Manuel Rivera

I just hear (and read!) this so much it has got me a bit worried that I am missing some subtlety :)

I have often heard in talks and discussions people say "...the algebra A is Calabi Yau". Hence it sounds like if this was a property, when it is certainly structure! Is this a sloppy use of language or am I missing some uniqueness issue?

Hi all, happy new year! I have a question which might be a bit silly but nonetheless it has been bothering me. In several classical references (including Ginzburg) a Calabi-Yau algebra of dimension d is defined as a homologically smooth dg algebra A "such that there is a quasi-isomorphism A \to A^![d]"... instead of "equipped with a quasi-isomorphism."

I meant to say is every simplicial set (Joyal/categorically) equivalent to one with this property...

Hi, I hope everyone is doing well! quick question: is every quasi-category (Joyal/categorical) weakly equivalent to one for which each non-degenerate simplex is embedded, namely, the representing map of each non-degenerate simplex is injective?

@AaronMazel-Gee hmmm, so can I obtain from this perspective a model structure on say simplicial bialgebras for which simplicial Hopf algebras are fibrant objects?

@AndreaMarino some of the explicit models we work out there might be useful in your context...

@AndreaMarino hi, you may be interested in checking this out: journals.mq.edu.au/index.php/higher_structures/article/view/92/…

hi! does anyone know if hopf algebras can be interpreted as (the?) fibrant objects of some homotopy theory for bialgebras? are there any precise statements of this sort in the literature? the analogy I have in mind is that one can model pointed homotopy types via a model category structure on simplicial monoids for which simplicial groups are fibrant.

I guess you can also deduce this by comparing the right adjoints for the functors I just mentioned, perhaps one may find it in this form in the literature.

Hi, Does anyone know a reference for an explicit proof of the following fact: If X is a simplicial set with a single vertex then |G(X)|, the geometric realization of the Kan loop group of X, is naturally weakly equivalent - as a topological monoid - to the space of Moore based loops in |X|, the geometric realization of X. I have an outline for a proof constructing an explicit zig-zag and some of the combinatorics seem to be surprisingly tricky- I did not find this in the literature.

@blank_space hey thank you for sharing this. I also think I am "homotopy-adjacent" in the sense that the techniques I know and the questions I am interested in are not really part of the "mainstream" (whatever that means). I'd be happy to learn more homotopy theory with you (at ANY level) if you wish. Feel free to email me if you want to talk math. You can feel comfortable asking any kind of questions with me (I am sure many people in this chat feel similarly). :)

and of course, the conjecture (which we are "close" to proving) is that the E-infinity coalgebra of integral singular chains under cobar-quasi-iso completely determines homotopy types :)

So to answer your question: if you consider the singular chains coalgebra in this homotopy theory of coalgebras you remember the dg algebra of chains on the based loop space up to quasi-iso (and consequently the fundamental group algebra) together with the coalgebra of chains on the universal cover up to quasi-iso. The latter can also be obtained algebraically from coalgebras under cobar-quasi-iso.

This last statement was proven by Adams in 1956 for simply connected spaces, but it turns out that essentially the same statement holds for path-connected spaces and oddly enough it was not formulated in the literature until 60 years later.

@AaronMazel-Gee In terms of what you are remembering, well algebraically by definition is the quasi-isomorphism type of the dg associative algebra of the cobar construction of the underlying coassociative structure. The key observation I guess is that the cobar construction of the dg coassociative coalgebraa of singular chains on a path-connected pointed space is naturally quasi-iso to the dg associative algebra of chains on the based loop space.

@AaronMazel-Gee Yes under a conilpotency condition on the coalgebra side the adjunction is a quillen equivalence between coassociative coalgebras under these cobar-quasi-isomorphisms and associative algebras under quasi-isomorphism.

@TimCampion can you make precise mathematical sense of what you mean by “algebraic data”? By this phrase I usually think of something like an algebra over an operad in R-mod for some fixed (discrete) ring R... but that is probably a definition I make that suits my taste.

@AaronMazel-Gee regarding Gabriel's theorem... yeah that is a crazy classical result. I recently read an article in the Notices about it which was very accesible and started wondering about possible (homotopical) generalizations. (replace a quiver by a simplicial set with some conditions). this relates to my message above since, if X is a simplicial set, you can think on the dg category obtained by applying (homwise) chains on \mathfrak{C}(X) as a generalization of the path algebra of a quiver.

2) the normalized chains on \mathfrak{C}(X) is (essentially) naturally isomorphic as a dg bialgebra to the cobar construction on the coalgebra of of normalized chains on X. The cobar construction of a dg coassociative coalgebra is a dg associative algebra but in this case the E-infinity coalgebra structure of the normalized chains on X (more precisely the E_2 part) induces a dg coalgebra structure on cobar of chains on X making it a dg bialgebra.

People in this chat will probably appreciate how we realized this since it was based on the following observations. Let X be a Kan complex with one vertex and let \mathfrak{C}(X) be the left adjoint of the homotopy coherent nerve applied to X, considered as a simplicial monoid. Then 1) the normalized chains (with coefficients on a fixed ring R) on \mathfrak{C}(X) yields a DG R-bialgebra whose 0-th homology is isomorphic as an R-bialgebra to R[ pi_1(X) ], and

We wrote a short paper is explaining this point here arxiv.org/pdf/1807.06410.pdf

@AaronMazel-Gee @AaronMazel-Gee hey! it is subtle, of course. first, what do we mean by chains? we mean the singular chains coalgebra (with all of its natural algebraic structure controlling homotopy cocommutativity) under a notion of weak equivalence which is stronger quasi-isomorphisms, i.e. maps that become quasi-isomorphisms after applying the Cobar functor. Then the first kind of surprising observation is that the fundamental group is completely determined by this algebraic structure!

@ArthurPanderMaat @DenisNardin I will take this opportunity to self advertise this version of Whitehead theorem (which I like a lot :) ) in which we describe when a map of spaces is a weak homotopy equivalence in terms of the induced map at the level of singular chains: ams.org/journals/proc/2019-147-11/S0002-9939-2019-14555-X/…

Hi all, this is my first time here... Covid19 and quarantine kind of brought me to this chat :)