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I studied some basic algebraic topology (homotopy/homology/cohomology groups). When reading about the Dold-Thom theorem, the fancier and more recent sources sooner or later all started to use homotopy limits, homotopy pushout squares etc. which I couldn't understand.

Hence, I was wondering which source suits best to start learning about homotopy theory. Any recommendations?

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    $\begingroup$ If you've already had some exposure to algebraic topology, Peter May's "Concise course..." and May and Ponto's "More concise..." could be good places to look. Also Jeff Strom's "Modern homotopy theory". $\endgroup$
    – Mark Grant
    Commented Jan 27, 2023 at 10:42
  • $\begingroup$ Not claiming to be the "best", but you could try this course. $\endgroup$
    – Z. M
    Commented Jan 27, 2023 at 16:42
  • $\begingroup$ If all you want is things like homotopy limits and push-outs, Volic's "Cubical homotopy theory" gives you quick access. I think it's available on-line. $\endgroup$
    – Ryan Budney
    Commented Jan 27, 2023 at 18:13
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    $\begingroup$ It's quite regrettable that the very basic tools of homotopy such as homotopy limits/colimits are essentially not present in the basic textbooks. Hatcher, for example, does treat a few special cases (mapping cone/cylinder, mapping telescope), but there is no mention of the general case. There are a few pages in the book of May–Ponto and a rather detailed treatment in Munson–Volić (unfortunately, only for topological spaces). A quick introduction can be found in Dugger's “A primer on homotopy colimits”, which is one of the more popular sources on this topic. $\endgroup$
    – Dmitri Pavlov
    Commented Jan 27, 2023 at 20:46
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    $\begingroup$ For limits, section 15 of Gray’s book should serve at least as an introduction bearing in mind that push out or pull backs could be seen as limit of some directed systems! The book is also a very nice introduction to many classic topics in homotopy theory. $\endgroup$
    – user51223
    Commented Jan 29, 2023 at 20:02

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