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Which is the model structure on $ \text{sSet} $ (category of simplicial sets) that makes $\text{sSet}$ Quillen equivalent to the category $ \text{Cat} $ (of small categories) by the adjunction realization $-|$ nerve?
Is it possible/reasonable to describe this model structure without using this adjunction?
Is there a reference you can give me about this kind of stuff?

Thank you in advance

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    $\begingroup$ What model structure are you putting on $\mathbf{Cat}$? $\endgroup$ – Zhen Lin Jan 28 '14 at 8:07
  • $\begingroup$ Sorry. The canonical model structure on Cat: isofibrations, isocofibrations and equivalences. $\endgroup$ – Fernando Jan 28 '14 at 13:23
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    $\begingroup$ There is a Quillen adjunction if you put the Joyal model structure on $\mathbf{sSet}$. I imagine you could make it a Quillen equivalence by doing a left Bousfield localisation. $\endgroup$ – Zhen Lin Jan 28 '14 at 13:38
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It is hard enough to describe this model structure even with reference to the realisation–nerve adjunction, so I will not even try to do it without.

As with the Joyal model structure, we take as our cofibrations all monomorphisms in $\mathbf{sSet}$. Of course, these are generated by the boundary inclusions $\partial \Delta^n \hookrightarrow \Delta^n$ (for $n \ge 0$). Then the trivial fibrations are the trivial Kan fibrations, and it is not hard to see that a functor $\mathbb{A} \to \mathbb{B}$ is a trivial isofibration (i.e. surjective on objects and fully faithful) if and only if $N (\mathbb{A}) \to N (\mathbb{B})$ is a trivial Kan fibration.

So far so good. Of course, in order for the left adjoint $\tau_1 : \mathbf{sSet} \to \mathbf{Cat}$ to be a left Quillen functor, (Ken Brown's lemma implies that) it is necessary that the weak equivalences go to categorical equivalences, so let us simply define the weak equivalences to be these morphisms.

One can use a standard argument to show this class of weak equivalences is accessible. It is easy (using $\tau_1$ and the canonical model structure on $\mathbf{Cat}$) to check that the class of trivial cofibrations is closed under pushouts and transfinite composition. We may therefore apply Smith's recognition principle to deduce that there is a cofibrantly generated model structure on $\mathbf{sSet}$ with the given cofibrations and weak equivalences.

Note that our model structure on $\mathbf{sSet}$ is constructed so that $\tau_1$ is a left Quillen functor. Since both $\tau_1$ and $N$ also reflect weak equivalences, it is straightforward to see that we have a Quillen equivalence between $\mathbf{Cat}$ and $\mathbf{sSet}$.

What remains to be done is to give a description of the trivial cofibrations in this model structure. Of course, they include all the inner horn inclusions $\Lambda^n_k \hookrightarrow \Delta^n$ (for $n \ge 2$ and $0 < k < n$). But they also include the boundary inclusions $\partial \Delta^n \hookrightarrow \Delta^n$ for $n \ge 3$. (In some sense, this is telling us that the $n$-simplices for $n \ge 3$ are not relevant to this model structure.) There are also more complicated things like the morphisms $\Delta^0 \to I$ where $K$ is any finite simplicial set such that $\tau_1 K$ is a contractible groupoid. I would conjecture that these together form a generating set of trivial cofibrations.

The class of fibrant objects strictly contains the class of (simplicial sets isomorphic to) nerves of categories. Certainly any simplicial set that has the unique extension property for $\Lambda^2_1 \hookrightarrow \Delta^2$ and $\partial \Delta^n \hookrightarrow \Delta^n$ (for $n \ge 3$) is a nerve of a category. On the other hand, consider the following pushout diagram: $$\require{AMScd}\begin{CD} \partial \Delta^2 @>>> \Delta^2 \\ @VVV @VVV \\ \Delta^2 @>>> \Delta^2 \cup_{\partial \Delta^2} \Delta^2 \end{CD}$$ There must exist a fibrant object $X$ and a trivial cofibration $\Delta^2 \cup_{\partial \Delta^2} \Delta^2 \to X$, but trivial cofibrations are monomorphisms, but any morphism $\Delta^2 \cup_{\partial \Delta^2} \Delta^2 \to N (\mathbb{C})$ must factor through the codiagonal $\Delta^2 \cup_{\partial \Delta^2} \Delta^2 \to \Delta^2$, so $X$ cannot be the nerve of a category.

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  • $\begingroup$ Thank you very much! It was a complete nice answer! $\endgroup$ – Fernando Dec 5 '14 at 15:07
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If you can understand written French, I warmly invite you to read the introduction (i.e. pages 3 to 19) of this book to find your answer and much more material.

And in case you can't, this book is a good reason to try learning it!

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    $\begingroup$ No! This is a description of the Thomason model structure on $\mathbf{Cat}$, which is different! $\endgroup$ – Zhen Lin Jan 29 '14 at 0:06

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