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November 23, 2007


Posted by Urs Schreiber

I am thinking about the notion of concordance of 2-coycles used by Baas, Bökstedt and Kro.

It seems more or less obvious how it is related to transformations of the ana-2-functors involved, but it also seems that there is a big interesting issue lurking here, once we try to get serious about thinking about ω\omega-anafunctors.

Influenced by the discussion about Sjoerd Crans’s generalization of the Gray tensor product to ω\omega-categories with Todd Trimble in Extended Worldvolumes, I had the following thoughts (unfinished, need to catch a train right now)

Homotopy, Concordance and Natural Transformation

Abstract. Transformations between (ω\omega-)functors are like homotopies between maps of topological spaces. This statement can be given a precise meaning using the closed structure of ωCat\omega\mathrm{Cat} in terms of the extension of the Gray tensor product from 2-categories to ω\omega-categories given by Sjoerd Crans. The analogous construction is familar in homological algebra from categories of chain complexes.

After recalling the basics, we turn to “anafunctors” (certain spans of functors) and highlight how the general relation between transformations, homotopies and concordances appears in the study of nonabelian nn-cocycles classifying nn-bundles.

One way to state the question that this document is concerned with is this:

Is there a definition of ana-ω\omega-functors such that the biclosed structure of ωCat\omega\mathrm{Cat} extends to a biclosed structure of ωCat ana\omega\mathrm{Cat}_{\mathrm{ana}}, where in the latter we take morphisms not to be ω\omega-functors but ana-ω\omega-functor?

A more gentle way to state what this document is concerned with is this:

A is a map h:X×IY h : X \times I \to Y which restricts on the two ends of the interval II to two fixed maps.

A natural transformation on the other hand is a functor h˜:Chom(I,D), \tilde h : C \to \mathrm{hom}(I,D) \,, where now II is the category with one single nontrivial morphisms.

Using adjointness of the internal hom with a tensor product, both pictures coincide: Hom(C,hom(I,D))Hom(CI,D). \mathrm{Hom}(C,\mathrm{hom}(I,D)) \simeq \mathrm{Hom}(C \otimes I, D) \,. For this to be true for higher categories, \otimes must not be the cartesian product, ×\times, but the Gray tensor product Gray\otimes_{\mathbf{Gray}} and its generalization to ω\omega-categories given by Sjord Crans: like the product of an nn-dimensional space with an mm-dimensional space is an (n+mn+m)-dimensional space, the \otimes-product of an nn-category with an mm-category is an (n+m)(n+m)-category.

Often, morphisms from XX to YY, need to be taken as “generalized morphisms”, namely spans Y f D C \array{ Y &\stackrel{f}{\to}& D \\ \downarrow^\simeq \\ C } where ff starts not at CC itself, but on a cover of CC.

Such morphisms have been termed anafunctors by Makkai, being closely related to profunctors, Morita morphisms and the like. An archetypical example for an anafunctor is a GG-cocycle classifying a principal GG-bundle, for GG some nn-group, to which we come in moment.

A homotopy between two anafunctors has to take the choice of cover into account. A concordance is a span Y^ f D C×I \array{ \hat Y &\stackrel{f}{\to}& D \\ \downarrow^\simeq \\ C \times I } which restricts to two given anafunctors over the endpoints of II.

Here I want to eventually talk about how concordance of anafunctors relates to natural transformations of anafunctors, as given originally by Makkai.

One motivation is to clarify how the notion of concordance of 2-bundles used in Baas-Bökstedt-Kro relates precisely to other natural notions of morphisms of 2-bundles, like those used in [X,Y,Z (see the references in the pdf)]:

given a space XX and given an nn-category SS, a (“nonabelian”“) SS-cocycle on XX is (this observation, which feels tautologous nowadays, has apparently first been made long ago by Roberts) a choice of regular epimorphism π:YX \pi : Y \to X together with an nn-functor g:Y S, g : Y^\bullet \to S \,, where Y Y^\bullet is the obvious nn-groupoid obtained from pp.

For instance when S=BG={g|gG}S = \mathbf{B}G = \left\lbrace \bullet \stackrel{g}{\to} \bullet | g \in G\right\rbrace is the one-object groupoid obtained from a group GG, then a functor g:Y BG g : Y^\bullet \to \mathbf{B}G is precisely a labelling of points in Y× XYY \times_X Y by elements in GG, such that the cocycle condition π 23 *gπ 12 *g=π 13 *g \pi_{23}^* g \cdot \pi_{12}^* g = \pi_{13}^* g familiar from principal GG-bundles is satisfied.

Analogously, by letting S=BAUT(G)S = \mathbf{B} \mathrm{AUT}(G), for AUT(G)\mathrm{AUT}(G) the automorphism \emph{2-group} of an ordinary group GG, one obtains the 2-cocycle classifiying a GG-gerbe \cite{X,Y,Z}.

In these cases, there is a global notion of the structure being classified by the cocycle. The cocycle itself, including the choice of cover it involves, is part of the descent data which describes the descent of a trivial nn-bundle on the cover down to a possibly nontrivial nn-bundle on the base.

Correspondingingly, there is then little choice for the right notion of morphisms of nn-cocycles: whatever these are, they need to reproduce the morphisms between the global objects that they come from.

Still, there is quite some leeway in making the details precise, as nn increases above n=1n=1. For some time this issue had found attention mainly in the context of \emph{bundle gerbes}:

a line bundle gerbe is, in our language used here, a smooth 2-functor g:Y B1dVect. g : Y^\bullet \to \mathbf{B}1d\mathrm{Vect} \,. Here B1dVect2Vect\mathbf{B}1d\mathrm{Vect} \subset 2\mathrm{Vect} is the 2-category with a single object such that the End-category of that object is the category of 1-dimensional vector spaces, with horizontal composition being the tensor product over the ground field.

While it is in principle clear how morphisms between bundle gerbes over different covers YY should behave (see the review of anafunctors in section \ref{anafunctors}), there are some subtleties involved in spelling this out.

Seeing exactly how the notion of concordance fits into this picture, which has been done to great effect in BBT, should be helpful.

By the above considerations, something like the following should be true:

Concordance is what becomes of transformations in the joint context of anafunctors and the closed structure on ω\omegaCat.

Since, at the time of this writing, it is Friday evening and I need to catch a train in a moment, the following is unfinished. But I guess the main point should already be visible.

Posted at November 23, 2007 5:49 PM UTC

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