Homotopy Theory

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1:24 AM

@AaronMazel-Gee I was pointing out the factual error in the statement that there are only a handful of (extremely selective/prestigious) departments with people who are serious about homotopy theory. There are probably 3-4 times as many departments that satisfy that requirement if one considers Paris (where one can collaborate with anyone in any department across the city), some other places in France, several extremely strong places in Germany,

several extremely strong places in the Netherlands, Australia, and probably a ton of others that I'm forgetting.

Also, I haven't had bad experiences with Algebraic Geometers. I was just saying that the culture of expertise in that field is much more overtly aggressive than anything I've seen in Homotopy Theory (something that I don't personally mind!).

7 hours later…

Harry Gindi

8:06 AM

Hey, has anyone ever done a proof of proper basechange by an appeal to the valuative criterion?

It's pretty easy to reduce to the case where you ask the question for something proper over a strict henselian local ring and you pull back along the inclusion of the closed point. This reduces to asking that the canonical map from global sections to global sections of the pullback to the special fibre is an equivalence.

so in this case, proper basechange becomes a kind of cofinality argument

with respect to étale specializations

(for constructible sheaves)

blank_space

8:22 AM

Thank you for responding so kindly. I am seriously considering your offer.

After a few minutes' struggle, I was able to use existing results to reduce your Moore loop problem to a possibly harder question which appears even more natural, but I'm afraid even this may be wrong in some trivial way.

Cockroaches are survivors too, but thank you. Feeling fraudulent is pretty ingrained by now. It actually gets worse the older I get and the more people expect or imagine of me.

I don't want to question how you feel, but I find it hard to account for your not feeling like a part of this community. Is there some profession-wide persecution complex we're all unwittingly imbibing?

You and I have indeed had interactions on the StackExchange sites, with what I'd characterize as mixed results. If there is some direct messaging aspect to this platform and if you'd like, I can recount them for you.

@AaronMazel-Gee I really appreciate the kindness of your response and am genuinely considering emailing. There are many things that I want to respond to, but I'm weighing whether it's more than anyone else is going to want to have to scroll past.

You didn't @ me, so I'm not sure how much of that was directed toward me, but for the portion I think was, I'm responding in as polite a manner as I can muster, after some half a dozen rewritings.

I could add three departments to what you said and argue the number up to perhaps two dozen, but as you point out, these are all extremely selective institutions, and I don't see what the point in trying to fix an exact number would be. Whether graduate opportunities in homotopy theory are extremely limited or merely very, I was nearly out of grad school before I got interested in it, and what I'…

Harry Gindi

=) I try to keep it light

Denis Nardin

@blank_space SE deliberately has no direct messaging. Feel free to email me though (possibly from an anonymous email) or if you want I can set up a private chatroom (although that would necessarily be accessible to all moderators of the SE network, if that is an issue).

Also, don't knock on cockroaches, they'll inherit the earth after the human race has gone to extinction :)

Harry Gindi

Hey Denis, question: Is there a way to reduce the question of whether the pullback of a map is cofinal by a kind of devissage using diagonals?

You can do this kind of thing when you're actually working with honest to goodness spaces

here by 'map', I mean functor

Denis Nardin

@HarryGindi I'm not sure, I'd have to see a concrete example

Harry Gindi

So suppose I have a functor F:Y→X, and X has an initial object x. I want to show that the fibre over x is initial in Y

so 'smoothness', more or less, is what I'm testing in Grothendieck's terminology

And I know that for every higher diagonal of the map Y→X, there exist lifts of all of the arrows x→f(y) for y in some (Y/X)^{S^n}

and also to Y itself

here (Y/X)^{S^n} means the target of the nth diagonal

7 hours later…

Edison

3:43 PM

heyyy

user2661923

i just created a separate chat room and have given a new user full access to it. does anyone know what i need to do to invite this specific user to the chat room that i created?

4 hours later…

lemiller

7:50 PM

I think I'm misunderstanding something very basic. In HTT 1.2.2, he gives a concrete model for the mapping space of an infinity category S as Hom_S^R. If S = sAlg_k where k is a ring and sAlg_k are the simplicial algebras, the n-simplicies in an object Hom_S^R( U_, V_) are n+1 simplicies in sAlg_k with n+1 vertex V_* and all others U_. What are the maps defining these simplicies? Here I'm taking an n-simplex in sAlg_k as an n-fold composition U_ -> U_* -> .... -> U_* -> V_*.

ack, I should have texed that better, hopefully its clear there should have been no italics, and U_ really is U_*.

Denis Nardin

8:04 PM

@lemiller The whole face spanned by the vertices 0,..,n is degenerate, so those maps U→U are the identity

I.e. your description of Hom^R is incomplete: its n-simplices are the (n+1)-simplices s.t. the face opposed to the (n+1)-st vertex is degenerate at U, not just that have U as its vertices

(Lurie says this by saying ".. the face $z|_{\Delta^{0,\dots,n}$ is a constant $n$-simplex at the vertex $x$.", although it's not clear to me why he says "a" constant $n$-simplex, when there's only one of those...)

Harry Gindi

8:32 PM

@lemiller It's definitely worth working out Lurie's explicit description from the following description: Hom^R(x,y) is the fibre of the right fibration C/y→C over the vertex x.

so giving an n-simplex here amounts to giving an n+1-simplex of C whose face opposite the terminal vertex factors through the constant map at x (this is just writing out the pullback diagram and flipping around the adjunctions).

as Denis described

lemiller

8:50 PM

@DenisNardin @HarryGindi Thanks a bunch, so the only non-trivial (i.e., non-identity) map in the n-fold composition is the last map U -> V or is that an erroneous oversimplification?

Harry Gindi

9:11 PM

@lemiller remember that you're working with a whole simplicial set of them

hang on can I post a picture in here

Denis Nardin

@lemiller You actually get n+1 maps U→V, corresponding to the map from the i-th vertex to the (n+1)-th, although they're all homotopic

For example let us see what happens for n=1: you get a 2-simplex whose edge [01] is the constant edge at U. Concretely this is the datum of two maps U→V and a homotopy between them (as it should be, given that it is a path in the mapping space!)

Harry Gindi

imgur.com/TJx76Wb

this is what a 2-simplex will look like

Denis Nardin

Harry, you can upload the images

Harry Gindi

oh well

it's not a work of art

anyway, so the top face in my picture, which is opposite the terminal vertex

is getting squished down to x

but you still have a the three maps to the terminal vertex

you can think of this as a kind of 'oriented suspension'