On the Freyd categories of an additive category

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

Keywords

contravariantly finite and reflective subcategories, Freyd and Auslander categories, abelian, weak abelian and triangulated categories, pure-semisimple categories, flat and homological functors, ind-objects and pro-objects

2010 Mathematics Subject Classification

Primary 16D90, 18E10, 18E30. Secondary 16G10, 16G60, 16L60.

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Published 1 January 2000