Question on Homotopical Structure on SimpSet

March 5, 2009

Question on Homotopical Structure on SimpSet

Posted by Urs Schreiber

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I’d like to better understand how the category SSet with its classical model structure sits inside $SSet$ equuipped with Joyal’s model structure.

More concretely, I’d like to get a better formal idea of the following situation:

in the standard model structure on $SSet$ , for every fibrant object $X \in Kan \subset SSet$ (= $\infty$ -groupoid) the object $[\Delta^1, X]$ is a path object for $X$ . So $Kan$ equipped with $[\Delta^1, -]$ is a category of fibrant objects in this sense with a functorial assignment of path objects.

In the Joyal model structure, for every fibrant object $X \in WeakKan \subset SSet$ (= quasi-category) the object $[\Delta^1, X]$ clearly plays the role of the right directed path space object for $X$ , it still factors the diagonal as $X \to [\Delta^1, X] \to X \times X \,,$ but it is no longer a path object in the standard model-theoretic sense, as $X \to [\Delta^1,X]$ need not be a weak equivalence anymore: $\Delta^1$ is a directed interval object.

I am thinking that there should be a good and nice relaxation of the axioms of category of fibrant objects such that $WeakKan$ becomes an example and such that the inclusion $Kan \hookrightarrow WeakKan$ becomes an inclusion of categories of fibrant objects in the relaxed sense.

There are some obvious guesses for how to try to relax the path space object axiom. But I am not entirely sure yet what the good way to do it really is. Has anyone thought about this?

Posted at March 5, 2009 8:13 PM UTC

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Re: Question on Homotopical Structure on SimpSet

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Well, a different thing to do would be to replace $\Delta^1$ by the nerve $N(J)$ of the walking isomorphism. Then you do get an interval object, in the model-category sense, for both model structures, so that $Kan \hookrightarrow QCat$ should then be an inclusion of categories of fibrant objects.

Posted by: Mike Shulman on March 5, 2009 11:49 PM | Permalink | Reply to this

Re: Question on Homotopical Structure on SimpSet

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Thanks, Mike, for the comment.

As discussed at interval object, there are two ways to ensure that the mapping space from the interval into an object is equivalent to that object:

either ensure that the object is undirected or groupoidal, or to ensure that the interval object is so (or both).

The category $\{a \to b\}$ (with nerve $\Delta^1$ ) with non-invertible morphism is the standard directed interval of (quasi-)categories, I’d think.

The groupoid $\{a \stackrel{\simeq}{\to} b\}$ (or its nerve) with invertible morphism is the standard un-directed interval of (quasi-)categories. You are saying one should use this to get a good notion of path objects.

Yes, I agree. But maybe my question was about something different:

I want to carry the path-space yoga which identifies homotopies (natural isomorphisms) between maps (functors) between $\infty$ -groupoids (Kan complexes) in terms of mapts into path objects over to the world of quasi-categories and have directed homotpies (i.e. natural transformations) encoded nicely in terms of maps into suitable directed path objects.

For this purpose $N(a \stackrel{\simeq}{\to} b)$ is not the right path object. Maybe $\Delta^1$ isn’t either, though, but something directed like $\Delta^1$ should be needed.

Posted by: Urs Schreiber on March 6, 2009 1:41 PM | Permalink | Reply to this

Re: Question on Homotopical Structure on SimpSet

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Good question. (As I said, my first comment was “a different thing to do.” (-: ) Here’s one possible approach. Consider, rather than merely a factorization of $C\to C^I \to C\times C$ , a factorization

$c_\bullet C \to C^{\Delta^\bullet} \to C^{\bullet+1}$

of simplicial objects in your category, where $c_\bullet C$ is the constant simplicial object at $C$ and $C^{\bullet+1}$ is the simplicial object whose $n$ -simplices are $C^{n+1}$ , with diagonals and projections giving the degeneracies and faces. Thus, we levelwise have factorizations

$C \to C^{\Delta^n} \to C^{n+1}.$

When $n=0$ we should probably demand that $C^{\Delta^0}=C$ . When $n=1$ this looks like the path-object factorization. In general, the simplicial object $C^{\Delta^\bullet}$ is a candidate to be an “internal quasi-category” that encodes directed homotopies into $C$ and their compositions.

