The notion of extriangulated category was introduced by Nakaoka and Palu giving a simultaneous generalization exact categories triangulated categories. Our first aim is to provide an extension Auslander's formula: for some $\mathcal{C}$, there exists localization sequence $\operatorname{def}\mathcal{C} \to \mod\mathcal{C} \operatorname{lex}\mathcal{C}$, where $\operatorname{lex}\mathcal{C}$ den...

We study Translation functors and Wall-Crossing functors on infinite dimensional representations of a complex semisimple Lie algebra using D -modules. This functorial machinery is then used to prove the Endomorphism-theorem and the Structure-theorem, two important results established earlier by W. Soergel in a totally different way. Other applications to the category O of Bernstein-GelfandGelfa...

Kato has constructed reflection functors for KLR algebras which categorify the braid group action on a quantum group by algebra automorphisms. We prove that these reflection functors are monoidal.

Extending Pirashvili’s work, we characterize small additive categories A such that the inclusion in the category of functors from A to abelian groups of the full subcategory of analytic functors induces an isomorphism between extension groups.

Accessible set functors can be presented by signatures and equations as quotients of polynomial functors. We determine how preservation of pullbacks and other related properties (often applied in coalgebra) are re ected in the structure of the system of equations.

Given a group G, we prove that the 2-category of small categories with G-action and G-equivariant functors is 2-equivalent to the 2-category of small Ggraded categories and degree-preserving functors.

We prove that any endofunctor on a class-theoretic category has a final coalgebra. Moreover, we characterize functors on set-theoretic categories which are identical on objects, and functors which are constant on objects.

We examine the cokernel of canonical homomorphism from trivial source ring a finite group to $p$-rational complex characters. use Boltje and Co\c{s}kun's theory fibered biset functors determine structure cokernel. An essential tool in determination this is Bouc's rational $p$-biset functors.

Recognizable graph languages are a generalization of regular (word) languages to graphs (as well as arbitrary categories). Recently automaton functors were proposed as acceptors of recognizable graph languages. They promise to be a useful tool for the verification of dynamic systems, for example for invariant checking. Since automaton functors may contain an infinite number of finite state sets...