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# Homology, Homotopy and Applications

## Volume 2 (2000)

### Number 1

### On the Freyd categories of an additive category

Pages: 147 – 185

DOI: http://dx.doi.org/10.4310/HHA.2000.v2.n1.a11

#### Author

#### Abstract

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.