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.