Morphism - Knowlpedia

Let $\mathcal C$ be a category and $X,Y \in \mathrm{Ob}(\mathcal C)$ . A morphism $f : X \to Y$ is an element of $\mathrm{Hom}_{\mathcal C}(X,Y)$ .

The source (or domain) of $f$ is $X$ , and its target (or codomain) is $Y$ . Morphisms can be composed when targets and sources match, and each object has an identity morphism.

Special morphisms

If $X=Y$ , then $f$ is an endomorphism.
If $f$ is invertible (has a two-sided inverse), then $f$ is an isomorphism.
If $f$ is left-cancellative under composition, then $f$ is a monomorphism (a “mono”).
Dually, one has epimorphisms (“epis”).

Examples

In $\mathbf{Set}$ , morphisms are functions between sets.
In $\mathbf{Grp}$ , morphisms are group homomorphisms.
In $\mathbf{Top}$ , morphisms are continuous maps.