On the Freyd categories of an additive category

Abstract

To any additive category C, we associate in a functorial way two additive categories A(C), B(C). The category A(C), resp. B(C), is the reflection of C in the category of additive categories with cokernels, resp. kernels, and cokernel, resp. kernel, preserving functors. Then the iteration AB(C) is the reflection of C in the category of abelian categories and exact functors. We call A(C) and B(C) the Freyd categories of C since the first systematic study of these categories was done by Freyd in the mid-sixties. The purpose of the paper is to study further the Freyd categories and to indicate their applications to the module theory of an abelian or triangulated category.