unrelatedly, perhaps this is already familiar to some of you, but i only just learned about an utterly ridiculous result called gabriel's theorem: a quiver is of finite type (i.e. has finitely many irreps) if and only if it is a dynkin diagram of type A, D, or E. what the hell?!?? i don't know whether the moral is that lie theory knows about way more than it should, or that dynkin diagrams themselves are more fundamental than we understand, or something else entirely...
@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!
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
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.
@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.
@AaronMazel-Gee yeah it was sort of weird to be like "oh actually I HAVE been to this part of the world before, specifically to study chromatic homotopy theory"
@AaronMazel-Gee yeah I think the classification of (semi)simple Lie algebras is pretty ridiculous. Maybe the moral of the story is that pictures can be pretty smart and algebras can be pretty much pictures?
@AaronMazel-Gee regarding the ADE thing, I think the common origin is positivity: these are the only graphs with positive definite Cartan matrix. Why the Cartan matrix is a really natural thing to attach to a graph is something I don't really understand.
About AmbiHeight: the idea is that we want to define the height (in the chromatic sense) of objects and categories in a way that uses only ambidexterity. The chromatic height is defined using the invertability or completeness of the v_i maps, but ambidexterity gives you another extrapolation of p into a sequence of "numbers". Namely, in an additive category, you can measure the cardinality of C_p, the cyclic group on p elements, and ask if it is invertible. If it does the height is 0.
If on the other hand p is complete, you are in height => 0
Now instead of taking the quotient by p and consider v_1, if you are 1-semiadditive you can consider the cardinality of |BC_p| as well, it is now "finite" so it have a "number of elements". The main idea is that the invertability of |BC_p| should be a height 1 phenomena, and the higher generalizations of this fact.
This notion of "semiadditive height" have several nice categorical properties: up to issues with height infinity the height filtration for this notion of height splits completely, some version of chromatic redshift holds for it easily (I would say it is easy enough to give the impression that this is the "reason" for redshift, but I'm super biased). Also, the property of being ambidextrous of height n is a "modalic" property: its the property of being a module over a commutative ring in Prl.
One last remark: we believed for a while that this notion of height is the same as the chromatic height for higher semi-additive stable infinity categories, but few weeks ago Allen Yuan found an ingenious counterexample based on the Segal conjecture (this is why I asked about it here :-)).
@TomBachmann for an object in a stable infinity category we can define its "stable height" to be the <=n if after tensoring with a type n complex, the v_n self map is invertible on it. if C is also ambidextrous, we can say its of semiadditive height <= n if the cardinality |B^nC_p| is invertible on it. We thought the two notions might coincide, but Allen gave an example of an object of semiadditive height 1 and stable height infinity in the the semiadditive stable p-complete mode.
Clearly a topos is locally $\infty$-connected iff it is generated under colimits or effective epimorphisms by some collection of objects of constant shape. Is it true that a topos is locally $\infty$-connected iff it is generated under colimits or effective epimorphisms by some collection of trivial shape? The one categorical analogue is true: An ordinary topos is locally connected iff it is generated by connected objects.
@dhy Thanks for this! It's very relevant. I guess this read will have to wait for when I have the time. It would be nice to have a good formal encapsulation of all this very old and non-conjectural weight theory on l-adic complexes.
It is a very confusing state of affairs that the theory of "weight structures" doesn't apply directly to classical weight theory. I wouldn't be surprised if others have made a similar mistake before us.
@S.carmeli Just a brief response as to "Why the Cartan matrix is a really natural thing to attach to a graph" from the point of view of quivers (perhaps you know this and this is implicit in your response): the (stacky) dimension of the space of representations of the corresponding quiver is given by something like -v^TAv, where A is the Cartan matrix. So A positive definite is the case where the stacky dimension is negative & you expect only finitely many representations
On the other hand I feel like explaining why positive definiteness is the key to classifying semisimple Lie algebras is a lot trickier (without going through the entire machinery of root data)... I think the "fastest" way nowadays is via Hall algebras:
(lots of technicality ahead) K^0 of the category of perverse (or just constructible) sheaves on the moduli of representations of the quiver has a natural Hopf algebra structure, and it contains as a subalgebra the universal enveloping algebra of a Borel of the semisimple lie algebra, so that gives a direct link between quivers & semisimple Lie algebras
but that's a gap of 20 years between Gabriel's theorem and Lusztig's work showing the above... I'm not sure how people thought about the situation in the intervening 20 years. Anyways this is all more or less irrelevant to your original question but its a great (unfinished) story & I just woke up so I felt like being verbose. Also I feel like the general mathematical public is underaware of just how much cool representation theory Lusztig did
I like to think of positivity in the context of semi-simple algebras as coming from positivity of the flag variety. That it's fano, or even that all it's chern rootsare nef. In fact I think the cartan matrix can be interpreted as the pairing between the basis of $H^{n-2}$ given by B-invariant curves basis of $H^2$ given by B-invariant line bundles. Both of these groups are canonically isomorphic to the weight lattice of the abstract cartan $T$ for different geometric reasons.
The cartan matrix is then the change of basis between the chern roots (which give a basis) and the irreducible $B$-invariant curves.
So it can be thought of as an explicit manifestation of this "nefness" of the chern roots.
@dhy Whenever I read about Luztig's work I get really excited. Recently I've been reading a bit about deligne luztig trying to understand what I can. There's a really beautiful story there I feel.
(the story above was for a simply connected semisimple group over C - I don't know how much of this story translates to the general case).
I have no idea if this has anything to do with the way positivity comes up for quivers. I am aware that there's some absurdly categorified version of lie theory for quiver varieties (although I no nothing about it) maybe that's the answer...
@TomBachmann @AaronMazel-Gee and by the way, thank you both for the interest in this work! really. It's so fun to know that people care about this stuff :-)
