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I am a former student of Mark Hovey's, and during grad school, I wrote a document giving an update on the status of the 13 problems (as of 2012 or 2013, I guess). I just briefly went through it a mome …

Abstract homotopy theory allows one to use the tools of homotopy theory (e.g. inverting weak equivalences, computing homotopy colimits, doing Bousfield localization, taking fibrant and cofibrant repla …

In response to Ryan Budney's comment, let me try to say something about topological data analysis, and other recent applications of algebraic topology outside of traditional mathematics. Applied Alge …

There is a whole slew of examples given by the $J$-semi model structures which arise in the study of operads and algebras over an operad. A $J$-semi model category satisfies most of the axioms of a mo …

Yes, this has been done in other settings. For example, Scott Balchin, Kyle Ormsby, Angélica M. Osorno, and Constanze Roitzheim wrote a paper, Model structures on finite total orders, that enumerates …

I guess I'm the canonical person to answer this question. I wrote those notes as a PhD student, a long time ago, to go along with a talk I was giving at a grad student conference. They were basically …

Given $f:A\to X$ and $g:B\to Y$, the map in your question is usually called the pushout product of $f$ and $g$, and often denoted $f\Box g$. The property in your question is very related to the pushou …

The injective model structure is monoidal (satisfies the pushout product axiom), see Hornbostel's paper "Localizations in motivic homotopy theory", Thm 1.9 and Lemma 1.10. The projective model structu …

An example of a different sort is the model structure on $R$-mod, whose homotopy category is the stable module category. A great reference is Theorem 2.2.12 in Hovey's book Model Categories. In this r …

Voevodsky won a Fields Medal for resolving the Milnor Conjecture in number theory. His work fundamentally used model categories, and kicked off the field of motivic homotopy theory. Model categories h …

sSet is not fibrantly generated. I originally thought the issue would be with the lack of cosmall objects, but it's even worse than that. In fact, there is no set of maps in sSet that can detect the a …

I agree that Hirschhorn is very complete, but it can be hard to find things in it. That's why I'd recommend Model Categories by Hovey instead (it also seems a more canonical reference). It's written t …

Yes, the structure you describe forms a model structure. However, some care is needed. First, topological groups are not algebras over an operad, because operads don't encode inverses. Topological mon …

(1) The main benefit of a simplicial model category structure is explicit formulas for (co)simplicial resolutions. Remark 5.2.10 in Hovey's book identifies the functor $A\mapsto \tilde{A}^m = A\otimes …

To me, the interest in model categories stems from Quillen's observation that the tools of topology (e.g., CW approximation) can be applied in so many different settings, especially in algebra. But no …