# On the Freyd Categories of an Additive Category

## Apostolos Beligiannis

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

Homology, Homotopy and Applications, Vol. 2, 2000, No. 11, pp. 147-185

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