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August 10, 2009

Question about ∞-Colimits

Posted by Urs Schreiber

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

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

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

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

Π(X)hocolim k[Δ C k,X]. \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 FF out of a domain that is itself a hocolimit:

hocolim(F)=hocolim((hocolim k[Δ C k,X])FA). 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:[Δ C k,X]AF_k : [\Delta^k_C ,X] \to A be the cocone components of FF induced by the universality property of the colimit. Is there an expression of the hocolim over FF 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

?hocolim k(hocolim([Δ C k,X]F kA)), \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 kF_k of FF, and the outer hocolim is over a resulting diagram in AA.

Here the trouble starts with saying precisely what that diagram in AA formed by the hocolim kF khocolim_k F_k actually is. Below is considered as one special case the case where all F kF_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 kk, where it looks like

Here on the right is indicated the functor F:Π(X)AF : \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 FF itself as well as its components maps may be pushed forward to the point, meaning that their hocolimit, denoted lim \lim_\to there, are computed.

It seems from this that the colimiting cocones over the component functors F kF_k want to assemble themselves to yield some kind of cocone over FF, much as the components F kF_k themselves assemble to yield FF. 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 Π(X)\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=C = CartSp or a similar category of smooth test spaces.

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

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

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

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

Π(X):=hocolim k[Δ C k,X]. \Pi(X) := \mathrm{hocolim}_k [\Delta^k_C,X] \,.

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

We know that morphisms

F:Π(X)A F : \Pi(X) \to A

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

We want to regard one of these as the background field for a charged brane σ\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 A Z_F : Bord_\infty \to A

given the input of FF. Here for simplicity I take Bord 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 (X)Bord_\infty(X) of bordisms with smooth maps into XX. Into that Π(X)\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 FF to Z FZ_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=const aF = \mathrm{const}_a that for a cobordism Σ inΣΣ out\Sigma_{\mathrm{in}} \to \Sigma \leftarrow \Sigma_{\mathrm{out}} with the induced diagram

[Σ,X] [Σ in,X] const a [Σ out,X] const a const a A \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

hocolim [Σ,X]const a hocolim [Σ in,X]const a hocolim [Σ out,X]const a \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 AA to produce

hocolim k(hololim(S kconst aA), hocolim_{k} ( \mathrm{hololim} (S_k \stackrel{\mathrm{const}_a}{\to} A ) \,,

where now kk denotes some indexing of the above diagram and S kS_k denotes the objects [Δ C ,X][\Delta^\cdot_C,X], just to emphasize the pattern.

They discuss how if AA is a suitable \infty-catgeory of algebras, the object thus obtained is naturally a bimodule that may be regarded as a morphism from hocolim([Σ in,X]const aA)\mathrm{hocolim}( [\Sigma_{\mathrm{in}}, X] \stackrel{\mathrm{const}_a}{\to} A) to hocolim([Σ in,X]const aA)\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 FF. 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 FF itself. Since FF maps out of an \infty-groupoid that is itself given by a homotopy colimit, it is of the form

hocolim(hocolim k[Δ C k,X]FA). hocolim ( hocolim_k [\Delta^k_C,X] \stackrel{F}{\to} A) \,.

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

Question: what is the relation between

hocolim((hocolim k[Δ C k,X])FA) \mathrm{hocolim}( (\mathrm{hocolim}_k [\Delta^k_C,X]) \stackrel{F}{\to} A )

and

hocolim k(hocolim([Δ C k,X]F kA)) \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.

Posted at August 10, 2009 8:39 PM UTC

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Re: Question about ∞-Colimits

This is a peculiar higher categorical situation which does not have a 11-categorical analog

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 22-colimits in CatCat.

Posted by: Toby Bartels on August 10, 2009 11:16 PM | Permalink | Reply to this

Re: Question about ∞-Colimits

This is a peculiar higher categorical situation which does not have a 1-categorical analog

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.