In any model category, there is a standard such factorization, namely the (acyclic-cofibration, fibration) factorization in the Reedy model structure on simplicial objects, and it is in fact an “internal Kan complex” in a suitable sense. I wouldn’t be much surprised if one could construct an analogous such factorization in any category of fibrant objects.

But now we care about cases when $c_\bullet C \to C^{\Delta^\bullet}$ is not a weak equivalence. It’s not clear to me whether in this case the whole simplicial object $C^{\Delta^\bullet}$ can be constructed from a factorization axiom for single maps. Also it seems less likely that the condition of $C^{\Delta^\bullet}$ being an “internal quasi-category” will come for free; we might have to assert it explicitly.

But it $QCat$ , at least, such a factorization does exist, where $C^{\Delta^n}$ is what it looks like it should be, and it is an “internal quasi-category” in some sense. Probably I mean here that $C^{\Delta^n} \to C^{\Lambda^n_k}$ is a weak equivalence (or maybe an acyclic fibration) for each inner horn $0\lt k\lt n$ , where $C^{\Lambda^n_k}$ is constructed out of $C^{\Delta^\bullet}$ as a suitable limit.

Posted by: Mike Shulman on March 6, 2009 6:21 PM | Permalink | Reply to this

Re: Question on Homotopical Structure on SimpSet

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Hi Mike,

thanks, yes, this is exactly the kind of thing we have been thinking about.

So in the terms of Hovey’s book, we want a modification of framing such that in degree 1 the condition that $C \to C^I$ is a weak equivalence is dropped.

But it QCat, at least, such a factorization does exist, where $C^{\Delta^n}$ is what it looks like it should be

Hm, so that was the point under debate, no? Is it good to take $C^{\Delta^n}$ to be the internal Hom $[\Delta^n, C]$ ?

(I don’t know, I am really asking.)

So it seemed that we want $C^{\Delta^1} = [\Delta^1, C]$ , but maybe $C^{\Delta^n}$ for $n \geq 2$ be something bigger?

Probably I mean here that $C^{\Delta^n} \to C^{\Lambda^n_k}$ is a weak equivalence

Since I was thinking about the category of locally weakly Kan simplicial presheaves or similar, I was thinkig of requiring that

- for every morphism

$X \to C^{\Lambda^{n}_k}$

there is an acyclic fibration $\hat X \to X$ such that we have a diagram

$\array{ &&&& C^{\Delta^n} \\ && \nearrow && \downarrow \\ \hat X &\to& X &\to& C^{\Lambda^n_k} } \,.$

Posted by: Urs Schreiber on March 6, 2009 6:46 PM | Permalink | Reply to this

Re: Question on Homotopical Structure on SimpSet

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Hm, so that was the point under debate, no?

I don’t know, I wasn’t part of this debate. (-: Did it take place somewhere where I could have read it? I didn’t see it at “path object” or “interval object” on the nlab, but maybe I missed it.

Do you have some reason for thinking that $C^{\Delta^n}$ should be anything different from the internal-hom $[\Delta^n,C]$ ? I don’t see any reason, at least not in $QCat$ , where it does give the right notion of composition of transformations.

Also, your condition on lifting modulo acyclic-fibration replacements doesn’t seem very different from requiring $C^{\Delta^n} \to C^{\Lambda^n_k}$ to be an acyclic fibration, since of course $\hat{X}$ could just be a cofibrant replacement for $X$ , or even just the pullback of $C^{\Delta^n}$ to $X$ . Are you just phrasing it that way so that you don’t have to talk about cofibrant objects? Saying that the map is an acyclic fibration seems a more natural condition to me.

Posted by: Mike Shulman on March 6, 2009 9:03 PM | Permalink | Reply to this

Re: Question on Homotopical Structure on SimpSet

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Just a very quick reply, since I am on the train:

Are you just phrasing it that way so that you don’t have to talk about cofibrant objects?

Yes, I am finding it useful here to stay within the axiomnatics of categories of fibrant objects, not assuming anything about cofibrations.

I may be misled, but the cat of fib objects structure for instance on simplicial presheaves seems to be useful in that the fibrations are tractable (they are just localy Kan fibrations).

