Title: Categorically closed topological groups,
Abstract: Let $\mathcal C$ be a subcategory of the category of topologized semigroups
and their partial continuous homomorphisms. An object $X$ of the category
${\mathcal C}$ is called ${\mathcal C}$-closed if for each morphism $f:X\to Y$
of the category ${\mathcal C}$ the image $f(X)$ is closed in $Y$. In the paper
we detect topological groups which are $\mathcal C$-closed for the categories
$\mathcal C$ whose objects are Hausdorff topological (semi)groups and whose
morphisms are isomorphic topological embeddings, injective continuous
homomorphisms, continuous homomorphisms, or partial continuous homomorphisms
with closed domain.