Homotopy Theory

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Eric Peterson

04:46

@MingcongZeng in general, idk, but the 'tate vanishing' type results sound like this?

4 hours later…

Saal Hardali

08:54

@TylerLawson Thanks! This is exactly the kind of example I needed. Do you know where can I read more about this? Can this happen for $E=H\mathbb{Z}$?

5 hours later…

Espen Nielsen

14:08

So if $C$ is a small $\infty$-category, the presheaf category $P(C)$ is accessible and the compact objects are precisely the representable functors, right? For some reason I'm having trouble finding this spelled out.

Denis Nardin

@EspenNielsen I don't think that the compact objects are $C$ in general. I think that is true only when $C$ has all small colimits and is idempotent complete. Let me take a look in HTT...

(silly example: C=Δ^0, then P(C) is the ∞-cat of spaces and its compact objects are bigger than just the point)

Dylan Wilson

the representables in P(C) are way more than compact... they're tiny. Mapping out of them preserves all colimits.

Denis Nardin

Right, I just realized. I don't that it will ever happen that the compacts are all representable

Espen Nielsen

Oh, I see. But can we still identify $P(C)$ with ind-objects in $C$?

Denis Nardin

No, it's the formal cocompletion, it's way bigger

Espen Nielsen

14:24

That's what I feared. Thanks!

Denis Nardin

Ind(C) is the subcategory of P(C) of those presheaves that preserve finite limits

Tyler Lawson

14:55

@SaalHardali That can't happen for HZ, but the proof that H_* (free(X)) is a functor of H_*(X) relies more on Z being a PID than it relies on Z being commutative.

I believe it fails for HZ/4 even when talking about E_1-algebras

2 hours later…