In the full model structure on simplicial presheaves the fibrant objects are the Kan-valued presheaves which satisfy descent, hence the fully $\infty$ -stackified things. This is much harder to handle in computations.

Posted by: Urs Schreiber on March 6, 2009 10:38 PM | Permalink | Reply to this

Re: Question on Homotopical Structure on SimpSet

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I’m not an expert on simplicial presheaves, but I’m happy to try to work in the axiomatics of a category of fibrant objects. $QCat$ is, of course, also a good example of a category of fibrant objects in which the fibrations and weak equivalences are much simpler than they are in the larger model structure.

But I think that even within that axiomatics, asking for $C^{\Delta^n}\to C^{\Lambda^n_k}$ to be an acyclic fibration is reasonable. In particular, it implies the condition you suggested, since given $X\to C^{\Lambda^n_k}$ you can take $\hat{X}$ to be the pullback $\array{\hat{X} & \to & C^{\Delta^n}\\ \downarrow && \downarrow\\ X& \to & C^{\Lambda^n_k}.}$

Posted by: Mike Shulman on March 7, 2009 3:42 AM | Permalink | Reply to this

Re: Question on Homotopical Structure on SimpSet

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Over here, I asked about differences between cubical “stuff” and simplicial “stuff”.

I suspect that directed stuff is where cubical stuff might have some advantages. How might the argument be different if the factorization was more like:

$C\to C^{\Box^n}\to C^{n+1}$

It is maybe more natural to put a global “direction” on cubes than it is for simplices.

Just a thought…

Posted by: Eric on March 6, 2009 6:59 PM | Permalink | Reply to this

Re: Question on Homotopical Structure on SimpSet

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I found the correct reference where I asked my question about simplices and cubes:

Cross Modules and Non-Abelian 2-Forms: Cubes vs Simplices

Posted by: Eric on March 6, 2009 8:57 PM | Permalink | Reply to this

Re: Question on Homotopical Structure on SimpSet

One direction on simplices is from 0 to n
cf. the paths on a simplex that run from 0 to n
this is indeed used in studying loop spaces

but maybe that’s not the kind of direction you meant

Posted by: jim stasheff on March 7, 2009 12:35 AM | Permalink | Reply to this

Re: Question on Homotopical Structure on SimpSet

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Hi Jim,

What I had in mind might be misguided, but whenever I think of directed spaces, I think of “spacetime”. The crucial aspects of spacetime can be encoded in a poset and a volume measure. I recently learned that directed spaces have a fundamental category.

With that in mind, I’m hoping to watch this drama play out so that one day, Urs or somebody will come out and say something like, “Ah ha! Spacetime IS a fundamental category!”

With that in mind, the picture I’m thinking of is a plane filled with either 2-cubes or 2-simplices. If we fill a plane with 2-simplices and then assign a direction to each 2-simplex as you suggested, i.e. $0\to 1\to 2$ , then many of these directions will wrap around on themselves and it is not clear that you could get a global sense of direction from these local directions that might represent the flow of time.

If you fill the plane with 2-cubes, then it is easy to see a global direction emerge, i.e. the direction across the diagonal of each 2-cube. Essentially, you have a directed binary tree (which Urs and I call a 2-diamond).

When I hear “directed space” I think “spacetime”. So although simplices have a local direction, it is not obvious to me that a global direction would emerge from a space filled with such directed simplices. I would interpret that global direction as a flow of time.

Like I said, maybe my thought was misguided, but I think that when you work with directed spaces, it might be helpful to think of a global direction as a flow (or direction) of time

Posted by: Eric on March 7, 2009 4:33 PM | Permalink | Reply to this

Re: Question on Homotopical Structure on SimpSet

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I am receiving some very useful comments by private email. So maybe I should give more details on what I am actually after:

given any category $C$ of fibrant objects, it seems it has a simplicial localization $\hat C$ described roughly as follows:

for $X,A$ two objects of $C$ , the 0-cells in the simplicial set $\hat C(X,A)$ are spans

$\array{ && \hat X &\to& A \\ & \swarrow \\ X }$

with the left leg an acyclic fibration (aka anafunctors, aka generalized morphisms)

and the 1-cells are diagrams

$\array{ && \hat X_0 &\to& A \\ &\swarrow& \uparrow && \uparrow \\ X &\leftarrow& \hat X_{01} &\to& A^I \\ & \nwarrow & \downarrow && \downarrow \\ && \hat X_1 &\to& A }$

with all the arrows between $X$ s acyclic fibrations, and with $A^I$ a path object of $A$ and the maps on the right two boundary evaluation maps out of the path object. So this encodes that the map out of $\hat X_0$ has a homotopty to the map out of $\hat X_1$ after both are pulled back to a joint refinement of their domain.

