Contents

Definition

An object $A$ in a category is a retract of an object $B$ if there are morphisms $i:A\to B$ and $r:B\to A$ such that $r \circ i = 1_A$.

In this situation, $r$ is a split epimorphism and $i$ is a split monomorphism; the composite $i \circ r$ is a split idempotent. Sometimes $r$ is called a retraction of $i$ and $i$ is called a section of $r$; these terms come from topology. The whole thing may also be called a splitting of $i$, $r$, or $i \circ r$.

Properties

• Retracts are clearly preserved by any functor.

• A split epimorphism $r; B \to A$ is the strongest of various notions of epimorphism (e.g., it is a regular epimorphism, in fact an absolute? coequalizer, being the coequalizer of a pair $(e, 1_B)$ where $e = i \circ r: B \to B$ is idempotent). Dually, a split monomorphism is the strongest of various notions of monomorphism.

This appears as (Borceux, lemma 6.5.6)

Examples

Of simplices

The includion of standard topological horns into the topological simplex $\Lambda^n_k \hookrightarrow \Delta^n$ is a retract in Top.

In arrow categories

In the theory of weak factorization systems and model categories, an important role is played by retracts in $C^{\mathbf{2}}$, the arrow category of $C$. Explicitly spelled out in terms of the original category $C$, a morphism $f:X\to Y$ is a retract of a morphism $g:Z\to W$ if we have commutative squares

such that the top and bottom rows compose to identities.

References

In

• Francis Borceux, Handbook of categorical algebra I

the definition appears as def. 1.7.3. Properties are discussed in section 6.5