thank you @Untitled I will see it :) :)

thank you

@DenisNardin that's bonkers, I never knew it was two papers a week. While teaching and doing other courses??

@JonathanBeardsley I took it on the first year when there aren't teaching obligations, but yeah it's insane. To be honest a lot of papers I skimmed more than read, and discussing them together after the talk helps a lot in figuring out the important parts to take home

@PraphullaKoushik No problem :)

There are also Yuri Berest's notes and Zhen Lin's Notes on Homotopical Algebra (the version I found while searching now had only 70 pages, but the notes have 1200 pages. I don't know what happened to them)

As I said it is a big, big story. Reading the preface to HTT (Higher Topos Theory) can also help with the big picture :)

@Untitled you mean zll22.user.srcf.net/writing/homotopical-algebra/… ..?

By the work of Bhargav Bhatt (i hope i'm not wrong) the Hodge-completed derived de rham complex of a finite type $\mathbb{C}$-algebra is quasi-somorphic to the singular co-chain complex on the corresponding analytic variety. What about localizations of finite type algebras? As a concrete example: Can one algebraically compute the cohomology of a link of an isolated singularity from the data of the punctured spectrum of the corresponding local ring?

Maybe more naive question would be what kind of intersting topological data does the hodge completed derived de rham complex of a local ring carry?

@DenisNardin Thanks :) I will read

@PraphullaKoushik Yes!

Is it true that $Alg_{\mathbb{E}_n} (CoAlg_{\mathbb{E}_m}(-))=CoAlg_{\mathbb{E}_n} (Alg_{\mathbb{E}_m}(-))$ for any $n$ and $m$ and any category you plug in? If its true is it something general for operads or is it something special about $\mathbb{E}_k$?

You need to be specific about which monoidal structures you're putting on Alg and CoAlg

Right... I'm assuming my underlying category is symmetric monoidal and then there's the canonical one which comes from the underlying monoidal structure on the category.

At least that's what i thought up until now. I'm not extremly well versed with this yet but I think I saw this in Higher Algebra a couple of days ago.

Actually I'm lying I have no idea what to do for coalgebras

(maybe its just taking the opposite twice)

Yeah, you have that monoidal structure. I think what you're claiming is true, but I'm not 100% sure how to prove it. If Jon shows up, this might be the kind of thing he knows

Do you know if its something general about operads or something ispecial?

I don't really have a clue, although I'm vaguely tempted to think that it should be something general

I think you swapped m and n unintentionally . I don’t see a clean direct argument here, but I think we should expect it to be true.

Hm. Does this work? Let P and Q be operads, and C a symmetric monoidal category. Then P-algebras in some sym mon D are symmetric monoidal functors Env(P)-->D and I think Q-coalgebras are symmetric monoidal functors Env(Q)^{op}-->D. So could we write P-Q-bialgebras as symmetric monoidal functors Env(P)\times Env(Q)^{op}--->C?

I think that describes pairs of a P-algebra and a Q-coalgebra, no?

I do think Props are the way to go though

You want something like the pushout in the category of symmetric monoidal categories...

hmmm... what do I replace \times with to get the right answer for Fun^{\otimes}(A,Fun^{\otimes}(B,C))=??

ah, right

except now I'm confused. symmetric monoidal categories are CAlg objects in Cat

so shouldn't the coproduct be the tensor structure in Cat, which is \times ?

Yes, I think the coproduct is given by the cartesian product

I guess the question/point you're raising is whether this thing actually provides an adjoint to Fun^{\otimes}

But the problem is that what you want is not the coproduct in symmetric monoidal categories, but the coproduct in props (a.k.a. symmetric monoidal categories generated by a distinguished object)

I'm not convinced that's what I want, but I'd be convinced if you told me that there isn't an left adjoint to A \mapsto B^A:= Fun^{\otimes}(A, B) in CAlg(Cat)

ah!

What is the symmetric monoidal structure on Fun^⊗(A,B)? Just the coproduct (i.e. pointwise ⊗)?

I'm convinced, this doesn't take coproducts to products

good question

I guess I should take whichever thing gives the right answer when A=Env(P)

which appears to be pointwise

Yeah, that's the only sensible thing to take

but anyway- you've already convinced me. now: what is a prop and do I really have to learn what it is?