And analogously for higher simplices.

First question: has this particular kind of construction of simplicial localization in cats of fib objects been considered before? Notice that using the morphisms given by the path object I can make a cosmetic modification to this diagram without changing its content

$\array{ && \hat X_0 &\to& A \\ &\swarrow& \uparrow&& \uparrow & \nwarrow \\ X &\leftarrow& \hat X_{01} &\to& A^I &\leftarrow& A \\ & \nwarrow &\downarrow&& \downarrow & \swarrow \\ && \hat X_1 &\to& A }$

and it begins to look more akin to a hammock. It’s not hammock localization, but supposed to be something more direct using the properties of path objects in cats of fib objects.

Now, if $C$ here is something like locally Kan simplicial presheaves or presheaves with values in some other model of $\infty$ -groupoids, then $\hat C(X,A)$ will be a Kan complex, too.

But now assume $C$ is something like (presheaves with values in) weak Kan complexes. Then we are still in a category of fibrant objects, but now we are inclined to allow in the construction above $A^I$ not be a path object in the standard sense (not being weakly equivalent to $A$ ), but be more generally of the form $[D,A]$ , where $D$ is some directed interval object. The point being that for objects in $C$ behaving (locally at least) like quasi-categories, we don’t want to demand the Hom $SSets$ $\hat C(X,A)$ to have outer fillers.

And there is another thing that makes me wonder, and triggers my question here:

if we drop the condition that $A^I$ in the above hammock-like gadget be weakly equivalent to $A$ , then it seems natural to also drop in 1-cells the requirement that all the morphisms between the $\hat X$ s be acyclic.

This is kind of remarkable, because the diagram

$\array{ && \hat X_0 &\to& A \\ && \uparrow && \uparrow \\ && \hat X_{01} &\to& A^I \\ & & \downarrow && \downarrow \\ && \hat X_1 &\to& A }$

without any condition on the left vertical morphisms is a bi-brane, namely a span of (generalized) spaces with a gerbe-like thing on each space and a morphism between the pullback of these to the correspondence space.

So I am wondering what’s going on. And if there is some general nonsense on directed path objects which would tell me which conditions to put here on the left vertical morphisms if on the right I allow directed path objects so that the whole thing somehow forms a nice structure.

Oh, I just see a new comment by Mike coming in …

Posted by: Urs Schreiber on March 6, 2009 6:36 PM | Permalink | Reply to this

Re: Question on Homotopical Structure on SimpSet

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Regarding this particular kind of simplicial localization, it’s of course very much like the construction of simplicial mapping spaces in a model category using simplicial framings/resolutions. I haven’t seen it written down in this context.

I will continue to think about this, but one general comment: I do not expect that there will be a condition characterizing “directed path objects” solely in terms of the category-of-fibrant-objects structure. I’m not sure whether that’s what you’re expecting/hoping, but I wouldn’t expect it. My intuition is that categories-of-fibrant-objects, like model categories, are a presentation of $(\infty,1)$ -categories, and if you want to recover a structure with noninvertible 2-cells from them, you need to impose extra structure, like considering an enriched or a closed-monoidal thingy (like $QCat$ ).

Posted by: Mike Shulman on March 6, 2009 9:12 PM | Permalink | Reply to this

Re: Question on Homotopical Structure on SimpSet

Thanks for all yor replies!

This here is just to let you know that the reason I am not reacting much currently is that I am spending a couple of days in Chert, Spain, which is alltogether a rather offline town.

Posted by: Urs Schreiber on March 9, 2009 2:40 PM | Permalink | Reply to this

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