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This is a soft question.

I am currently learning higher category theory in the context of stable homotopy theory and find many concepts more intuitive now that I have taken a basic course in algebraic topology and homotopy theory.

Constructions in higher category theory are strongly tied to ideas in homotopy theory. I have heard various different opinions on whether one should have a very solid background in homotopy theory before learning higher category theory.

Question: How much homotopy theory would you recommend to learn before learning higher category theory and what branches (rational, (un-)stable, equivariant,...) in particular?

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    $\begingroup$ Are you asking the question "in general" or in the context of someone learning this stuff "for stable homotopy theory" ? $\endgroup$
    – Maxime Ramzi
    Commented Sep 30 at 9:44

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I guess it depends on what you mean by homotopy theory and on what kind of mathematician you are.

As a general statement, I think it's good to have some background in algebraic topology before embarking on Homotopy theory, as many constructions come from there (this is -- say -- the content of Hatcher's book or May's concise book). That said though, nowadays many people enter this topic directly from where Quillen left it, say from the content of Hovey's book on model categories.

If you are somewhat familiar with the topics above, you are very much ready to enter higher category theory and personally I recommend the book by Cisinski, Higher Categories and Homotopical Algebra.

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  • $\begingroup$ Do there really exist people who learn about model categories with first learning algebraic topology? What would their motivation be? $\endgroup$
    – Andy Putman
    Commented Sep 29 at 15:29
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    $\begingroup$ @AndyPutman I guess I exaggerated, but I have met many people from Bonn who speak the language of model categories much better than they speak the language of Postnikov towers, or spectral sequences. Of course you will hear something (!) about algebraic topology before model categories... $\endgroup$
    – Ivan Di Liberti
    Commented Sep 29 at 15:31
  • $\begingroup$ @Andy Putnam: “with” means “without”? $\endgroup$
    – ThiKu
    Commented Sep 29 at 16:09
  • $\begingroup$ @ThiKu: Yes, sorry for the typo. $\endgroup$
    – Andy Putman
    Commented Sep 29 at 16:41
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    $\begingroup$ @AndyPutman: this agrees with my observations as well. The shortest route to training a homotopy theorist to where they can successfully publish papers seems to often avoid getting familiar with spaces and manifolds. Very important topics, yes, but the market often creates perverse pressures. $\endgroup$
    – Ryan Budney
    Commented Sep 29 at 18:02

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