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To complete the argument you need to apply K. Brown's Lemma. Call your model category $\mathcal{M}$, then the map $Z \to Y$ induces a pullback functor $\mathcal{M} \downarrow Y \to \mathcal{M} \downar …

This is not an answer in a full generality, but it is certainly not the case if the domains of generating acyclic cofibrations are cofibrant. (I have a hard time thinking of an example of a model cate …

The results on homotopy invariance of the geometric realization of simplicial spaces go back at least to May's The Geometry of Iterated Loop Spaces, although he doesn't explicitly mention model catego …

EDIT: This is an old question, but I have stumbled upon it by accident and realized that my answer is wrong. It turns out that genuine homotopy equivalences cannot be characterized in terms of their h …

Here's a suggestion. Hopkins uses the name flat map for a morphism such that every pushout along it is a homotopy pushout. Rezk uses the name sharp map for the dual notion. (Roughly speaking, I think …

Yes, assuming that by "expected $G$-homotopy extension property" you mean the Serre $G$-cofibrations, not the Hurewicz $G$-cofibrations. A very explicit reference is Proposition A.1.18 in Schwede's Gl …

$\newcommand{\Ex}{\mathrm{Ex}}\newcommand{\Sd}{\mathrm{Sd}}$Consider the square $$\begin{CD} X @>{\sim}>> \Ex^\infty X \\ @V{f}VV @VV{\Ex^\infty f}V \\ Y @>>{\sim}> \Ex^\infty Y \textrm{.} \\ \end{CD …

It is true for arbitrary products of model categories (or just cofibration categories) as proven in Theorem 7.1.1 of http://arxiv.org/abs/math/0610009v4. It is also true for finite products of arbitr …

You may be thinking of Johnson, Mark W. On modified Reedy and modified projective model structures. Theory Appl. Categ. 24 (2010), No. 8, 179–208. but his constructions (Definitions 3.3 and 5.2) hav …

It seems to me that there is a relatively simple answer to this question but perhaps I am overlooking something. The category of K-spaces does satisfy the monoid axiom. (If I read the question correc …

I don't think we can expect to have one general answer to this question, only a collection of unrelated specialized results. Here are two more: In the category of simplicial sets all filtered colimi …

As Tyler Lawson points out you can use the category of all diagrams on $[1]$. Then the projective and injective model structures are both instances of Reedy model structures. This is discussed in Sect …

It doesn't make a difference as long as we restrict attention to compactly generated saturated classes, i.e. cofibrantly generated ones where generators can be chosen to have $\aleph_0$-small domains. …

I don't know if this will work with the definition of $(n, 1)$-categories exactly as stated by Lurie, but it will work with a reasonable modification. Definition 2.3.4.1 from HTT basically says that …

I'm not sure about the case of general monoidal categories. (Although I seem to recall a remark that there is no such structure since the category of monoidal categories is not cocomplete and a suitab …