Morphism

Definition

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

Special cases:

• 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

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