#
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.

http://www.tac.mta.ca/tac/volumes/16/13/16-13.dvi

http://www.tac.mta.ca/tac/volumes/16/13/16-13.ps

http://www.tac.mta.ca/tac/volumes/16/13/16-13.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/16/13/16-13.dvi

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