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?

fibrantobjects are the DK-equivalences? $\endgroup$ – Charles Rezk Jun 28 '10 at 16:07toa Segal category from a Segal space, so fibrant replacement would be problematic. But it's easier to mapfroma Segal category to a Segal space, so it seems plausible that "Segal categorification" can be implemented as a kind of cofibrant replacement. $\endgroup$ – Charles Rezk Jun 28 '10 at 17:23cofibrant-fibrant objects on a model structure on simplicial spaces. $\endgroup$ – Chris Schommer-Pries Jun 28 '10 at 17:51