Let $s^2Set$ denote the category of bisimplicial sets, i.e. simplicial objects in the category of simplicial sets. Recall that in the Moerdijk model structure on $s^2Set$, weak equivalences are "point-wise" weak equivalences of simplicial sets (in the usual sense) and cofibrations are the injections.

Is this model structure combinatorial?


1 Answer 1


That's not the Moerdijk model structure. In the Moerdijk model structure, weak equivalences and fibrations are created by the diagonal simplicial set construction. Your model structure resembles the Bousfield-Kan model structure, where weak equivalences and fibrations are defined pointwise as in simplicial sets. You rather define weak equivalences and cofibrations pointwise as in simplicial sets. This is known as the injective model structure in the diagram category of simiplicial objects simplicial sets (which are the same as bisimplicial sets). This model structure is known to be combinatorial since that of simplial sets is. See Lurie's book.

  • $\begingroup$ Ah silly of me. Should I edit the question or leave it as it is? $\endgroup$
    – user84563
    Jul 27, 2016 at 11:58
  • $\begingroup$ I think it's irrelevant, only a matter of terminology. I've just clarified it in case aonyone gets confused. $\endgroup$ Jul 27, 2016 at 12:57
  • $\begingroup$ And did you mean that the Bousfield Kan model structure is combinatorial or that injective model structure on bisimplicial sets is? $\endgroup$
    – user84563
    Jul 27, 2016 at 20:05
  • $\begingroup$ Both, actually. $\endgroup$ Jul 27, 2016 at 20:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.