For questions about co-hopfian groups, rings, modules, etc. Namely, objects which are not isomorphic to any proper subobject.
A set $S$ is finite if and only if every surjective map $S \to S$ is a bijection, if and only if every injective map $S \to S$ is a bijection.
We say an object $A$ in a category is a Hopfian object if every epimorphism $A \to A$ is an automorphism, while we say it is a co-Hopfian object if every monomorphism $A \to A$ is an automorphism. Unlike in the category of sets, the two notions do not coincide in general.
These concepts were first studied in the category of groups. Some examples of Hopfian groups include finite groups, $\mathbb{Z}$, finitely generated free groups, and $\mathbb{Q}$. Examples of co-Hopfian groups include finite groups, $\mathbb{Q}$, $\mathbb{Q}/\mathbb{Z}$, and fundamental groups of closed hyperbolic manifolds. Just as there are groups which are both Hopfian and co-Hopfian (e.g. finite groups, $\mathbb{Q}$), there are also groups which are neither, such as $\mathbb{R}$.
Other categories where these notions have been considered include the categories of rings, modules, and topological spaces. See the Wikipedia article for more information. For questions about Hopfian objects, see hopfian-groups.