nLab Hopfian group

Definition

In sets

Definition

A discrete group $G$ is called a Hopfian group if every surjective endomorphism $\phi \colon G\to G$ is an isomorphism.

Dually, it is called coHopfian if any injective endomorphism of $G$ is an isomorphism.

In concrete categories

As the epimorphisms and monomorphisms in Grp are precisely the surjections and injections (see epimorphisms of groups are surjective), the definition generalises immediately to that of a Hopfian object in any category. In other words, we could define an object $X$ to be Hopfian if every epimorphic endomorphism is an isomorphism (cf. Dedekind finite object).

Whatever that generalization is worth, much of the literature (such as Varadarajan 1992) adopts the more concrete notion:

Definition

Given a concrete category $U \colon C \to Set$ with $U$ a faithful functor, say that an object $X$ of $C$ is Hopfian if every morphism $\phi \colon X \to X$ in $C$ with $U(\phi)$ surjective is an isomorphism.

Remark

Notice that if $U$ if faithful, then $\phi$ is epimorpic if $U(\phi)$ is surjective.

Remark

For monadic functors $U \colon C \to Set$, this surjectivity assumption is the same as the assumption that $\phi$ is a regular epimorphism.

Of course there is a dual notion of being co-Hopfian; here the hypothesis that $U(\phi)$ is injective frequently coincides simply with $\phi$ being monomorphic – certainly that is true if $U$ preserves finite limits (which is frequently the case in application).

Examples

Example

Clearly all finite groups are both Hopfian and coHopfian.

Example

Using Nielsen’s method, one can show that every finitely generated free group and the union of any ascending chain of such free groups (for example, $\mathbb{Q}$) are Hopfian.

Example

Every torsion-free hyperbolic group? is Hopfian.

References

• K. Varadarajan, Hopfian and Co-Hopfian Objects, Publicacions Matemátiques, Vol. 36 (1992), 293-317. (web)