(repost from math.stackexchange)

Let $\mathbf{D}$ be a small category, $\mathbf{Set}$ the category of sets and $\mathbf{sSet}$ the category of simplicial sets with its usual model structure. The category of functors $\mathbf{D}\to \mathbf{sSet}$ (diagrams of shape $\mathbf{D}$) has a model structure (called the projective model structure) in which the weak equivalences and fibrations are the object-wise such and the cofibrations are determined by the left lifting property with respect to the trivial fibrations. We call the the cofibrant objects in this model structure "projectively cofibrant diagrams/functors". This is used for example to calculate homotopy colimits.

My question is about a sufficient condition for a diagram to be projectively cofibrant. There are some results about it on nLab and also in Dugger's paper, specifically in the beginning of section 9, where he defines "free degeneracies". The formulation is a bit more involved than what I would expect so perhaps I am missing something. To be specific, I would like to know if the following is true:

A functor $X:\mathbf{D}\to \mathbf{sSet}$ is projectively cofibrant if for every $n\in\mathbb{N}$, the non-degenerate simplices form a subfunctor of $X_n:\mathbf{D}\to\mathbf{Set}$ and this subfunctor is a coproduct of representables.

Namely, the condition is that the maps in the diagram take non-degenerate simplices to non-degenerate simplices and at every level they form a disjoint union of "free orbits". This differs from Dugger's definition, since he seems to require the existence of subfunctors $N_n\subseteq X_n$ with some properties and he does not say that these are precisely the non-degenrate simplices (are they?). Anyways, any clarification of those matters would be appreciated.


marked as duplicate by Dmitri Pavlov, Community Mar 18 '15 at 9:16

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  • $\begingroup$ And of course, N is the subobject of nondegenerate simplices, which follows from Definition 9.2 and the fact that every simplex is the degeneration of some nondegenerate simplex via some unique surjective map of simplices. $\endgroup$ – Dmitri Pavlov Mar 18 '15 at 9:07
  • $\begingroup$ I really don't know how I missed it. The answer is precisely what I wnanted! Thank you very much for pointing this out and for your second comment as well. It confirms what I thought, it just seemed like a strange formulation so I wasn't sure that I am not missing something. I will delete this question soon. $\endgroup$ – KotelKanim Mar 18 '15 at 9:10