Yes, that’s what part of what I am saying: it doesn’t arise in a purely 1-categorical context, you need at least a notion of colimits over categories.

On the other hand, there may be more good 1-categorical analogs than it might seem on first sight:

the crucial fact that Charles Rezk kindly emphasizes in his comment below is that the hocolim over an (,1)(\infty,1)-presheaf F:XGrpdF : X \to \infty Grpd happens to coincide with what for 1-categorical presheaves would be its category of elements in that it is the pullback of the universal fibration of (,1)(\infty,1)-categories

Z| GrpdGrpd Z|_Grpd \to \infty Grpd

whose homotopy fiber over each CGrpdC \in \infty Grpd is just CC itself.

So we have a homotopy pullback

hocolimF Z| Grpd X F Grpd \array{ hocolim F &\to& Z|_{Grpd} \\ \downarrow && \downarrow \\ X &\stackrel{F}{\to}& \infty Grpd }

by the theorem summarized at limit in a quasi-category.

For a 1-categorical presheaf F:CSetF : C \to Set we would similarly take the universal SetSet-bundle Set *SetSet_* \to Set and consider the pullback

F Set * X F Set \array{ \int F &\to& Set_* \\ \downarrow && \downarrow \\ X &\stackrel{F}{\to}& Set }

which is the category of elements of FF.

(You, Toby, know all this, I am just saying it for completeness).

So a useful 1-categorical analog of my question should be obtained with one of the two colimit notions replaced by “pullback of universal fibration”.

Posted by: Urs Schreiber on August 11, 2009 1:29 PM | Permalink | Reply to this

Re: Question about ∞-Colimits

If I understand your question correctly, I think it is exactly correct. Here is the picture that I have of it:

For me, I want to thnik of the \infty-groupoid Π(X)\Pi(X) as just being a space XX. The functor F:XAF: X\to A I think of as a “bundle” of objects of AA over the space XX.

If AA is the (,1)(\infty,1)-category of spaces, this is literally true: the functor FF corresponds to a fibration E FXE_F\to X. In this case, the homotopy colimit of FF is the space E FE_F.

Now, I’ll suppose that XX is a homotopy colimit. Using simplicial homotopy colimits shouldn’t be crucial, so I’d rather take XX to be a homotopy colimit of a diagram of spaces X 1X 0X 2X_1 \leftarrow X_0 \rightarrow X_2.

The restriction of FF to X iXX_i\to X is a functor F i:X iAF_i: X_i\to A, and so corresponds to a fibration E F iX iE_{F_i}\to X_i. But this fibration is exactly the homotopy pullback of E FXE_F\to X along X iXX_i \to X; restriction of functors between \infty-groupoids is homotopy pullback of fibrations.

So I think your formula amounts to saying:E F=hocolim(E F 1E F 0E F 2). E_F = \mathrm{hocolim}(E_{F_1} \leftarrow E_{F_0} \rightarrow E_{F_2}).This is certainly true if A=A=spaces (it is an instance of the “descent” property of an (,1)(\infty,1)-topos).

Well, that argument assumed that A=A=spaces. But it certainly can be extended to the case when AA is an (,1)(\infty,1)-category of presheaves of spaces. I think you can prove your formula even if AA is a Bousfield localization of a presheaf (,1)(\infty,1)-category, which includes almost anything you’d like. But I’m too sleepy to check it right now …

Posted by: Charles Rezk on August 11, 2009 7:13 AM | Permalink | Reply to this

Re: Question about ∞-Colimits

Thanks for your reply!

If I understand your question correctly, I think it is exactly correct. […]

(it is an instance of the “descent” property of an (,1)(∞,1)-topos)

Thanks, that was the nudge that I needed.

