### Question about ∞-Colimits

#### Posted by Urs Schreiber

Here is a question on a certain peculiar configuration of $(\infty,1)$-colomits also kown as homotopy colimits, where one hocolimits is taken *over another* hocolimit:

Suppose $\Pi(X)$ is an $\infty$-groupoid and $A$ an $(\infty,1)$-category and $F : \Pi(X) \to A$ is an $(\infty,1)$-functor of which we would like to compute some push-forward, such as, for simplicity, its $(\infty,1)$-colimit

$hocolim ( \Pi(X) \stackrel{F}{\to} A) \in A \,.$

But let there be the following extra piece of datum: say the $\infty$-groupoid $\Pi(X)$ itself comes to us exhibited as a homotopy colimit in $\infty$-Grpd of a simplicial $\infty$-groupoid $\Delta^{op} \to \infty Grpd$ which I’ll denote $[\Delta^k_C,X]$ for reasons to be explained below: so suppose that we have a hocolimit expression

$\Pi(X) \simeq hocolim_k [\Delta^k_C, X] \,.$

This is a peculiar higher categorical situation which does not have a 1-categorical analog: we have a hocolimit over a functor $F$ out of a domain that is itself a hocolimit:

$hocolim(F) = hocolim( (hocolim_k [\Delta^k_C, X]) \stackrel{F}{\to} A ) \,.$

**Question**: Are there any general useful facts one can know about such “hocolimits over hocolimits”?

In particular, let $F_k : [\Delta^k_C ,X] \to A$ be the cocone components of $F$ induced by the universality property of the colimit. Is there an expression of the hocolim over $F$ in terms of that over its components?

By looking at the relevant diagrams, one is inclined to expect that – to some maybe imperfect extent – there might be a certain commutativity rule of the sort

$\cdots \stackrel{?}{\simeq} hocolim_k ( hocolim( [\Delta^k_C ,X ] \stackrel{F_k}{\to} A ) ) \,,$

where now the *inner* holcolim is over the component functors $F_k$ of $F$, and the outer hocolim is over a resulting diagram in $A$.

Here the trouble starts with saying precisely what that diagram in $A$ formed by the $hocolim_k F_k$ actually is. Below is considered as one special case the case where all $F_k$ are constant on a single object, in which case there is a simple answer.

To see why one might expect some such “hierarchical commutativity” rule for hocolimits or similar, maybe it is helpful to contemplate the relevant diagram in low degree of $k$, where it looks like

Here on the right is indicated the functor $F : \Pi(X) \to A$ and the way it factors into components maps as a morphism out of a hocolimit.

Then to the left is indicated how $F$ itself as well as its components maps may be pushed forward to the point, meaning that their hocolimit, denoted $\lim_\to$ there, are computed.

It seems from this that the colimiting cocones over the component functors $F_k$ want to assemble themselves to yield some kind of cocone over $F$, much as the components $F_k$ themselves assemble to yield $F$. But so far I keep getting a headache when trying to extract a precise statement along these lines.

By the way: in the above the map from $\Pi(X)$ to the point is factored in the curious way as indicated due to the way this question arises in the discussion of Quantization by Kan extension. More background information on that motivational aspect in the following.

In the motivating example we are in the context of Lie $\infty$-groupoids modeled by $\infty$-stacks on $C =$CartSp or a similar category of smooth test spaces.

Write $[-,-]$ for the internal Hom in that context.

Write $\Delta^k_C$ for the standard smooth $k$-simplex regarded as a smooth $\infty$-groupoid.

Then, in our example, we are presented with a smooth $\infty$-groupoid $X$ that plays the role of some physical space

To every such smooth $\infty$-groupoid $X$ we get the smooth $\infty$-groupoid $[\Delta^k_C,X]$ of smooth $k$-paths in $X$. These fit together to form the smooth path $\infty$-groupoid defined by

$\Pi(X) := \mathrm{hocolim}_k [\Delta^k_C,X] \,.$

Let now $A$ be an $(\infty,n)$-category whose underlying $\infty$-groupoid $Core(A)$ is equipped with the structure of a smooth $\infty$-groupoid. We shall abusively write $A$ for $Core(A)$, too, when the context makes the distinction clear.

