@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
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)
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?
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 $\mathbb{E}_k$?
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
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
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?
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)
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
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
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
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}?
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
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).