  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
$\mathbb{C}$ is a Mal'tsev category if and only if every object in $\mathbb{C}$
is a Mal'tsev object. We show that for a well-powered regular category
$\mathbb{C}$ which admits coproducts, the full subcategory of Mal'tsev objects
is coreflective in $\mathbb{C}$. 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.
