Jonathan Beardsley

@MartinSleziak Thanks for your continued service Martin. It seems that the Algebraic Topology Discord server has basically completely taken over.

@TimCampion good to know. Yeah I think it told me that pin was actually unpinned by a mod

room topic changed to Homotopy Theory: A room for anyone interested in homotopy theory, or any nearby fields (e.g. category theory, algebraic geometry). To activate chatjax in this room go to meta.math.stackexchange.com/questions/1088/…. See also the related discord server here: nodorek.net [homotopy-theory]

But maybe you can =P

Haha, well, I'm no longer able to. Once my pin was unpinned by a mod, it's toast.

@TimCampion eh i mean, it's not really a big deal. just seemed like a nice way to make sure nobody got left out of the migration

altho tbh the LaTeX rendering in here is so much better....

Maybe I'll just periodically come back in here and be like "There is a Discord for algebraic topology.... let me know if you're interested...." lol

Thanks for the info though. Good to know.

I see. Weird.

I wonder if a mod can change the description on a room I created....

room topic changed to Homotopy Theory: A room for anyone interested in homotopy theory, or any nearby fields (e.g. category theory, algebraic geometry). To activate chatjax in this room go to meta.math.stackexchange.com/questions/1088/…. See also the related discord server here: discord.gg/CDNKyEZaK6 [homotopy-theory]

Perhaps I should just add a link to the room description.

Yeah and it won't let me re-pin the message.

so i think a moderator had to come in actually remove the pin

and other starred comments get pushed below it

so now that's pinned over there

Like this.

It was marked by having a star outlined in black.

I don't know, I mean... yeah I have no idea. I suppose "pin" here is not quite the right word. It should have been "pinned" as a faved comment on the right hand side of the screen (on the computer at least).

I tried to pin a link to the Discord chat.... but a moderator removed my pin??

@TimCampion the shadow of the Discord

But I think it's slightly worse because it's harder to find, and harder to just anonymously learn things.

I know for some folks this is a plus, because they don't want their discussions to be totally public.

@WilliamBalderrama yeah that is a problem. you've got to give Discord your information... so it's less public

This may be the death knell of the MO chat, but that's okay...

I think I set the link to last forever, but who knows...

Just in case people are happening in here and wondering where everyone went.

There's a pretty active algebraic topology Discord server. Here's a link: discord.gg/CDNKyEZaK6

@CharlesRezk do you have any sense of what the "generating cells" or "generating cofibrations" for an arbitrary ∞-topos might be? Or do you think it's really case dependent?

Surprised, but also not surprised, to find out that my brain is an ∞-topos.

@CharlesRezk hahah Charles I found the exact same paper yesterday.

Isn't there some paper about cell structures, or CW-decompositions, or something like that, in ∞-topoi? Does anyone remember seeing something like this?

@kiran I don't know, but I feel like this must be true, and if it is there will be some counterexample by looking at MU_*MU operations or something

However, it does seem like one would need to extremely motivated to engage with this stuff.

@MikeMiller Perhaps. I mean, it's also evidence that my claim that it's impossible is inaccurate.

@MikeMiller it might be worth checking this out arxiv.org/abs/math/0406270

Or at the very least, extraordinarily tedious

@MikeMiller right, I think writing down the generators/relations POV for bialgebras or Hopf algebras in homotopy theory is effectively impossible

And a PROP

I guess there's also a Lawvere theory for Hopf-algebras

There's also the PROP of bialgebras BiAlg, and monoidal functors BiAlg→Ch(R) will be equivalent to bialgebras in Ch(R), so that gives a description of bialgebras at least which is somewhat explicit.

I guess by "explicitly" I mean in terms of something like a list of operations, in the same way that we describe A_∞-algebras in chain complexes.

But there's this definition at least: ncatlab.org/nlab/show/differential+graded+Hopf+algebra

E.g. there won't be an operad for it.

@MikeMiller I think this is going to be a huge mess to try to suss out explicitly.

@RuneHaugseng oh right.... thanks. forgot about that. that's another one Chris Rogers suggested to me.

Speaking of which, anyone have a good elementary reference for the TQFT/Frobenius algebra stuff?

(i.e. the goal would be to get to monoidal categories and then apply them)

I think the direction we're going to end up going, to make it at least a little relevant to the low-dimensional topologists' students who might take the course, is a primer in very basic category theory capped off by a discussion of (1+1)-TQFTs and their characterization by Frobenius algebras.