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I am trying to prove that the category of simplicial sets is a combinatorial model category by using Proposition A.2.6.15 of Lurie's book, and this requires proving that the weak equivalences are generated by a (small) set of weak equivalences under filtered colimits. It seems so doable, yet I am at a loss. The only references I have found on the matter, state it as a consequence of the fact that sSet is a combinatorial model category so that's no help.

Maybe his alternative characterisation of accessibility (Proposition A.2.6.5) can help?

Any help and references would be much appreciated.

Edit for clarification: I need to prove that sSet is a combinatorial model category via Smith's theorem, meaning that I can only use that sSet is locally presentable and that I have to prove a few conditions for the classes of weak equivalences and cofibrations. One of these conditions is that weak equivalences are generated by filtered colimits from a set. Therefore I cannot use that sSet is a model category, and I have to prove this statement in some more direct way.

Also, weak equivalences are the morphisms that become weak homotopy equivalences in the realisation.

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  • $\begingroup$ This question is not really meaningful unless you specify which description of the weak equivalences of sSet you take as the definition. $\endgroup$ – Reid Barton May 31 '20 at 9:25
  • $\begingroup$ A morphism is a weak equivalence if it is a weak homotopy equivalence in the realisation. $\endgroup$ – Sebastian H. Martensen May 31 '20 at 15:15
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See Corollary 5.1 in The accessibility rank of weak equivalences, which shows that weak equivalences of simplicial sets are finitely accessible.

See also Theorem 4.6 in Model Structures on Ind Categories and the Accessibility Rank of Weak Equivalences, which establishes the same result in a different way.

This proves that sSet is a finitely combinatorial model category, because it has a set of generating (acyclic) cofibrations with compact (co)domains (compact simplicial sets are precisely simplicial sets with finitely many nondegenerate simplices), and its underlying category is locally finitely presentable because it admits a set of compact generators given by all representables, i.e., simplices.

If your goal is to prove that sSet is merely a combinatorial model category (without finiteness), then this follows much more directly and immediately from the fact that sSet is a cofibrantly generated model category (by construction), whose underlying category is a locally presentable category (just like any category of presheaves of sets on a small category). The accessibility of weak equivalences is then an automatic consequence of these facts.

$\def\Exi{{\rm Ex}^\infty}$

Finally, if one merely wants to prove the accessibility of weak equivalences directly, without referring to model structures, this can be done as follows. Recall the original definition of simplicial weak equivalences due to Kan: a simplicial map $f$ is a weak equivalence if $\Exi(f)$ is a simplicial homotopy equivalence. The functor $\Exi$ is an accessible functor between locally presentable categories because $\Exi$ is the filtered colimit of ${\rm Ex}^n$ and each of these preserves filtered colimits. Thus, it suffices to show that simplicial homotopy equivalences between Kan complexes form an accessible subcategory.

Indeed, by the simplicial Whitehead theorem, simplicial homotopy equivalences between Kan complexes can be characterized by a relative-homotopy-lifting property with respect to the set of morphisms ∂Δ^n→Δ^n. The relative-homotopy-lifting property can be reformulated as an ordinary extension property with respect to a certain morphism in the category of simplicial maps, see the formula (3.3) in Dugger and Isaksen's paper Weak equivalences of simplicial presheaves. But such a lifting property singles out an accessible subcategory of the category of simplicial maps between Kan complexes, see the proof of Corollary A.2.6.8 in Lurie's Higher Topos Theory.

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  • $\begingroup$ Thanks for the references! But both of those articles seem to conclude that weak equivalences are accessible on the condition that sSet is already a finitely combinatorial model category, and I can't assume that. I think the point in both articles is to show that it is finitely accessible, whereas I am comfortable with any level of accessibility. Do you know any work around? Or maybe you have another reference? $\endgroup$ – Sebastian H. Martensen May 31 '20 at 0:46
  • $\begingroup$ @SebastianH.Martensen: The “assumption” of being finitely combinatorial is a completely elementary fact: the generating (acyclic) cofibrations of sSet have as their (co)domains compact simplicial sets, i.e., simplicial sets with finitely many nondegenerate simplices. $\endgroup$ – Dmitri Pavlov May 31 '20 at 0:51
  • $\begingroup$ @SebastianH.Martensen: I amended my answer accordingly. $\endgroup$ – Dmitri Pavlov May 31 '20 at 1:22
  • $\begingroup$ I am sorry for the communication error but I realise I may not have been clear (or maybe I am just not understanding you correctly); I want to prove that sSet is a combinatorial model category via Smith's theorem, i.e. I only know that sSet is locally presentable, and I have to prove that a bunch of conditions are true for sSet equipped with the chosen cofibrations and weak equivalences. Among these conditions, I need to prove that weak equivalences are generated by filtered colimits from a set. Therefore I cannot use that sSet is a model category directly. Do you know an argument for this? $\endgroup$ – Sebastian H. Martensen May 31 '20 at 7:22
  • $\begingroup$ @SebastianH.Martensen: Yes, I added an answer to this question as the last paragraph to my answer. $\endgroup$ – Dmitri Pavlov May 31 '20 at 17:06

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