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I'm not actually a category theorist, but assume I have some background in category theory and am pretty comfortable with it; and I want to learn higher category theory (say in the sense of Boardmann-Vogt-Joyal-Lurie) but don't want to have to delve into the intricacies of simplicial sets and anodyne extensions and other scary-looking things (to the eyes of someone who isn't very good in combinatorics and dealing with indices)

I've read in several places (but there was just a sentence or 2) that it would be possible to actually just learn "the language" of $(\infty,1)$-categories, how they are manipulated etc. without having to go through said intricacies.

A comment by Qiaochu Yuan on another post had the same idea, but explicitly stated that he didn't know where to find such a treatment. The passage that mainly sums up what I'm looking for is :

" learn some model-independent things that ought to be true in all models of (∞,1)-categories (e.g. formal statements about the behavior of homotopy limits and colimits)"

I would add "behavior of functor categories, adjoints, (Kan extensions if they make sense ?)" and other nifty $1$-categorical stuff, done homotopically. Emily Riehl's model-independent approach has been mentioned a bunch of times, but when I tried to see what it was about (the title sounded promising) I realized that it relied (I don't know how heavily, but at least it required some stuff) on the already existing theory of $\infty$-categories, which I would therefore have to learn (with the afore-mentioned intricacies) before learning this model-independent treatment, which would defeat (some of) the purpose of learning the model-independent story (I guess a subquestion would be : how much of "model-dependent" $\infty$-category theory does one have to go through before reading Riehl's approach ?)

Is there such a treatment ?

To be more precise, I would be fine with a document that says stuff like "this works well because we can check it simplicially" and then moves on to build on these "simplicial stones hidden under the rug" and actually do some proofs that are more "categorical" in a sense, based on these accepted principles.

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  • $\begingroup$ You could try Carlos Simpson's book Homotopy Theory of Higher Categories. $\endgroup$
    – Tyrone
    Jul 11, 2019 at 10:30

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No, there's no such treatment, if Riehl-Verity doesn't work for you. You can learn certain aspects of the theory by reading about derivators, such as in Moritz Groth's thesis. But this is not by any means a replacement for reading Riehl-Verity or Lurie. There is no known way to get a fully functioning treatment of homotopy coherent category theory without some use of simplicial stuff. This is basically natural, since $(\infty,1)$-categories are categories weakly enriched in homotopy types and simplicial sets are the best behaved way of modeling those. Ideally you would really just be working with raw homotopy types, but at that point you're probably working in homotopy type theory and no such treatment of higher category theory is anywhere close to extant. Riehl and Shulman have a relevant paper but I doubt one can understand it without some comfort with the intended model in bisimplicial sets.

EDIT: To answer the sub-question, the intention of Riehl and Verity's work is that one should be able to read it without prior exposure to higher category theory. I don't personally know anyone who's read it from that stage, so I can't guarantee that there aren't hidden challenges.

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  • $\begingroup$ Thank you for your answer. If you have the time, and if you know the answer, could you also comment on my "subquestion" which is : how much "model-dependent" $\infty$-category theory must one learn before embarking on Riehl-Verity ? $\endgroup$
    – Maxime Ramzi
    Jul 11, 2019 at 8:07
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    $\begingroup$ " [...] and simplicial sets are the best behaved way of modeling those." This is a bold thing to say. Just a superficial reading of Pursuing Stacks will tell you that Grothendieck himself did not seem to feel the same way (I think that he would argue that small categories are way better as modelling homotopy types and that simplicial sets just happen to be presheaves on a test category). I agree with the rest of your answer though. $\endgroup$
    – Pece
    Jul 11, 2019 at 12:35
  • $\begingroup$ @Pece It's not bold at all. You may have taken this as a metaphysical claim, but I meant it empirically. As mathematics has developed in our world, homotopy types have been understood mostly via the simplicial set and topological space models, and the latter has major disadvantages for categorical development. The Thomason presentation of homotopy types via small categories, beautiful as it is, has seen rather limited use in homotopy theory as it actually exists. It is perhaps relevant to remark that Grothendieck did homotopy theory only in his post-career. If he had started while he was... $\endgroup$
    – Kevin Arlin
    Jul 11, 2019 at 14:02
  • $\begingroup$ ...still producing research in the style of the '60s, perhaps we would have ended up making more use of categories for homotopy types. But there's but no effort at all to study $\infty$-categories using the Thomason model. $\endgroup$
    – Kevin Arlin
    Jul 11, 2019 at 14:04

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