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No, there are far too many "canonical cofibrations" for that. For example, let $A$ be a category with two objects and two parallel arrows between them. Then there is a unique functor $A \to [1]$ that …

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 …

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 …

The condition that appears in the assumption of what you call "Relative Theorem A" was introduced by Grothendieck in Pursuing Stacks. It is a part of the definition of a basic localizer, i.e. a class …

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 …

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 …

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 …

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 …

$\newcommand{\Ex}{\mathrm{Ex}}\newcommand{\Sd}{\mathrm{Sd}}$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 …

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 …

I don't remember how Baues does this exactly, but all facts of this sort follow from the Gluing Lemma (see Lemma 1.4.1 in this paper) and "Brown type factorization". By this I mean the following const …

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 …

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 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. …

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 …