Okay, so let me recall the relevant facts to see if I get them straight:

First of all, it is a theorem – now recalled at limit in a quasi-category – that for F:XGrpdF : X \to \infty Grpd an \infty-functor from an \infty-groupoid XX to the (,1)(\infty,1)-category of \infty-groupoids its (homotopy) colimit is equivalent to the pullback of the universal fibration of (,1)(\infty,1)-categories Z(,1)CatZ \to (\infty,1)Cat along FF, in that

hocolimF Z| Grpd X F Grpd \array{ hocolim F &\to& Z|_{\infty Grpd} \\ \downarrow && \downarrow \\ X &\stackrel{F}{\to}& \infty Grpd }

is a (homotopy) pullback.

Next, it is a big theorem by you – which appears as theorem 6.1.6.8 in HTT – that in a (Grothendieck-Rezk-Lurie-)(,1)(\infinity,1)-topos all colimits are universal (and in fact this is part of the characterization of them), which is a funny way of saying that they are stable under base change.

In particular this is true in the archetypical (,1)(\infty,1)-topos TopTop, where it should be a classical fact.

Here we use it with the above theorem to deduce the formula

hocolim((hocolim kC k)FD)hocolim k(hocolim(C k)F kD) hocolim ( (hocolim_k C_k) \stackrel{F}{\to} D) \simeq hocolim_k (hocolim (C_k) \stackrel{F_k}{\to} D)

from the fact that the pullback along the universal (,1)(\infty,1)-categorical fibration Z(,1)CatZ \to (\infty,1)Cat preserves colimits, so that

hocolimF hocolim k(hocolimF k) Z| Grpd X hocolim kX k F Grpd. \array{ hocolim F &\simeq& hocolim_k (hocolim F_k) &\to& Z|_{\infty Grpd} \\ \downarrow && \downarrow && \downarrow \\ X &\simeq & hocolim_k X_k &\stackrel{F}{\to}& \infty Grpd } \,.

So that clarifies the situation in the case where the functor in question has as domain an \infty-groupoid and as codomain the (,1)(\infty,1)-category of \infty-groupoids.

I think you can prove your formula even if AA is a Bousfield localization of a presheaf (,1)(\infty,1)-category, which includes almost anything you’d like. But I’m too sleepy to check it right now…

I’d be quite grateful if you could help me with this question indeed for the case that \infty-Grpd in the above is replaced with some (,1)(\infty,1)-category of (,1)(\infty,1)-sheaves, yes.

I am somewhat puzzled about where in that case I should think of the analog of the morphism F:XGrpdF : X \to \infty Grpd to actually live.

My other puzzlement at the moment is that for the applications that I indicated in the above entry it’s not really that the coefficient object is Grpd\infty Grpd or some other (,1)(\infty,1)-topos. It’s rather some incarnation of somthing that deserves to be called nVectn Vect or the like. Maybe Pr LPr^L.

I’d be grateful for whatever comments you might have.

Posted by: Urs Schreiber on August 11, 2009 2:03 PM | Permalink | Reply to this

Re: Question about ∞-Colimits

I have another question, this time on how to re-express hocolims over hocolims as hocolims over holims.

First, start with the setup as before. Let CC be a category, regarded as an (,1)(\infty,1)-category, and

H:CGrpd op H : C \to \infty Grpd^{op}

some (,1)(\infty,1)-functor.

Write X:=hocolimHX := hocolim H for its colimit.

Then consider another functor F:XGrpdF : X \to \infty Grpd and its hocolim.

As now recalled at limit in a quasi-category this yields in total a diagram of the form

hocolimF X=hocolimH F Grpd C H Grpd, \array{ hocolim F \\ \downarrow \\ X = hocolim H &\stackrel{F}{\to}& \infty Grpd \\ \downarrow \\ C &\stackrel{H}{\to}& \infty Grpd } \,,

where the two vertical morphisms are the coCartesian fibrations arising as the pullback of the opposite of the universal fibration of \infty-groupoids.

Now, as recalled at these links, coCartesian fibrations of course compose to coCartesian fibrations. Therefore the vertical composite in the above diagram is a coCartesian fibration. So it must come from some classifying (,1)(\infty,1)-functor

tot:CGrpd tot : C \to \infty Grpd

top yield a pullback diagram

hocolimX Z| Grpd op C tot Grpd \array{ hocolim X &\to& Z|_{\infty Grpd}^{op} \\ \downarrow && \downarrow \\ C &\stackrel{tot}{\to}& \infty Grpd } Question 1: Is there a formula for tottot?

