Homotopy Theory

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Harry Gindi

09:13

@DylanWilson if you are thinking about quasicategorical operads, the equivalence of the quasi-operads in Dendroidal sets from the work of Moerdijk and his various collaborators with Lurie's notion from HA, I just learned at this conference

the original tensor product that is isomorphic to the Boardman-Vogt tensor product is not associative for them, but they found a weakly homotopy equivalent construction that is associative, so it's ok

this failure of on the nose associativity maybe exists in Lurie's version as well

6 hours later…

Bogdan

15:12

Just trying to understand the left/right completion of a t-structure: Is the standard t-structure on Spectra (where left/right bounded objects are co/connective spectra) left or right complete ? If no, what are the left/right completions ?

Feels like one of the should be pro-spectra maybe ?

Saul Glasman

If I understand HA correctly, then you're right - spectra are left complete but not right complete

The right completion isn't pro-spectra but the full subcategory of Postnikov systems therein

sorry, I may have right and left transposed there

yes, please swap right and left in what I just said

the left completion somehow feels like the one that should be called "right" to me

Bogdan

15:30

yeah I feel the same, I guess it's because of the homological indexing convention here

Thanks for that. How do you see that they are right complete ?

Saul Glasman

the way I think of it, spectra are equivalent to ind-finite spectra

and the ind-system of finite spectra mapping to a spectrum is a refinement of the ind-system of bounded-below truncations

maybe that's sort of a logically convoluted way of saying it, though

you should also be able to directly see that the mapping space between the ind-systems of truncations computes the mapping space of spectra

Bogdan

but the two notions of left and right complete are both homotopy limits and not colimits, right ? Aren't those pro-systems instead of ind ?

Saul Glasman

hmm, I thought the right completion was the colimit of the \geq n categories

Is it actually something else?

Bogdan

sorry gotta go, I ll try to think more. (I think both are homotopy limits, just on the other side of the t0-structure)

Saul Glasman

Lurie just says "oh it's the dual"

Tyler Lawson

15:42

i'm really confused, I thought it was left + right complete

Saul Glasman

uh oh

ok, let's see - is it the case that left completeness is equivalent to the mapping space between the Postnikov systems being equivalent to the mapping space of spectra?

if so, is that, in fact, true?

Tyler Lawson

yeah, i thought it was essentially stating that for any X the map X -> holim tau_{leq n} X is an equivalence

if you have that, then

Bogdan

makes sense yes

the fact that Postnikov systems converge or something

Tyler Lawson

Map(Y,X) = Map(Y, holim tau_{leq n X}) = holim Map(Y, tau_{leq n} X) = holim Map(tau_{leq n} Y, tau_{leq n} X)

Saul Glasman

but maps between pro-objects aren't necessarily the same as maps between the limits, right?

Tyler Lawson

15:47

right, so maps between the pro-objects should be

holim_n hocolim_m Map(tau_{leq m} Y, tau_{leq n} X)

but that's constant in m as soon as m >= n

Saul Glasman

ah, right

good - sorry for misleading with my mistake earlier

Tyler Lawson

is the "dual" taking the opposite category with the t-structures flipped from less to greater and so on?

Saul Glasman

that's exactly the argument I was going to give that spectra are right complete, so I don't know why I didn't notice that it works both ways

Tyler Lawson

ah, here, HA 1.2.1.19 says: if you've got countable products that preserve connectivity, then being left complete is the same as having the intersection of the ">= n" categories being empty

seems like the dual to that is: if you've got countable sums that preserve coconnectivity, then being right complete is the same as the intersection of the "<= n" categories being empty

Saul Glasman

right

presumably examples of non-left-complete t-structure come from stabilizing non-hypercomplete topoi or something

Tyler Lawson

15:53

but yeah, I agree, the right completeness is basically because everything's a filtered colimit of finite things

yikes. yeah, maybe

that seems likely to be a place where Postnikov towers don't converge

Dylan Wilson

16:06

y'all have it right but for an explicit reference there's HA.1.4.3.6

(so Sp is both left and right complete)

1 hour later…

skd

17:25

@YuriSulyma i've been attending the seminar, and it seems really cool. as i understand it, there's a fully faithful functor {p-divisible groups over k} -> {vector bundles on the FF curve}, where k is a perfect field of characteristic p

moreover, a deformation of a p-divisible group over k is the same as a modification of the vector bundle on the FF curve, away from oo

so, if you're interested in the deformation theory of p-divisible groups, studying the geometry of the FF curve seems to be one way to go

also --- and i don't understand this yet --- from a cursory reading of scholze's icm address, it seems like point on the FF curve parametrize untilts of a perfectoid fields of positive characteristic

another thing: this idea of deformations being the same as modifications of vector bundles gives a description of the lubin-tate space at infinite level (6.3.10 of scholze-weinstein); but the lubin-tate tower doesn't lift to homotopy theory, so idk how useful this is

Jonathan Beardsley

@skd the lubin Tate tower does actually lift to homotopy theory in a certain sense. Andrew Salch has worked this out but not, afaik, written it down anywhere.

skd

@JonathanBeardsley could you tell me more about this? (it doesn't lift under the most naive interpretation of the word "lift".)