by the Quillen adjunctions involving Ω ∞ with the Quillen adjunction of the unstable Dold-Kanconstruction (365), and hence the commuting diagram of derived functors (A.55) follows (Prop. A.20).104 eferences [Ad62] J. F. Adams, Vector fields on spheres , Bull. Amer. Math. Soc. (1962), 39-41,[ euclid:bams/1183524456 ].[Ad75] J. F. Adams, Stable homotopy and generalized homology , The University of Chicago Press, 1974,[ ucp:bo21302708 ].[Ad78] J. F. Adams,
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