So, all I know about props is what I already wrote: they are symmetric monoidal categories generated (as symmetric monoidal categories) by a distinguished object

ah, sorry I missed that

Ok, I also know that you can give a description of them in terms of graphs, but I don't really know how you're supposed to do that and I don't think that's relevant for the question at hand

But the point is that, as you've corrently pointed out, you want to find an object that is a P-algebra and a Q-coalgebra at the same time

I see. So we'd like a better description of this coproduct in props

to convince ourselves that it does the right thing

Now an object of a symmetric monoidal category is the same thing as a symmetric monoidal functor Fin→C

sure, so first thing to try is pushout in CAlg with those structure maps

So I think that what we want is the pushout Env(P)⊔_Fin Env(Q)^{op}

And I guess we need to convince ourselves that this object represents both P-Alg(Q-Coalg(C)) and Q-Coalg(P-Alg(C))

incidentally we could write that as a little tensor product gadget over Fin, thinking of both sides as modules

Indeed

Actually, no this is not the right thing

:(

This is an object that is both a P-algebra and Q-coalgebra, but we're not asking any compatibility between the two structures

While P-Alg(Q-Coalg(C)) has tons of compatibilities

ah!

it's worth noting that everything we've said so far works in algebra-algebras

in which case we know we want whatever we get when we do

Env

to the BV tensor product

Env(P\otimes Q) = Env(P)??Env(Q)

Right, but in that case, is the BV tensor product symmetric? I thought not...

maybe I'm using the wrong name

but there is a symmetric monoidal structure on infty-operads

and it does the thing we expect if we didn't say "coalgebras" above: Alg_P(Alg_Q)=Alg_Q(Alg_P)=Alg_{P\otimes Q}

No, no that's exactly what the BV tensor product is. Are you sure it's symmetric?

HA.2.2.5.13

Apparently yes. I don't know where I got the notion that it was only associative

Also: do we think what we're doing is special for operads with one color? (that seems to be what you've privileged with your definition of prop)

No, no. It's just traditional but colored props also exist

So... arxiv.org/abs/1207.2773 claims that the BV tensor product exists for (classical) colored props

Proposition 34 is the statement we're after

I think I see how to soup up that proof to work also for symmetric monoidal ∞-categories

well that's good! it'd be nice to not have to port all that over piece by piece though...

Agreed...

I think you can define bilinear functors (C,D)→E as functors C^⊗×D^⊗→E^⊗ above a certain map Fin_\ast×Fin_\ast→Fin_\ast and work your way from there

the other strategy would be to just wee what happens when you try to write down the total-category "Alg(coAlg(C))^{\otimes}" you're gonna be writing down sections of some monstrosity over Fin_*. It's possible some six-functory stuff involving pushforward/pullback along cocartesian fibrations and such could be put to use in comparing them...

but if what you're saying works wouldn't it produce an adjoint to Fun^{\otimes}(-,B) in CAlg? And I thought we thought that couldn't happen

Well, now that I discovered that it exists for symmetric monoidal 1-categories I'm willing to recant that conclusion

? but it doesn't- the assumptions in that paper are that everything is a prop

Well, colored props. I'm actually a bit fuzzy on the distinction between symmetric monoidal categories and colored props

Oh I guess the condition is that it is freely generated by a set of objects?

Yeah, I guess that's it

Although, how do you even define "freely generated" when you can add arbitrary arrows?

you're quicker than me- I don't see from their definition how to encode a prop as any kind of special symmetric monoidal category...

So, the strings $\langle a_1,...,a_n\rangle$ correspond to the object $a_1⊗...⊗a_n$

Well, they're doing ordered props, which should correspond to monoidal categories

Then they give you a mapping set $Hom(\langle a_1,...,a_n\rangle, \langle b_1,...,b_m\rangle)$

And a "vertical composition" (a.k.a. composition of maps) and a "horizontal composition" (a.k.a. tensoring of maps)

Plus a boatload of identities

so you're saying that, in particular, a colored prop should have a symmetric monoidal envelope. that's fine. but I don't see how to encode a colored prop as a special kind of gadget over Fin_{\ast}... is it just like... any map which restricts to a cocartesian fibration over Fin^{inert}_{\ast}?

I'm slowly convincing myself that a colored prop is literally the same thing as a symmetric monoidal category with a distinguished set of generators

That is, that the symmetric monoidal envelope is essentially an equivalence

neat. and weird. So we have this weird set up where we have our distinguished generators S-->P and T--->Q and the BV tensor product buys us a prop with generators S\times T

er

There's a tricky thing, where morphisms of props are given by maps S→T and symmetric monoidal functors P→Q making the diagram commute

I'm still not able to distill a useful definition of bilinear map- even (and maybe especially) in the case when we focus on props with just one color

I'm having problems understanding why this is not just the cartesian product of the symmetric monoidal categories with the pointwise ⊗ structure

anyway- I should probably get going, but thanks for teaching me about props!

(and we're back where we started...)

;)

Have a nice evening!

Hello everyone! I'm wondering if there are examples of weak monoidal quillen pairs between weak monoidal model categories? This means the left Quillen functor is colax and the colax structure are weak equivalences between cofibrant objects. The only example i know is the Dold-Kan correspondance, but are there others ? (where the left functor is not simply a strong monoidal functor).