Homotopy Theory

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skd

12:51 AM

cool, thanks @ArunDebray

4 hours later…

Jonathan Beardsley

5:16 AM

So Comm is the terminal symmetric operad in, say, Top, or Set or Cat, but this is not true in, say, Ch_k or Vect_k, right?

(which may be the reason that algebraic people like Fresse and Vallette often restrict to Hopf operads)

Anyway I'm just sort of thinking about the fact that, when Comm is terminal, this means that for any operad O, "every commutative algebra has an underlying O-algebra," which seems... strange to me.

And I'm wondering if there are funny k-linear operads such that commutative algebras do NOT necessarily pull back to them. And what such things look like.

Denis Nardin

5:42 AM

@JonathanBeardsley Not a perfect example, but what about the Lie operad?

I mean, you have the zero map Lie→Comm (zero on all operations of arity >=2), giving you the trivial Lie algebra, but that doesn't really use Comm

Jonathan Beardsley

5:58 AM

Right, I guess in k-linear examples you actually always can factor through the zero operad (which I guess is a zero object in the category of operads there?)

But yeah I was thinking about the Lie operad there for a second. What about the commutator construction that takes the associative algebra structure and produces a Lie algebra structure?

Maybe that's not "underlying" because the binary operation isn't the same?

It's also sort of interesting to me that, despite Comm not being terminal here (BECAUSE of the k-linear structure), it's precisely the k-linearity that allows one to take commutators

Additivity giveth, and additivity taketh away.

Denis Nardin

6:21 AM

@JonathanBeardsley I don't know if the trivial operad is the zero object in k-linear operads, but the composite of the "underlying" functors Lie→Ass→Comm factors through Triv and in fact is what I called the "zero" map

Jonathan Beardsley

6:36 AM

@DenisNardin what's the map from Lie to Ass?

Oh I'm being silly, haha,the commutator is zero.

So we're talking about the same thing here.

3 hours later…

Tom Bachmann

10:05 AM

@MarcHoyois great, thanks!

Tom Bachmann

10:21 AM

I guess I should seriously try to read SAG...

6 hours later…

Saal Hardali

4:44 PM

I'm terribly confused about finite and smashing localizations. I thought being a finite localization is equivalent to the inclusion of acyclics preserving compact objects but I just now realized that I can "prove" with this definition that smashing implies finite, like this:

Let $L$ be a smashing localization functor and let $G$ be the corresponding acyclization (both as endofunctors of spectra). By the equivalent characterization of smashing $G$ preserves colimits and therefore the right adjoint to the inclusion of the acyclics must preserve colimits as well. This is equivalent to the inc…

Saal Hardali

5:06 PM

Maybe this has to do with Idempotent completion issues...

1 hour later…

Drew

6:35 PM

@SaalHardali How do you know that the compact objects in the category of acyclics are actually finite spectra?

In any case, for this type of thing I would always check Hovey--Palmieri--Strickland, Section 3.3 (Smashing and finite localizations!)

Saal Hardali

7:26 PM

@Drew If the inclusion of acyclics preserves compacts then then compact acyclic spectra must be compact spectra and hence finite.

Drew

8:12 PM

To conclude that the inclusion preserves compacts, you need that the category of acyclics is compactly generated (one way of phrasing the telescope conjecture is precisely this - the acyclics of any smashing localization are compactly generated)

skd

@Drew, is it true that any smashing localization must necessarily be E(n)-localization?

Eric Peterson

i don't think so; p-localization is given by smashing with S_(p), whereas any E(n)-localization (including "E(infty)") will factor through BP-localization

Drew

Even ignoring those, an answer to that question still involves the telescope conjecture

Q: Smashing localizations in the category of spectra

Let $E$ be a spectrum. Then $E$ determines an idempotent localization functor $L_E: \mathrm{Sp} \to \mathrm{Sp}$ sending each spectrum to its $E$-localization. The functor $L_E$ generally does not commute with homotopy colimits. (It does send homotopy colimits in spectra to homotopy colimits in...

skd

8:34 PM

(sorry, yeah, I meant localizations which are not localizations at spheres)

thanks for that link

Saal Hardali

8:47 PM

@Drew That does solve the issue but now I'm surprised to discover that there's now a huge source of examples for non presentable categories. Just take any colocalization of spectra and most probably it won't be presentable (as it sounds like this is kind of a rare condition from your answer)

In particular what you're saying implies that the category of $E(n)$-acyclic spectra is not known to be presentable. I find this hard to believe somehow, but I guess such is life...

@skd What i'm pretty sure about is that any smashing localization sits between telescopic localization and $E(n)$-localization for some $n$

So if these coincide they all coincide.

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