In Hirschhorn's Model categories and their localizations he gives a sufficient condition to induce a cofibrantly generated model structure on a category $\mathcal{N}$, given an adjoint pair of morphisms $\mathcal{F:M\rightleftarrows N:U}$ (where $\mathcal{M}$ is a cofibrantly generated model category) such that the adjoint functors induce a Quillen equivalence. Using Kan's characterisation of cofibrantly generated model categories.

Is there a way to make a similar statement about combinatorial model categories along the lines of Jeff Smith's theorem?

My main motivation for this is to see whether I can use the diagonal functor from the category of $n$-simplicial sets to the category of simplicial sets to define a combinatorial model structure on $n$-simplicial sets.


1 Answer 1


Yes, there is such a criterion for combinatorial model categories, though it is quite similar to Hirschhorn's: the only difference is that the smallness conditions (11.3.2.(1) in Hirschhorn) can now be dropped.

The transferred model structure on bisimplicial sets (and multisimplicial sets are no different) is the subject of Moerdijk's paper “Bisimplicial sets and the group-completion theorem”.

  • $\begingroup$ I was hoping for criteria that could be proved just from Jeff Smith's theorem without knowing the characterisation of cofibrantly generated categories. For example as the conditions for Jeff Smith's theorem does not talk about the set that generates the trivial cofibrations, can we get a condition which does not depend on this set? $\endgroup$
    – Arun Kumar
    Commented Mar 17, 2016 at 11:08
  • $\begingroup$ @ArunKumar: You can certainly formulate a condition without any generating sets. To show that the transferred model structure on N exists it is necessary and sufficient to show that transfinite compositions of cobase changes of elements in F(AC_M) are weak equivalences in N. Here AC_M is the class of all acyclic cofibrations in M. $\endgroup$
    – Dmitri P.
    Commented Mar 17, 2016 at 11:26
  • $\begingroup$ I was trying to prove that fact but I couldn't show the solution set condition for weak equivalences. $\endgroup$
    – Arun Kumar
    Commented Mar 17, 2016 at 12:09
  • $\begingroup$ @ArunKumar: If you are referring to the fact that weak equivalences in N form an accessible subcategory of the category of arrows in N, this is the content of Corollary A.2.6.5 in Lurie's Higher Topos Theory. $\endgroup$
    – Dmitri P.
    Commented Mar 17, 2016 at 12:56

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