# Categorically closed topological groups

## TL;DR

This paper investigates the properties of topological groups that are closed under various categories of topological semigroups and homomorphisms, identifying conditions for categorical closedness.

## Contribution

It characterizes topological groups that are closed in specific categories of topological semigroups with different types of morphisms.

## Key findings

- Identifies conditions for $ ext{C}$-closedness in various categories
- Characterizes topological groups that are $ ext{C}$-closed for different morphism types
- Provides a framework for understanding closedness in topological group categories

## 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.