Euler characteristic, character of group representation and Riemann Roch theorem

I am considering the following set up:Let G be a finite group,let Rep(G) denote the category of finite dimensional representations over C. Let V,W be representations of G in Rep(G). One can define a bilinear form on Rep(G) or inner product in K0(Rep(G)) (in Teleman’s notes) as dimCHom(V,W)G which is G invariant of Hom(V,W).Then there … Read more

Categorified probability and statistics?

To put it simply, what if the sample space underlying our probability space is a category instead of a mere set. Has a theory or probability and statistics been developed for such a situations in which the samples can have relations between each other? This questions is somewhat similar to this one however I do … Read more

Categorification of spaces and models for set theory

One aspect of topos theory is that it provides an enlarged view of the classical concept of space. Indeed, one may thought that the notion of topos is a sort of categorification of the notion of space. On the other hand, a topos can be thought as well as a model for set theory. Q: … Read more

Categorifying skein algebras?

We can obtain the Jones polynomial by the Temperly-Lieb algebra and the HOMFLYPT polynomial from the Hecke algebra. Were there attempts to categorify the algebras itself and obtain the Khovanov homology or HOMFYLPT homology from there? When googling, one can find a lot of papers containing certain categorifications of algebras but I find it hard … Read more

The 2-group of extensions

Let A,B objects of an abelian category. Then we can define the abelian group Ext1(A,B) as the set of isomorphism classes of extensions 0→B→E→A→0, endowed with the Baer sum. Following the principle of categorification, a finer and hopefully better invariant is the category of extensions, where morphisms are commutative diagrams as usual. Actually it is … Read more

Categorification request

Possible Duplicate: Can we categorify the equation (1 – t)(1 + t + t^2 + …) = 1? Can you give a categorification of the geometric series identity: 1/(1−x)=1+x+x2+… Categorifications of partial sum identities (1−xn+1)/(1−x)=1+x+x2+…+xn would also be nice. Answer I tried to discuss this geometric series example of categorification in one of my answers … Read more

What is the mathematical structure of 2d TQFT from the 2d foam category (instead of 2d cobordism category)?

It is well-known that the category of 2d TQFTs is equivalent to the category of commutative Frobenius algebras. What about functors from the 2d foam category (instead of 2d cobordism category) to the category of vector spaces? (Here I am thinking of the foams as just abstract spaces, not embedded in any Euclidean spaces.) Is … Read more

categorifying induction in homotopy type theory

In trying to understand homotopy type theory, I stumbled upon the following silly question, which is likely to be trivial for the experts. Let $B=\sqcup_{n\in\Bbb N} BS_n$, which I’d like to think of as the categorified version of the natural numbers $\Bbb N$. There is an obvious map $\sigma:B\to B$ that covers the successor map … Read more

Categorifying Hyperoperations

Is there some categorical version of tetration or higher hyperoperations? This is motivated by the fact that coproducts categorify addition of finite cardinals, and products/exponentials categorify multiplication/exponentiation in the same way. (exponentials categorify infinite cardinal exponentiation too) Hunting for intuition in $Sets$ works for the first three operations, so it seems reasonable to look there … Read more