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So in Julie Bergner's work on (infty, 1)-categories arXiv:0610239, she considers several model categories which model (infty, 1)-categories, which are known to be equivalent. I'm guessing that there is another model which is equivalent to these. It is probably known to experts, and probably exists for easy reasons, but I'm not seeing it so I'm asking here.

Background

In Julie's survey, which reviews some of her own work in arXiv:0504334, the work of Joyal and Tierney arXiv:0607820, and the work of others, she compares four models of (infty, 1)-categories. They are model categories of Segal Categories, Complete Segal spaces, Quasi-categories, and Simplicial Categories.

All of these models are related in multiple ways, but some are more closely aligned than others. In particular the Complete Segal spaces and the Segal categories are very similar objects. The underlying categories for these two model categories are almost the same. They are the category of simplicial spaces and the category of simplicial spaces whose zeroeth space $X_0$ is discrete, respectively.

Among the simplicial spaces we have those which satisfy the Segal condition. These are the (Reedy fibrant) simplicial spaces such that the Segal map $$ X_n \to X_1 \times_{X_0} \cdots \times_{X_0} X_1 $$ is a weak equivalence. These are called Segal spaces. There is a model structure on the category of simplicial spaces such that the Segal spaces are the fibrant objects. It is a localization of the Reedy model structure. But in this model structure the weak equivalences between Segal spaces are just the level-wise weak equivalences.

The model category of complete Segal spaces is a further localization of this. The weak equivalences between fibrant objects (complete Segal spaces) in this model category are pretty easy. They are the level-wise weak equivalences. More generally the weak equivalences between Segal spaces in this new model structure can also be identified, without too much trouble. They are called Dwyer-Kan equivalences or DK-equivalences for short.

A Segal category is a Segal space where the space of objects $X_0$ is discrete. There is a Quillen equivalent model structure on the category of those simplicial spaces whose zeroeth space is discrete. In this model category, a map between two Segal categories is a weak equivalence if and only if it is a DK-equivalence.

Question

I'm wondering if there is an intermediary model category which is equivalent to both the above model categories (hopefuly in an obvious way) which has the following properties:

  1. It should be a model category on the category of simplicial spaces.
  2. The fibrant objects should be the Segal spaces. (Not necessarily complete).
  3. The weak equivalences should be the DK-equivalences.

Does such a model category exist? is it well known?

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I don't understand property 3. Do you mean the weak equivalences between fibrant objects are the DK-equivalences? –  Charles Rezk Jun 28 '10 at 16:07
    
um, yes. What I'd really like is for the weak equivalences to be exactly the same as the weak equiv in your complete Segal space model structure. That might be asking too much. I'd settle for just the weak equivalences between the fibrant objects (i.e. the Segal spaces) to be the same, i.e. the DK-equivalences. –  Chris Schommer-Pries Jun 28 '10 at 16:59
    
I'm imagining a situation which is reminiscent of the projective, injective and Reedy model structures, where there are three equivalent model structures on a single category (in this case simplicial spaces), all with the same weak equivalences. However these three model structures should have different fibrant objects. In one it is the Segal spaces, in another it is the Segal categories and in the third it is the complete Segal spaces. Is this fantasy too much to hope for? –  Chris Schommer-Pries Jun 28 '10 at 17:12
    
I don't see how Segal categories (or something like them) can be picked out as fibrant objects in simplicial spaces; it's hard to map to a Segal category from a Segal space, so fibrant replacement would be problematic. But it's easier to map from a Segal category to a Segal space, so it seems plausible that "Segal categorification" can be implemented as a kind of cofibrant replacement. –  Charles Rezk Jun 28 '10 at 17:23
    
Okay. I see. So I guess that's one reason people look for a model structure for Segal categories on the category of Segal Pre-categories (Simplicial spaces with zeroeth space discrete). Then you can get the fibrant objects to be Segal cats, but not otherwise. After a moment reflection I think I am starting to really like this idea that the Segal Cats should actually be the cofibrant-fibrant objects on a model structure on simplicial spaces. –  Chris Schommer-Pries Jun 28 '10 at 17:51

1 Answer 1

I'm not aware that the model category you want has been constructed. But it seems like an interesting question. You should ask Julie Bergner if she has thought of anything along these lines.

I don't have an answer, but I will think out loud for a bit. (I'll be a little vague by what I mean by "space", but I probably mean "simplicial sets" here.)

I will assume that your property 3 should say: "The weak equivalences between fibrant objects (i.e., between Segal spaces) are the DK-equivalences." I would also like to throw in an additional property:

4. The trivial fibrations between Segal spaces are maps $f:X\to Y$ which are DK-equivalences, Reedy fibrations, and such that the induced map $f_0:X_0\to Y_0$ on $0$-spaces is surjective. (Note: since $f$ is a Reedy fibration, $f_0$ is a fibration of spaces.)

Yes, this comes out of thin air ... but it's modelled on the trivial fibrations in the "folk model structure" on Cat. You could go further, and posit that fibrations between Segal spaces are Reedy fibrations such that $f_0$ is surjective.

Given a space $U$, let $cU$ denote the "$0$-coskeleton" simplical space, with $(cU)_n=U^{\times (n+1)}$. If $U$ is a fibrant space, then $cU$ is Reedy fibrant; if $U\to V$ is a fibration, $cU\to cV$ is a Reedy fibration. Furthermore, $cU$ clearly satisfies the Segal condition.

Thus, if $g:U\to V$ is a surjective fibration of spaces, $cg: cU\to cV$ should be a trivial fibration in our model category, according to 4.

The functor $c$ is right adjoint to $X\mapsto X_0$: that is, maps of simplicial spaces $X\to cU$ are naturally the same as maps $X_0\to U$ of spaces.

Putting all this together, we discover that, if such a model category exists, a cofibration $f: A\to B$ should have the following properties: the map $f_0 : A_0\to B_0$ is a cofibration of spaces, and $B_0=B_0'\amalg B_0''$ so that $f_0$ restricts to a weak equivalence $A_0\to B_0'$, and such that $B_0''$ is homotopy discrete (i.e., has the weak homotopy type of a discrete space).

In particular, a necessary condition for $B$ to be cofibrant is that $B_0$ is homotopy discrete.

This is a pretty restrictive condition on cofibrations, but it does not seem impossible. If there actually was a model category with all these properties, it appears that the class of fibrant-and-cofibrant objects would be what you might call the quasi Segal categories. These are the Segal spaces $X$ such that $X_0$ is homotopy discrete. Cofibrant replacement of a Segal category would give a DK-equivalent quasi-Segal category. That would be a pleasing outcome, and probably along the lines of what you're looking for.

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Your property 4 seems reasonable enough to me, and your conclusion is undeniable. It is not exactly the picture I was imagining, but I think this new perspective is growing on me. So this hypothetical model category seems like a homotopy version of Segal categories, where the space of objects is homotopically discrete. It also seems to be a place where general Segal spaces reside. –  Chris Schommer-Pries Jun 28 '10 at 17:57

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