On the Freyd categories of an additive category

Apostolos Beligiannis

Homology, homotopy, and applications, Tome 2 (2000) no. 1, pp. 147-185.

Résumé

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.

Détail

DOI : 10.4310/HHA.2000.v2.n1.a11

Classification : 16D90, 18E10, 18E30, 16G10, 16G60, 16L60
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