The category of finite dimensional (type 1) representations of a quantum affine algebra Uq(ĝ) is not semisimple. However, as any abelian category with finite-length objects, it admits a unique decomposition in a direct sum of indecomposable subcategories (blocks). We define the elliptic central character of a finite dimensional (type 1) representation of Uq(ĝ) and show that the block decomposit...

Quasicoherent sheaves should be thought of as the scheme-theoretic analogs of vector bundles. This analogy might lead one to think that vector bundles over a scheme X should be locally free OX-modules, but this turns out not to be an abelian category, and the category of quasicoherent sheaves contains the category of locally free OX-modules. There are two different different definitions we coul...

We describe Somekawa’s K-group associated to a finite collection of semi-abelian varieties (or more general sheaves) in terms of the tensor product in Voevodsky’s category of motives. While Somekawa’s definition is based on Weil reciprocity, Voevodsky’s category is based on homotopy invariance. We apply this to explicit descriptions of certain algebraic cycles.

Let A be an abelian category such that every object has only finitely many subobjects. From A we construct a semisimple tensor category T . We show that T interpolates the categories Rep(Aut(p), K) where p runs through certain projective pro-objects of A. This extends a construction of Deligne for symmetric groups.

In this paper we show that the Baues-Wirsching complex used to define cohomology of categories is a 2-functor from a certain 2-category of natural systems of abelian groups to the 2-category of chain complexes, chain homomorphism and relative homotopy classes of chain homotopies. As a consequence we derive (co)localization theorems for this cohomology.

We develop the Tannaka-Krein duality for monoidal functors with target in the categories of bimodules over a ring. The Coend of such a functor turns out to be a Hopf algebroid over this ring. Using a result of [4] we characterize a small abelian, locally finite rigid monoidal category as the category of rigid comodules over a transitive Hopf algebroid.

Herschend-Liu-Nakaoka introduced the notion of n -exangulated categories. It is not only a higher dimensional analogue extriangulated categories defined by Nakaoka-Palu, but also gives simultaneous generalization -exact in sense Jasso and ( + 2 ) -angulated Geiss-Keller-Oppermann. Let C be an category with enough projectives injectives, X cluster tilting subcategory . In this article, we show t...

We investigate the existence of normalizers of subobjects in pointed categories defined in the expected way, as motivated by the standard definition used in the category of groups. We show that, for a semi-abelian category C: (a) if the category C2 of morphisms in C is action representable, then so is the category Mon(C) of monomorphisms in C; (b) if Mon(C) is action representable, then normali...

We first introduce the notion of an Auslander-Reiten sequence in a Krull-Schmidt category. This unifies the notion of an almost split sequence in an abelian category and that of an Auslander-Reiten triangle in a triangulated category. We then define the Auslander-Reiten quiver of a Krull-Schmidt category and describe the shapes of its semi-stable components. The main result generalizes those fo...

We give an equivalence between the derived category of a locally finitely presented category and the derived category of contravariant functors from its finitely presented subcategory to the category of abelian groups, in the spirit of Krause’s work [H. Krause, Approximations and adjoints in homotopy categories, Math. Ann. 353 (2012), 765–781]. Then we provide a criterion for the existence of r...