Retract

of an object of a category

A concept generalizing the corresponding concepts in algebra and topology. An object $R$ of a category $\mathfrak{K}$ is called a retract of an object $A$ if there exist morphisms $$\mu : R \rightarrow A \ \ \ \text{and}\ \ \ \nu : A \rightarrow R$$ such that $\nu\mu = 1_R$. The morphism $\mu$ in this case is a monomorphism and, moreover, the equalizer of the pair of morphisms $1_A$, $\mu\nu$. Dually, the morphism $\nu$ is an epimorphism and also the co-equalizer of the pair of morphisms $1_A$, $\mu\nu$. $\mu$ is sometimes known as a section and $\nu$ as a retraction.

If $R$ is a retract of an object $A$ and an object $R'$ is isomorphic to $R$, then $R'$ is a retract of $A$. Therefore an isomorphism class of retracts forms a single subobject of $A$. Each retract of $A$, defined by morphisms $\mu : R \rightarrow A$ and $\nu : A \rightarrow R$, corresponds to an idempotent morphism $\phi=\mu\nu : A \rightarrow A$. Two retracts $R$ and $R'$ of an object $A$ belong to the same subobject if and only if they correspond to the same idempotent. The retracts of any object of an arbitrary category form a set.

Comments

The last sentence above is true only if one assumes that all categories involved are locally small (i.e. "have small hom-sets" ) (cf. also Small category).