The n-Category Café

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

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

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

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

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

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

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

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 $G$-cocycle classifying a principal $G$-bundle, for $G$ some $n$-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 $$\array{ \hat Y &\stackrel{f}{\to}& D \\ \downarrow^\simeq \\ C \times I }$$ which restricts to two given anafunctors over the endpoints of $I$.

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

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

Analogously, by letting $S = \mathbf{B} \mathrm{AUT}(G)$, for $\mathrm{AUT}(G)$ the automorphism \emph{2-group} of an ordinary group $G$, one obtains the 2-cocycle classifiying a $G$-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 $n$-bundle on the cover down to a possibly nontrivial $n$-bundle on the base.

Correspondingingly, there is then little choice for the right notion of morphisms of $n$-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 $n$ increases above $n=1$. 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 $$g : Y^\bullet \to \mathbf{B}1d\mathrm{Vect} \,.$$ Here $\mathbf{B}1d\mathrm{Vect} \subset 2\mathrm{Vect}$ 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 $Y$ 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 $\omega$Cat.

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.