On the Freyd categories of an additive category

Abstract

To any additive category $\mathfrak{C}$, we associate in a functorial way two additive categories $\mathcal {A}\mathfrak{C})$, $\mathcal B(\mathfrak{C})$. The category $\mathcal {A}(\mathfrak{C})$, resp. $\mathcal {B}(\mathfrak{C})$, is the reflection of $\mathfrak{C}$ in the category of additive categories with cokernels, resp. kernels, and cokernel, resp. kernel, preserving functors. Then the iteration $\mathcal {A}\mathcal {B}(\mathfrak{C})$ is the reflection of $\mathfrak{C}$ in the category of abelian categories and exact functors. We call $\mathcal {A}(\mathfrak{C})$ and $\mathcal {B}\mathfrak{C})$ the Freyd categories of $\mathfrak{C}$ 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.