By just looking at the diagram, there is an obvious guess for what tottot must roughly be like:

a given cc in CC it must send to the \infty-groupoid which is the union of all the sections of hocolimFXhocolim F \to X over all sections of XCX \to C over cc.

So we want to choose sections σ:CX\sigma : C \to X, postcompose these with FF and sum up the result of that.

hocolimF X=hocolimH F Grpd σ C = C H Grpd, \array{ &&hocolim F \\ &&\downarrow \\ &&X = hocolim H &\stackrel{F}{\to}& \infty Grpd \\ &{}^{\sigma}\nearrow&\downarrow \\ C&\stackrel{=}{\to}&C &\stackrel{H}{\to}& \infty Grpd } \,,

So i’d expect some sub (,1)(\infty,1)-category

Γ C sortof(X)Func(C,X) \Gamma_C^{sortof}(X) \to Func(C,X)

of the full functor category Func(C,X)Func(C,X) such that the sought for classifying morphism of the total vertical coCartesian fibration is the hocolim

hocolim(Γ C sortof(X)Func(C,X)Func(C,F)Func(C,Grpd))Func(C,Grpd). hocolim ( \Gamma^{sortof}_C(X) \to Func(C,X) \stackrel{Func(C,F)}{\to} Func(C,\infty Grpd)) \in Func(C, \infty Grpd) \,.

Question 2: Is there a choice for Γ C sortof(X)\Gamma_C^{sortof}(X) such that this is true?

There is one natural candidate that comes to mind: the (,1)(\infty,1)-category of cartesian sections Γ C cart(X)\Gamma_C^{cart}(X) of XCX \to C.

We know that this happens to be the limit over HH. I thought that would help me see what happens when I feed it into the above question. But it didn’t so therefore I finally have

Question 3: What can we say about

hocolim(Γ C cart(X)Func(C,X)Func(C,F)Func(C,Grpd))Func(C,Grpd). hocolim ( \Gamma^{cart}_C(X) \to Func(C,X) \stackrel{Func(C,F)}{\to} Func(C,\infty Grpd)) \in Func(C, \infty Grpd) \,.

And by now it is inevitable that I also ask

Question 4: How is all this related to the (,1)(\infty,1)-categorical Kan extension (if any) of FF along XCX \to C?

Posted by: Urs Schreiber on August 13, 2009 12:17 AM | Permalink | Reply to this

Re: Question about ∞-Colimits

I wrote:

Question 1: Is there a formula for tot?

Sorry, I realize that my question is, or should be, another one:

let

X F Grpd p B \array{ X &\stackrel{F}{\to}& \infty Grpd \\ \downarrow^p \\ B }

be a diagram of (,1)(\infty,1)-categories. Adjoin to this the hocolim coCartesian fibration of FF

hocolimF X F Grpd p B \array{ hocolim F \\ \downarrow \\ X &\stackrel{F}{\to}& \infty Grpd \\ \downarrow^p \\ B }

Now while hocolimFXhocolim F \to X is a coCartesian fibration, the total vertical morphism hocolimFXpBhocolim F \to X \stackrel{p}{\to} B is in general not.

So within the model structure on marked simplicial over-sets for coCartesian fibrations find a fibrant replacement fib(hocolimF)Bfib(hocolim F) \to B of hocolimFBhocolim F \to B. This will be classified by some functor p *F:BGrpdp_* F : B \to \infty Grpd.

Is there any kind of useful formula or other tools for determining p *Fp_* F here?

How does p *Fp_* F relate to an (,1)(\infty,1)-categorical Kan extension (if any) of FF along pp?

Posted by: Urs Schreiber on August 13, 2009 10:07 PM | Permalink | Reply to this

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