Mathematics - 2025-02-18

EE18

00:01

@BenSteffan Gotcha here--thank you!

Ben Steffan

welcome :)

somebody publicly shared his personal notes for the algebraic geometry class this term, which is very kind

however, they are... bilingual, with spanish and english mixed haphazardly (?!)

sometimes I wish I could peek into the minds of other people to see what's going on in there

Thorgott

00:50

a major point of confusion for me has been that the "iterated Bar construction" does not appear to be the result of iterating the Bar construction

Ben Steffan

that would be too obvious wouldn't it

@Thorgott see page 842, right before the proposition

Thorgott

I kid you not

I just alt-tabbed into HA and that was the page I had already open

that said, I don't see where he actually makes the thing precise he claims to be making precise

Ben Steffan

yeah, I was looking for that as well

I guess it's Example 5.2.3.14?

See also the discussion on page 848

Thorgott

ah yes

this almost sorta kinda makes sense

Ben Steffan

"under good circumstances the iterated bar construction is an iterated bar construction" is a perfectly normal and sensible statement to have

I'm sad that we skipped everything about the $\mathbb{E}_k$-operads in our HA seminar now :(

much fun could've been had

Thorgott

01:07

the operads themselves are pretty reasonable, the stuff that's being done with them...

speaking of, do you have any intuition for why colim E_k = E_infty? the proof is surprisingly easy, but not particularly enlightening

I wager it's some sort of 'infinitary Eckman-Hilton principle', but that's also kind of just saying words

Ben Steffan

one way to characterize $\mathbb{E}_\infty$ is as the unique operad (let's say classical for the moment, and then with values in $\mathrm{Top}$) with each operation space contractible

(up to homotopy)

and then a computation shows that the $\mathbb{E}_k(r)$ "stabilize" to something contractible when $k \to \infty$, q.e.d.

Thorgott

right, that's the geometric perspective, but what's the algebraic one?

an E_k-algebra consists of k compatible associative algebra structures

why are infinitely many compatible associative algebra structures the same as one commutative associative algebra structure?

Ben Steffan

hmm

I think one should think of "$k$ compatible associative algebra structures" as being coherently commutative up to "level" $k$ (insert something about being $k$-truncated)

the translation happening via some form of Eckmann-Hilton

a grouplike $\mathbb{E}_1$-algebra is a loop space, so in particular a monoid in the homotopy category. a grouplike $\mathbb{E}_2$-algebra is a two-fold loop space, so by Eckman-Hilton in particular a commutative monoid in the homotopy category

in particular there is some square in the homotopy category witnessing this commutativity, with the $\mathbb{E}_2$-algebra structure providing the homotopy necessary to make it commute

an $\mathbb{E}_3$-algebra structure should then correspond to additional coherence data making certain cubes commute, ect.

Thorgott

01:27

I think the different algebra structures themselves also ought to agree up to some "level", but I can't quite make this precise

so e.g. in 1-categories, an E_2-algebra actually gives two copies of the same associative algebra structure + a braiding

what's, say, an E_2-algebra in 2-categories?

I presume the two associative algebra structures don't need to fully agree, but perhaps on objects and 1-morphisms?

Ben Steffan

not sure

In particular I don't know what model of $\mathbb{E}_2$ you'd use to make sense of that in the first place

Thorgott

01:45

theres multiple?

Ben Steffan

there's a bunch: cubes, disks, ... :)

but that's not what I mean. What I mean is that, to the best of my knowledge, to have an algebra over some operad $O$ in a category $C$ this category $C$ should be... enriched over the category $O$ takes values in (I think)?

but $\mathbb{E}_k$ is an operad in $\mathrm{Top}$

perhaps we just forget to $\mathrm{Set}$

I guess that's what people mean when they talk about $\mathbb{E}_k$-algebras in things like $\mathrm{Cat}$ (?)

Thorgott

02:01

@BenSteffan you don't need that

an O-algebra in C should just be a map of operads O -> C

(C being a symmetric monoidal category here)

Ben Steffan

you're implicitly forgetting to $\mathrm{Set}$, but then it's the same thing, yes :)

Thorgott

where does Set enter the picture?

Ben Steffan

maybe I'm misunderstanding what you mean by map of operads, but if you view $\mathbb{E}_k$ as some form of colored operad a la Lurie, then you will have spaces $\mathbb{E}_k(r)$, and if you package this up into a category of operators this will be naturally $\mathrm{Top}$-enriched.

Now either a map of operads (in the first description) consists of giving maps $\mathbb{E}_k(r) \to \mathrm{Mul}_C(...)$, but the domain is a space and the codomain a set, or it consists of giving a functor with certain properties from $\mathbb{E}_k^\otimes$ to $C^\otimes$, but the first category is $\mathrm{Top}$-enriched and the second isn't.

In either case you have to explain to me how to do that, and in both cases the obvious thing to do is to simply forget the topological structure on the domain :)

Thorgott

02:22

sorry, I meant to view it as an infty-operad

Ben Steffan

ah ok

then I don't know how to model 2-categories

(nevertheless this discussion should have the unfortunate upshot that an $\mathbb{E}_k$-algebra in $NC$ may not be the same thing as an $\mathbb{E}_k$-algebra in $C$ when $C$ is a 1-category, unless $k = 1, \infty$.)

Thorgott

let's say (2,1)-categories

i.e. (infty,1)-categories with 1-truncated mapping spaces

and I believe those can all be strictified actually

Ben Steffan

I will take your word for it

one potato two potato

05:12

there is a concept of "categroid" presumably a generalization of category.

Rakshith PL

07:22

Cleo's mystery has been finally solved!

3

Soham Saha

@PM2Ring interesting patterns

Soham Saha