Search Results

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 …

This is far from a comprehensive answer, but since you ask for references, here are two. The result for simplicial sets is obtained by an explicit computation in Appendix H of Joyal's The Theory of Q …

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 …

I was quite sure that this was explicitly written down by Rădulescu-Banu or by Hirschhorn, but I cannot find this exact statement. However, this follows directly by combining Theorems 7.5.10 and 7.6. …

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. …

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 …

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 …

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 …

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

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 …

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 …

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 …

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 …