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It is often said that model categories are but a shadow of an $\infty$-category. It is also often said that model categories should give rise to an $\infty$-category via their homotopies. In fact, this ought to be the quintessential example of an $\infty$-category. Unfortunately I have yet to find a place where the author bothered to give the definition. I am also unable to give the definition myself, as things start to get fuzzy beyond 2-simplices. What is the definition?

Edit: The only place that gets close is Hirschhorn's book on model categories, which defines a simplicial structure on the Hom-sets of a model category. The definition spans multiple chapters, and involves Reedy model structures, cosimplicial approximations, fibrant replacements, and function complexes, none of which I understand. I am certain that this is just another case of making things a thousand times harder than it needs to be.

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  • $\begingroup$ Maybe the answers here, mathoverflow.net/questions/8663/…, will be interesting to you. $\endgroup$
    – Tyrone
    Nov 12, 2018 at 10:48
  • $\begingroup$ Thanks but I'd already seen that question. Everybody seems to know the definition and works from that point on, but really the definition is all I'm concerned about. $\endgroup$
    – user542740
    Nov 12, 2018 at 10:54
  • $\begingroup$ If the model structure is simplicial then you can use the homotopy coherent nerve, I belive. See, for instance, Riehl's "Categorical Homotopy Theory". In any case there is always the simplicial localisation of a model structure, which gives a simplicial category. After taking a suitable fibrant replacement (in the category of simplicial categories) you can then apply the homotopy coherent nerve. I found much of this discussed on the nlab page, and in Barwick's paper with Schommer-Pries "On the Unicity of the Homotopy Theory of Higher Categories". $\endgroup$
    – Tyrone
    Nov 12, 2018 at 11:28
  • $\begingroup$ So if the model category is not simplicial, then what everybody really means is the simplicial localization of the model category you started out with. Is that correct? If so, what does simplicial localization intuitively do? $\endgroup$
    – user542740
    Nov 12, 2018 at 11:32
  • $\begingroup$ I would suggest following up Barwick's paper. It might be necessary to take fibrant replacements in a suitable sense, so really there might be many choices involved at each stage, and the overall process may be a little more subtle. $\endgroup$
    – Tyrone
    Nov 12, 2018 at 11:35

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Most model categories of interest are Quillen equivalent to a combinatorial model category, each of which is in turn Quillen equivalent to a simplicial model category by a fundamental theorem of Dugger. So in practice one is most often thinking of the homotopy coherent nerve of the category of fibrant-cofibrant objects.

The simplicial localization doesn't use the model category structure, and instead just builds the simplicial set of maps between two objects out of zigzags of maps and weak equivalences of arbitrarily large length. It's essentially impossible to compute anything directly with the simplicial localization. One can use the model category structure to more straightforwardly construct the mapping spaces, via the admittedly complex work of Hirschhorn you cite, but there is no currently known method to leverage this into a construction of the whole $\infty$-category.

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