# Where to learn about model categories?

A model category (introduced by Quillen in the sixties) is a category equipped with three distinguished classes of morphisms (weak equivalences, fibrations and cofibrations) satisfying some axioms designed to mimic the structure of the category of topological spaces. They provide a framework to study homotopy theory which applies to topological spaces (algebraic topology), chain complexes (homological algebra), and much more. In particular the notion of Quillen equivalence allows one to say when two categories are the same "up to homotopy".

What are good places (book, lecture notes, article...) to learn about model categories?

Assume that the reader is reasonably familiar with category theory, and perhaps with the basics of algebraic topology or homological algebra if some motivation is needed. I would also be interested in more advanced books that deal with finer points of model category theory (localizations come to mind).

This question is inspired by this previous one where the OP asked whether to learn $\infty$-category theory. I believe knowledge of model categories is rather essential to learn $\infty$-category, as for example many coherence results are stated in the form of "Such and such categories of $\infty$-categories are Quillen equivalent". As far as I understand much of $\infty$-category is also designed with model categories in mind, basically "what happens if I'm in a category of bifibrant objects and derived hom spaces". See also the MO question Do we still need model categories?

If possible try to argue why the book/paper you're mentioning is a good place to learn (and not just assert so), please. By now I think the subject is settled enough that there are several good books about it, and I think it would be nice to condense information about all these books in one place, instead of relying on hearsay.

• Heh, I just stumbled upon this old question. Six years later, and I'm the one teaching model categories :) I gave a lecture at the University of Paris on the subject: idrissi.eu/class/20-21-homotopie/ (the videos are in English but the notes are in French though). Jun 8, 2021 at 7:53

I like

Philip S. Hirschhorn. Model categories and their localizations. Mathematical Surveys and Monographs 99. Providence, RI: American Mathematical Society, 2003, pp. xvi+457. ISBN: 0-8218-3279-4. MR1944041.

First, a word of warning: the book is divided into two parts, but the first part depends logically on the second part. Thus, beginners should start at Chapter 7, not Chapter 1. The great thing about [Hirschhorn] is that all the definitions and results are stated carefully and clearly, so it serves very well as a reference. It also goes as far as constructing mapping spaces and homotopy (co)limits for a general model category, which might be considered advanced topics.

One downside of [Hirschhorn] is that it takes the standard model structures on $\mathbf{Top}$ (or, more precisely, a convenient subcategory) and $\mathbf{sSet}$ for granted. It also assumes some familiarity with basic homotopy theory for some proofs, but this is probably fair enough.

Now, as for whether you need model category theory for $(\infty, 1)$-category theory, that's a discussion for another day...

W. G. Dwyer and J. Spaliński. “Homotopy theories and model categories”. In: Handbook of algebraic topology. Amsterdam: North-Holland, 1995, pp. 73–126. DOI: 10.1016/B978-044481779-2/50003-1. MR1361887.

This paper is an introduction to the theory of model categories. The prerequisites needed to read it are very limited, and most (if not all) the proofs are detailed. All the basics of model categories are explained: basic definitions, homotopy category, derived functors, homotopy pushouts and pullbacks... Two fundamental examples (topological spaces and bounded below chains complexes) are also made explicit.