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:
- It should be a model category on the category of simplicial spaces.
- The fibrant objects should be the Segal spaces. (Not necessarily complete).
- The weak equivalences should be the DK-equivalences.
Does such a model category exist? is it well known?
Best Answer
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.