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I am looking for a general reference text on model categories, that contains all the basic results and definitions. I'm perfectly happy to be pointed towards a textbook, and I'm not looking for something with modern fancy theorems, just standard results. I know the definition of "model category", but not much else. I have some experience with one or two particular model categories, and I can prove any result I really need "by hand" for my particular examples, but much better would be to cite standard facts than to reprove special cases of them.

Here's the type of fact that I'd like to find in such a reference (if it is in fact true). By this I mean also that the reference should include all necessary definitions, since I'm not even sure how to make precise the following claim:

Suppose that $A$ is a cofibrant object, and $\hat B \to B$ is an acyclic fibration. For each $f: A \to B$, the space of lifts $\hat f : A \to \hat B$ covering $f$ is contractible.

The definition I know just guarantees that it is non-empty, but surely contractibility (if correctly defined) follows. Anyway, where can I find this and similar results? Something like "Model Categories for the working mathematician"?

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    $\begingroup$ I really like the presentations in May's More Concise Algebraic Topology and Riehl's Notes on Categorical Homotopy Theory. The latter is freely available at math.harvard.edu/~eriehl/cathtpy.pdf $\endgroup$
    – Justin Hilburn
    May 28, 2013 at 19:27
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    $\begingroup$ Thanks very much for the plug, but my coauthor's name, Kate Ponto, should be mentioned! (She was my student, and so was Emily Riehl). $\endgroup$
    – Peter May
    May 29, 2013 at 3:05
  • $\begingroup$ Quillen's book is in fact mentioned in an answer below. $\endgroup$
    – Sean Tilson
    May 29, 2013 at 17:51
  • $\begingroup$ If you want, you can also take a look at these notes: math.jussieu.fr/~vezzosi/seminar/notes.pdf, realized during a seminar organized this year by Gabriele Vezzosi. There might be mistakes, but they should also offer an overview of the theory (except for some more advanced results). (Beside that, if you find errors reading them, you can contact me freely) $\endgroup$
    – Mauro Porta
    Jun 14, 2013 at 12:36

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For an introductory textbook I will recommend Homotopy theories and model categories by Dwyer and Spalinski. This 56-page paper is one chapter of the book "Handbook of algebraic topology" and gives a reader-friendly and comprehensive introduction to model category.

I think the book "Model Category" by Hovey, as tetrapharmakon and David has already recommended, is also very good and it contains some deeper results. However on the other hand it is a little bit more difficult to read (at least to me).

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Hirschhorn's book, Model categories and their localizations, is a very thorough reference with many basic results explicitly stated and proved. The result you want is implied by axiom SM7 for simplicial model categories (see Proposition 9.6.1 in op. cit.), and for general model categories I suppose Corollary 16.5.4 is the closest analogue.

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  • $\begingroup$ Hirschhorn's book is an excellent, thorough reference for model categories. I would only warn to start Hirschhorn book at the second part. The first part is maybe too specialized? $\endgroup$
    – Agustí Roig
    Jul 13, 2020 at 6:56
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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 to maximize intuition for a (higher) category theorist, and focuses on developing a 2-category of model categories while simultaneously laying down all the basic theory in as concise a way as possible. Many people like this book because it strikes a good balance on the scale of readability vs. too-many-details. It manages to develop all the basic theory one would need and to contain all the major classical examples (including chain complexes and comodules over a Hopf algebroid, which don't seem to appear in Hirschhorn).

By the way, you should tell us what kind of application you have in mind. If you're interested in things like operads and algebras over an operad, then there are good arguments that what you really want are semi-model structures. In this case, the book Modules over Operads and Functors, by Benoit Fresse is very good.

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  • $\begingroup$ Hi, I am enjoying reading Hovey's Model categories book. Thanks for the suggestion... The book seems to have some typos. For example, in page $3$ before the definition $1.1.3$ of model structure, it should be $h:B\rightarrow X$ instead of $h:B\rightarrow Y$.. There is an errata hopf.math.purdue.edu/Hovey/model-err.pdf by Mark Hovey but it does not contain the typo I have mentioned.. Do you know any place which collects some more typos? Do you have any specific suggestion to read that book? $\endgroup$
    – Praphulla Koushik
    May 23, 2020 at 3:27
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Hovey's book seems exactly what you need. But take a look at http://folk.uio.no/paularne/SUPh05/DS.pdf which is nothing but a self-contained rewriting of Quillen's original work "Homotopical Algebra"; if you want to learn a thing from the basics, nothing better than reading the work of the people who invented that thing! :)

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I like the book "Abstract Homotopy Theory and Simple Homotopy Theory" by K.H. Kamps and T. Porter, which gives many examples, such as groupoids and crossed complexes, and also gives quite a bit on cylinder objects in the examples. It also gives an account of axioms on cubical sets which control homotopy behaviour, following the lead of a paper by Kamps, "Kan-Bedingungen und abstrakte Homotopietheorie", Math. Z. 124 (1972) 215-236.

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If you're interested in model categories in the context of operads, the book Homotopy of Operads and Grothendieck-Teichmuller Groups Part 2: The Applications of Rational Homotopy Theory Methods by Benoit Fresse has some relevant discussions of model categories near the beginning.

Also the survey paper by Dwyer-Spalinski which someone else mentioned is nice.

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    $\begingroup$ Dwyer-Spalinski was already mentioned by Zhaoting Wei. But I agree that Fresse's writings are great. $\endgroup$
    – David White
    Jul 13, 2020 at 12:00

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