@CharlesRezk hahah Charles I found the exact same paper yesterday.

Surprised, but also not surprised, to find out that my brain is an ∞-topos.

@CharlesRezk do you have any sense of what the "generating cells" or "generating cofibrations" for an arbitrary ∞-topos might be? Or do you think it's really case dependent?

That question is 5+ years old now. I guess several people now have experience teaching higher algebra graduate courses. I think it'd be great to hear about that (and to share resources, perhaps)

@BrunoStonek Well it's not my place to answer that question, but Yonatan Harpaz has been giving a course entitled "Higher Algebra" for a few years (I'm not exactly sure how many, maybe 2 or 3) with lecture notes available here math.univ-paris13.fr/~harpaz/#notes ("Little cube algebras and factorization homology")

although it doesn't quite follow the road that the initial question seemed to be "advocating" because it does deal with oo-categories in a non axiomatic way

(and it focuses on operads rather than stable things)

@BrunoStonek i guess now is as good a time as any to make it known that i'm throwing my hat in the ring. i've been teaching "homological algebra" this past quarter, and am planning to make the notes into a (very discursive and informal) book. the working title is "an invitation to higher algebra".

i'd be grateful for any tips (here or by email) for navigating the world of book publishing, which is even more opaque than the world of journal publishing.

my understanding from preliminary conversations is that i shouldn't make the notes publicly available, at least until i've formalized a book agreement that allows for that. nevertheless, google exists. on that note, i'd be grateful for any feedback on the existing (still incomplete) draft, as well.

i should mention that i both do and don't treat ∞-categories in an axiomatic way. specifically, i think it's important to understand what exactly one is actually talking about (i.e. so that one is doing math, not philosophy), and so there's some material on various models for ∞-categories. however, after that (skippable) material they are used in what i'd call an axiomatic/model-independent way.

@AaronMazel-Gee That's awesome ! I don't know if I'll have the time to actually read it (and give feedback), but thanks a lot!

it's great that you're writing that, @AaronMazel-Gee , thanks!

@AaronMazel-Gee You may at least let your notes publicly available on your webpage and give links to it anywhere you want. It might be better not to make your notes available on the arXiv right away, but this is not a strict rule, and you can always negotiate with the publisher to put them there eventually for sure (my experience with Cambridge University Press and Springer is that they are quite open on these matters).

@AaronMazel-Gee By skimming your notes I have just one comment about paragraph 6.3.6, where you mention universal properties of derived categories. I think it is a pity not to mention the universal property of the bounded derived category of an abelian category A: this is a universal stable category in which A goes in an exact way (i.e. via a functor that preserves finite sums and which sends short exact sequences to cofiber sequences).

@Denis-CharlesCisinski Wouldn't it instead be a universal stable $\infty$-category equipped with a $t$-structure, receiving a functor as you described? Can the universal property be phrased entirely in terms of stable $\infty$-categories?

@PiotrPstrągowski no there is in fact a universal property of the stable oo-category - I'm pretty sure it's actually in one of Denis-Charles' paper, but I don't remember which. In any case, the idea is that you don't need the t-structure: a bounded complex can always be seen as filtered (in the stable sense) by some elements of the abelian category, and so if you know where to send them, you know where to send each bounded complex, and there's no need for a t-structure there

(of course this is not a formal argument)

Actually, I think the paper I'm thinking about also proves a universal property for bounded chain complexes in exact categories

Ah I found the paper, it's "Controlled objects in left exact oo-categories and the Novikov conjecture", section 7.6

specifically corollary 7.59

@BastiaanCnossen Yeah, it should say any class L where f_* restricts

Okay, thanks!

where can i find an explicit description of the map of operads Lie[n-1] -> E_n (namely, the map at each arity)? by this i mean the map of spectral operads which is the shift + koszul dual of the map E_n -> E_oo (so Lie is the spectral lie operad)