One of the earliest theorems in category theory stated that an abelian category could be represented faithfully by exact functors into the category Ab of abelian groups [Freyd, 1964], [Lubkin, 1961] and [Heron, unpublished]. Then Mitchell [1965] showed that every such category had a full exact embedding into a module category. An equivalent formulation is that every abelian category into a cate...

2 The category of small abelian categories and exact functors 4 2.1 Categorical properties of ABEX . . . . . . . . . . . . . . . . . . . 5 2.2 Pullbacks in ABEX . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 ABEX is finitely accessible . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Abelian categories as schemes . . . . . . . . . . . . . . . . . . . . 16 2.4.1 The functor of point...

In this paper, we give the properties of tensor product and study the relationship between Hom functors and left (right) exact sequences in FS-Act. Also, we get some necessary conditions for equivalence of two fuzzy Sacts category. Moreover, we prove that two monoids S and T are Morita equivalent if and only if FS-Act and FT -Act are equivalent.

Let Λ and Γ be artin algebras and ΛUΓ a faithfully balanced selforthogonal bimodule. We show that the U -dominant dimensions of ΛU and UΓ are identical. As applications to the results obtained, we give some characterizations of double dual functors (with respect to ΛUΓ) preserving monomorphisms and being left exact respectively.

We study the K0 and K1-groups of exact and triangulated categories of perfect complexes, and we apply the results to show how determinant functors on triangulated categories can be used for the construction of Euler characteristics in relative algebraic K0-groups.

Let G be a locally compact, σ-compact group. We prove that the equivariant KK-theory, KK, is the universal category for functors from G-algebras to abelian groups which are stable, homotopy invariant and split-exact. This is a generalization of Higsons characterisation of (non-equivariant) KK-theory.

Maschke functors, semisimple functors and separable functors of the second kind. Abstract We introduce separable functors of the second kind (or H-separable functors) and H-Maschke functors. H-separable functors are generalizations of separable functors. Various necessary and sufficient conditions for a functor to be H-separable or H-Maschke, in terms of generalized (co)Casimir elements (integr...

One reason for the universal interest in Frobenius algebras is that their characterisation can be formulated in arbitrary categories: a functor K : A → B between categories is Frobenius if there exists a functor G : B → A which is at the same time a right and left adjoint of K; a monad F on A is a Frobenius monad provided the forgetful functor AF → A is a Frobenius functor, where AF denotes the...

All Mackey functors over a finite group G are built up by short exact sequences from Mackey functors arising from modules over the integral group rings of appropriate subquotients W H of G. The equivariant cohomology theories with coefficients in Mackey functors arising from W H-modules admit particularly simple descriptions. Let G be a finite group. The notion of a Mackey functor plays a funda...