We know that morphisms

$F : \Pi(X) \to A$

of smooth $\infty$-groupoids characterize $\infty$-bundles with flat connection on $X$, whose typical fiber is an equivalence class in $A$. Sometimes one calls these $A$-local systems on $X$.

We want to regard one of these as the background field for a charged brane $\sigma$-model and find an abstract-nonsense way to characterize the worldvolume quantum field theory as a functor on the $\infty$-category of cobordisms

$Z_F : Bord_\infty \to A$

given the input of $F$. Here for simplicity I take $Bord_\infty$, which should be an $\infty$-groupoid. I regard this as a smooth $\infty$-groupoid with trivial smooth structure.

There is also the mixed version $Bord_\infty(X)$ of bordisms with smooth maps into $X$. Into that $\Pi(X)$ should embed as the sub $\infty$-groupoid of bordisms that are topologically just disks.

Now, one may imagine several abstract-nonsense operations to get from $F$ to $Z_F$.

One proposal is indicated in

Freed-Hopkins-Lurie-Teleman, Topological Quantum Field Theories from Compact Lie Groups.

It seems, but is a bit hard to tell, that essentially they are proposing for the
special case of *constant* $F = \mathrm{const}_a$
that for a cobordism $\Sigma_{\mathrm{in}} \to \Sigma \leftarrow \Sigma_{\mathrm{out}}$
with the induced diagram

$\array{ && [\Sigma,X] \\ & \swarrow && \searrow \\ [\Sigma_{in},X] &&\downarrow^{const_a}&& [\Sigma_{out}, X] \\ & {}_{const_a}\searrow && \swarrow_{const_a} \\ && A }$

we are to take the hocvolimit over each of the vertical constant functors

$\array{ && hocolim_{[\Sigma,X]} const_a \\ & \swarrow && \searrow \\ hocolim_{[\Sigma_{in},X]} const_a &&&& hocolim_{[\Sigma_{out}, X]} const_a }$

and then in turn the hocolimits over the resulting diagram in $A$ to produce

$hocolim_{k} ( \mathrm{hololim} (S_k \stackrel{\mathrm{const}_a}{\to} A ) \,,$

where now $k$ denotes some indexing of the above diagram and $S_k$ denotes the objects $[\Delta^\cdot_C,X]$, just to emphasize the pattern.

They discuss how if $A$ is a suitable $\infty$-catgeory of algebras, the object thus obtained is naturally a bimodule that may be regarded as a morphism from $\mathrm{hocolim}( [\Sigma_{\mathrm{in}}, X] \stackrel{\mathrm{const}_a}{\to} A)$ to $\mathrm{hocolim}( [\Sigma_{\mathrm{in}}, X] \stackrel{\mathrm{const}_a}{\to} A)$

Now, there is *another* abstract nonsense that one may consider subjecting the
functor $F$. It looks like it leads to something similar. But I am not sure
what the precise relation is. My **question** is, if anyone
can help me see the relation between the two different double hocolim constructions
presented below.

So consider the homotopy colimit over $F$ itself. Since $F$ maps out of an $\infty$-groupoid that is itself given by a homotopy colimit, it is of the form

$hocolim ( hocolim_k [\Delta^k_C,X] \stackrel{F}{\to} A) \,.$

So we can take the corresponding diagram apart into the components $F_k$ of $F$ as depicted here:

**Question**: what is the relation between

$\mathrm{hocolim}( (\mathrm{hocolim}_k [\Delta^k_C,X]) \stackrel{F}{\to} A )$

and

$\mathrm{hocolim}_k ( \mathrm{hocolim} ( [\Delta^k_C,X] \stackrel{F_k}{\to} A) )$

?

Actually, as the diagram is supposed to indicate, I don’t really expect that this is quite the right question yet. More precisely one should probably ask the question not for push-forwards to the point, but for the intermediate push-forwards indicated above.

But I’ll leave it at that level of vagueness for the moment and see if anyone can help me out with a useful comment on the material as indicated so far.

## Re: Question about ∞-Colimits

I think it does, just not quite so circularly. I don't know anything about it, but if there is anything to be known, there should be something known about colimits over categories that are themselves $2$-colimits in $Cat$.