# Introductory book for homotopical algebra

I'm interested in learning homotopical algebra (by which I mean: model categories, simplicial methods, etc.) However, I've been unable to make heads or tails of any of the "standards" (Jardine&Goerss, Hovey, Hirschorn); they seem to presuppose knowledge of the subject material. What are some accessible introductions to this subject? (+ reading paths to get to the aforementioned "classics"?)

Background: I'm very comfortable with category theory and homological algebra, am learning enriched category theory, and have had a course in algebraic topology (and am currently studying more).

• I found the appendix to "Higher Topos Theory" helpful (as a supplement). Also, another thing you might enjoy, while learning this, is working out (since you've studied homological algebra) the fact that the (unbounded) derived category of $R$-modules is the homotopy category of the category of chain complexes of $R$-modules with the usual model structure. Many of the techniques when working with the derived category are kind of analogous to this model category formalism (e.g. the "derived functors" work out the same). Sep 15, 2011 at 2:29
• Also, I found the article by Goerss on the subject at jdc.math.uwo.ca/summerschool really fun. One application of the fact (explained in this article) that one can construct model structures on simplicial $R$-algebras for $R$ a commutative ring is the construction of the "cotangent complex": it is a nice concrete example of a (non-abelian!) derived functor. Sep 15, 2011 at 2:31

Have you tried to read Hirschhorn, but starting on Part 2? -The first part is the real purpose of the book -localization of model category structures-, but more specialized and advanced. The second part is designed to serve as a support of that, more advanced, first part, and contains all the basics of homotopy theory (model categories). I would try, at least, with chapters 7, 8 and 9 -see what happens: I think it's not intended to be a "pedagogical" book on model categories, but a reference for the results on the first part. Nevertheless it is, first of all, systematic, and secondly, quite readable.

• I second this. Let me add: A gentle introduction is in the survey by Dwyer and Spaliński, Homotopy theories and model categories. Handbook of algebraic topology, North-Holland, 1995, MR55014. It's availablee.g. on Dwyer's home page. There is a sequel to May's Concise course in press, written by May and Ponto, More Concise Algebraic Topology: Localization, completion, and model categories.
– t.b.
Sep 14, 2011 at 10:40
• I've started reading Part 2, and it's perfect! Thanks; that'll teach me to skip introductions. Sep 21, 2011 at 4:20
• I'm glad you're liking the second part. Don't forget Theo's advice: Dwyer and Spalinski is a great introduction. I particularly liked their very specific example about the necessity of deriving colim -and thus giving raise to hocolim. Sep 21, 2011 at 8:04
• The survey Homotopy theories and model categories can be found here now. Nov 8, 2022 at 13:59

I don't know if it is quite an introductory book but Quillen is not bad at all.

Dwyer and Spalinski is good as well.

There is a section in the Motivic homotopy theory book written by Bjorn Dundas (the section is by Dundas, the whole book is by a few other people as well). This might give the overall picture before you look for something more detailed.

• Have you tried "Abstract Homotopy and Simple Homotopy Theory" By K H Kamps , T Porter (World Scientific) 1997? It gives examples not considered in the other books. Dec 2, 2016 at 12:24
• Maybe you should add this as a separate answer. Dec 5, 2016 at 13:33