Schreiber
omega-anafunctor

In view of the folk model structure on omega-categories the notion of anafunctor has a straightforward generalization from 1-categories to $\omega$-categories:

Definition

For $C$ and $D$ $\omega$-categories, an $\omega$-anafunctor $g:C\to D$ is a span

$$\begin{array}{ccc}\hat{C}& \stackrel{g}{\to }& D\\ \downarrow \\ C\end{array}$$\array{ \hat C &\stackrel{g}{\to}& D \\ \downarrow \\ C }

of $\omega$-functors whose left leg is an acyclic fibraiton with respect to the folk model structure, i.e. an $\omega$-functor which is k-surjective for all $k\in \mathbb{N}$, i.e. a hypercover.

Given two $\omega$-anafunctors $C\stackrel{g}{\to }D$ and $D\stackrel{h}{\to }E$ their composite $\omega$-anafunctor is given by the span

$$\begin{array}{ccccc}{g}^{*}\hat{D}& \to & D& \stackrel{h}{\to }& E\\ \downarrow & & \downarrow \\ \hat{C}& \stackrel{g}{\to }& D\\ \downarrow \\ C\end{array},$$\array{ g^* \hat D &\to& D &\stackrel{h}{\to}& E \\ \downarrow && \downarrow \\ \hat C &\stackrel{g}{\to}& D \\ \downarrow \\ C } \,,

where the top left square is a pullback square. Notice that acyclic fibrations are preserved under pullback and closed under composition, so that the total vertical $\omega$-functor on the left is indeed again an acyclic fibration and hence the total span here indeed again an $\omega$-anafunctor.

While this definition of composition is in itself well defined, it is not quite associative: the two ways of composing three spans representing $\omega$-anafunctors tis way in general produces two spans out of different – albeit isomorphic – hypercovers.

Of course the precise choice of hypercover $\hat{C}$ of $C$ is not an essential datum: two $\omega$-anafuntors should be regarded as equivalent if they become equal after pulled back to a joint hypercover. This is formalized in the following definition:

category of $\omega$-categories and anafunctors

Define the category ${\mathrm{Ho}}_{\omega \mathrm{Categories}}$ – or just $\mathrm{Ho}$ for short – (the “_fat_ homotopy category”, to be distinguished from the true homotopy category) as follows:

• its objects are those of $\omega \mathrm{Categories}$;

• its morphisms are colimits over hypercovers of spans as above:

for $C$ an $\omega$-category, let $\mathrm{Covers}(C)$ be the category whose objects are acyclic fibrations $\hat{C}\to C$ and whose morphisms are commuting triangles

$$\begin{array}{ccccc}\hat{C}\prime & & \stackrel{r}{\to }& & \hat{C}\\ & \searrow & & \swarrow \\ & & C\end{array}.$$\array { \hat C' &&\stackrel{r}{\to} && \hat C \\ & \searrow && \swarrow \\ && C } \,.

Notice that by the two-out-of-three property of weak equiavlences this implies in particular that the horizonal morphism $r$ is a weak equivalence.

Given an $\omega$-anafunctor

$$\begin{array}{ccc}\hat{C}& \stackrel{g}{\to }& D\\ \downarrow \\ C\end{array}$$\array{ \hat C &\stackrel{g}{\to}& D \\ \downarrow \\ C }

with respect to the hypercover $\hat{C}\to C$ and given a morphism of covers as above, we obtain a new $\omega$-anafunctor with respect to the hypercover $\hat{C}\prime$ by pulling back along $r$, which just means that we precompose with $r$ to form:

$$(C\prime \stackrel{r^{*}g}{\to }D):=\begin{array}{ccccc}\hat{C}\prime & \stackrel{r}{\to }& \hat{C}& \stackrel{g}{\to }& D\\ & \searrow & \downarrow \\ & & C\end{array}.$$(C' \stackrel{r^* g}{\to} D ) \;\; := \;\; \array{ \hat C' &\stackrel{r}{\to}& \hat C &\stackrel{g}{\to}& D \\ &\searrow&\downarrow \\ && C } \,.

For fixed $D$ this yields a functor

$$\mathrm{Anafunc}(C,D):\mathrm{Covers}(C{)}^{\mathrm{op}}\to \mathrm{Spans}(\omega \mathrm{Categories}).$$Anafunc(C,D) : Covers(C)^{op} \to Spans(\omega Categories) \,.

So finally the morphisms of the category $\mathrm{Ho}$ are given by the colimit over this functor

$$\mathrm{Ho}(C,D):={\mathrm{colim}}_{\mathrm{Covers}(C{)}^{\mathrm{op}}}\mathrm{Anafunc}(C,D)$$\mathbf{Ho}(C,D) := colim_{Covers(C)^{op}} Anafunc(C,D)

which we also write as

$$\mathrm{Ho}(C,D):={\mathrm{colim}}_{\hat{C}\in \mathrm{Covers}(C{)}^{\mathrm{op}}}\begin{array}{ccc}\hat{C}& \stackrel{g}{\to }& D\\ \downarrow \\ C\end{array}.$$\mathbf{Ho}(C,D) := colim_{\hat C \in Covers(C)^{op}} \array{ \hat C &\stackrel{g}{\to}& D \\ \downarrow \\ C } \,.

Composition in $\mathrm{Ho}$ is given by the composition of representative spans as defined above: in the colimit the dependence on the choice of cover vanishes and hence the composition becomes associative and unital.

$\omega$-category valued hom on the category of $\omega$-categories and anafunctors

The above 1-category of $\omega$-anafunctors is only the first step towards a model for a proper $\mathrm{\infty }$-category of $\omega$-categories and $\omega$-anafunctors between them.

As first step to modelling this full $\mathrm{\infty }$-structure, we can consider the following functor which is to be thought of as assigning to any two $\omega$-categories the $\omega$-category of $\omega$-anafunctors between them:

(– my notebook battery is low… need to recharge… –)

References

see also Differential Nonabelian Cohomology and Nonabelian cocycles and their quantum symmetries.