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Mal'tsev objects, $R_1$-spaces and ultrametric spaces

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Thomas Weighill

In this paper we introduce a notion of Mal'tsev object, and the dual
notion of co-Mal'tsev object, in a general category. In particular, a
category C is a Mal'tsev category if and only if every object
in C is a Mal'tsev object. We show that for a well-powered
regular category C which admits coproducts, the full
subcategory of Mal'tsev objects is coreflective in C. We show
that the co-Mal'tsev objects in the category of topological spaces and
continuous maps are precisely the $R_1$-spaces, and that the co-Mal'tsev
objects in the category of metric spaces and short maps are precisely the
ultrametric spaces.

Keywords:
Mal'tsev object, Mal'tsev category, $R_1$-space, ultrametric space

2010 MSC:
18A05, 18A32, 18B30, 54D10, 54E35

*Theory and Applications of Categories,*
Vol. 32, 2017,
No. 42, pp 1485-1500.

Published 2017-11-20.

http://www.tac.mta.ca/tac/volumes/32/42/32-42.pdf

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