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Suppose I have a simplicial model category $M$. Then I can take the homotopy coherent nerve of $M$ to obtain a quasicategory. This, however, only depends on the fact that $M$ is a category enriched in simplicial sets (i.e. the homotopy coherent nerve is a right Quillen functor from $Cat_{sSet}$ to $QCat$). On the other hand, given a model category we can take the hammock localization and the apply the coherent nerve to obtain a quasicategory. How do these two constructions relate? What about the case where we take the simplicially enriched category of bifibrant objects and THEN apply the coherent nerve?

Thanks! :-)

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    $\begingroup$ For a simplicial model category, the full subcategory of bifibrants is equivalent (via a zigzag of Dwyer--Kan equivalences) to the hammock localization of the underlying model category -- this is Proposition 4.8 of Function complexes in homotopical algebra. $\endgroup$
    – Aaron Mazel-Gee
    Commented Apr 14, 2015 at 0:53
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    $\begingroup$ Also, the homotopy-coherent nerve is right Quillen, not left. So you should only apply it to fibrant objects. This is why one generally restricts to the subcategory of bifibrants in a simplicial model category before applying it (since these form a Kan complex-enriched category). $\endgroup$
    – Aaron Mazel-Gee
    Commented Apr 14, 2015 at 0:54
  • $\begingroup$ @AaronMazel-Gee thanks! But what if we only look at the underlying simplicial category of a simplicial model category? How does the coherent nerve of this category relate to the other constructions? $\endgroup$
    – Jonathan Beardsley
    Commented Apr 14, 2015 at 4:39
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    $\begingroup$ Another way to say the same thing is that even in a simplicial model category, the simplicial enrichment is not usually the "correct" simplicial enrichment. Instead you should use some version of the derived simplicial enrichment (left-derived in one variable, right-derived in the other). Then this will be DK-equivalent to the hammock localization, while the original simplicial category is (usually) not DK-equivalent to the hammock localization. One way to derive the simplicial enrichment is to pass to the bifibrant objects. $\endgroup$
    – Chris Schommer-Pries
    Commented Apr 14, 2015 at 7:46
  • $\begingroup$ @JonBeardsley If you want to use all the objects, my sense is that you are really best of forgetting the simplicial structure entirely. You can always take the hammock localization, Bergner-fibrantly replace (e.g. by locally applying $\mathrm{Ex}^\infty$, as Zhen Lin suggested), and then take the h.c.-nerve, and this will once again give you the ("correct") "underlying quasicategory" of your model category. $\endgroup$
    – Aaron Mazel-Gee
    Commented Apr 14, 2015 at 18:05

1 Answer 1

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We can detect the difference between the two constructions using the homotopy category. Given any simplicially enriched category $\mathcal{C}$, we can construct an ordinary category $\pi_0 [\mathcal{C}]$ by applying $\pi_0$ to the hom-spaces. (This makes sense because $\pi_0 : \mathbf{sSet} \to \mathbf{Set}$ preserves finite products.) One might call the morphisms in $\mathcal{C}$ that become isomorphisms in $\pi_0 [\mathcal{C}]$ "(simplicial) homotopy equivalences".

Now, in general, $\mathcal{C}$ is not fibrant (i.e. a Kan-enriched category) – but we can fibrantly replace it by applying $\mathrm{Ex}^\infty : \mathbf{sSet} \to \mathbf{sSet}$ to its hom-spaces. Of course, the induced functor $\pi_0 [\mathcal{C}] \to \pi_0 [\mathrm{Ex}^\infty [\mathcal{C}]]$ is an isomorphism, and the homotopy coherent nerve of $\mathrm{Ex}^\infty [\mathcal{C}]$ is a quasicategory whose homotopy category is isomorphic to $\pi_0 [\mathcal{C}]$.

On the other hand, if $\mathcal{C}$ is the hammock localisation of a model category $\mathcal{M}$ (not necessarily simplicial), then $\pi_0 [\mathcal{C}]$ is isomorphic to $\operatorname{Ho} \mathcal{M}$. But even if $\mathcal{M}$ is a simplicial model category, $\pi_0 [\mathcal{M}]$ and $\operatorname{Ho} \mathcal{M}$ need not be isomorphic. (Exercise: Show this is the case for $\mathcal{M} = \mathbf{sSet}$ with the Kan–Quillen model structure.) Thus, $\mathcal{C}$ and $\mathcal{M}$ need not be weakly equivalent, so the homotopy coherent nerves of their fibrant replacements need not be equivalent. (Recall that the homotopy coherent nerve is a right Quillen equivalence!)

Finally, if $\mathcal{M}_\mathrm{cf}$ is the (simplicially enriched) full subcategory of cofibrant–fibrant objects in a simplicial model category $\mathcal{M}$, then $\mathcal{M}_\mathrm{cf}$ is fibrant, and as Aaron says, it is a theorem of Dwyer and Kan that $\mathcal{M}_\mathrm{cf}$ and $\mathcal{C}$ are weakly equivalent. Hence the homotopy coherent nerves of $\mathcal{M}_\mathrm{cf}$ and $\mathrm{Ex}^\infty [\mathcal{C}]$ are equivalent.

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