@SeanTilson Yes I've read that paper. The proof I'm thinking of is different (and to my eyes much more natural). However I'm not going to claim anything more until I've actually finished it :)

i know i'm late to the party on this one, but i just added a comment to the discussion between peter may and lennart meier about the utility of $\infty$-categories for studying tmf (specifically in akhil & vesna's paper) mathoverflow.net/questions/185997/why-higher-category-theory/…

lennart pointed out one thing they do $\infty$-categorically that might be hard model-categorically, but there is a far more serious usage of $\infty$-categories in there that looks to be more or less impossible only using model-categorical machinery

here's a fun exercise I just noticed: for fixed n, which spheres have the property that there are countably many isomorphism classes of n-dimensional real vector bundles on them?

yeah, you basically just rediscovered derivators, Aaron

@QiaochuYuan idk, which ones

@AaronMazel-Gee (To your remark to me yesterday) Thanks. That means this construction is even more horrible than I anticipated. :P

Ugh, nevermind, the argument is probably as ugly as I thought

@DenisNardin thanks, I just can't keep lots of things straight when it comes to this stuff.

When people speak of the "category of spectra", do they usually mean with morphisms given by maps, or equivalence classes of maps (the whole cofinal subspectrum business)?

Specifically I'm wondering about definition 3.9 here: arxiv.org/abs/0810.0581

Alternatively, if anyone knows Josh Genauer's email address, I can ask him personally. :)

If you work with $\Omega$-spectra you can take maps on the nose

(defining homotopies between maps still requires the cofinal business though)

@DenisNardin Thanks. I am mainly interested in (diagrams/colimits of) Thom spectra, but I guess it will work out if I omega-replace everything.

Honestly you can usually assume that you are working in the $\infty$-category so that there's no more ambiguity (and you get a better description of Thom spectra as a bonus). But I guess it depends on what you're trying to do

Right. I haven't really made my peace with stable infinity-categories yet, but if it offers computational advantages, then it is definitely worth looking into.

I don't think you're going to get computational advantages. But it is liberating to be able to talk about "the category of spectra" without feeling the urge to air-quoting it

@Adeel awesome -- please feel free to explain more as an answer there!

(or @ZhenLin, whoever gets to it first -- or both of you, the more the merrier)

I already gave a brief explanation in a previous email...

ah right, i'll go back and re-check to see how it lines up with my question

okay, so the point is that the derivator is essentially the data of this cartesian fibration that i called $X_C$ in my question?

correct. the axioms, which are complicated, express the compatibility conditions that we expect for colimits

or maybe it's the functor $\mathrm{Fun}(-,C)$ really

i see

conventionally, derivators are defined as 2-functors because they are so in practice

sure

so then, how does this connect to the 2-functoriality of colimits? is there the functor $X_C \to C$ that i suggest?

I believe so, but in your situation the existence can be directly verified anyway

in a cartesian square (say a pullback of vector bundles over a map of spaces), if the downstairs map is X --f--> Y, does anyone have any good suggestions for the induced upstairs map on vector bundles? i'd almost like to say f*, but really that should be the name for the functor VB(Y) --> VB(X) between fibers of the cartesian fibration for which this arrow is cartesian

@QiaochuYuan nevermind

@AaronMazel-Gee um, f'?

@DenisNardin Hello.

Hello

@DenisNardin Nice meeting you the other day in MIT.

@DenisNardin I didn't end up meeting Jacob Lurie as I fell ill.

Uh I'm sorry

Cool

Do you know if ppl do like this derived stuff there?

No name springs to my mind, but that's probably just ignorance on my part

Ok. It seems quite big in harvard and possibly at MIT too.

There's certainly a very good homotopy theory group there, there's bound to be someone there thinking about this stuff