# Closedness properties of internal relations II: Bourn localization

## Zurab Janelidze

We say that a class $\mathbb{D}$ of categories is the Bourn localization of a class $\mathbb{C}$ of categories, and we write $\mathbb{D} = \mathrm{Loc}\mathbb{C}$, if $\mathbb{D}$ is the class of all (finitely complete) categories $\mathcal{D}$ such that for each object $A$ in $\mathcal{D}$, $\mathrm{Pt}(\mathcal{D}\downarrow A) \in \mathbb{C}$, where $\mathrm{Pt}(\mathcal{D}\downarrow A)$ denotes the category of all pointed objects in the comma-category $(\mathcal{D}\downarrow A)$. As D. Bourn showed, if we take $\mathbb{D}$ to be the class of Mal'tsev categories in the sense of A. Carboni, J. Lambek, and M. C. Pedicchio, and $\mathbb{C}$ to be the class of unital categories in the sense of D. Bourn, which generalize pointed Jónsson-Tarski varieties, then $\mathbb{D} = \mathrm{Loc}(\mathbb{C})$. A similar result was obtained by the author: if $\mathbb{D}$ is as above and $\mathbb{C}$ is the class of subtractive categories, which generalize pointed subtractive varieties in the sense of A. Ursini, then $\mathbb{D} = \mathrm{Loc}(\mathbb{C})$. In the present paper we extend these results to abstract classes of categories obtained from classes of varieties. We also show that the Bourn localization of the union of the classes of unital and subtractive categories is still the class of Mal'tsev categories.

Keywords: Mal'tsev, unital and subtractive categories; fibration of points

2000 MSC: 18C99, 08B05, 18A25

Theory and Applications of Categories, Vol. 16, 2006, No. 13, pp 262-282.

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