Section 12.1 Preliminary Categorical Definitions

A map \(f\) in \(\mathcal{C}\) is a retract of a map \(g\) if there is a commutative diagram

in which both horizontal composites are the identity.

The arrow category \(\mathrm{Ar}(\mathcal{C})\) has as objects the morphisms \(f \colon X \to Y\) of \(\mathcal{C}\text{,}\) and as morphisms \(f \to g\) the commutative squares

A functorial factorization is a pair of functors \((\alpha, \beta) \colon \mathrm{Ar}(\mathcal{C}) \to \mathrm{Ar}(\mathcal{C})\) such that \(f = \beta(f) \circ \alpha(f)\) for every \(f \in \mathrm{Ar}(\mathcal{C})\text{.}\)

Let \(i \colon A \to B\) and \(p \colon X \to Y\) be morphisms in \(\mathcal{C}\text{.}\) We say that \(i\) has the left lifting property (LLP) with respect to \(p\) — equivalently, \(p\) has the right lifting property (RLP) with respect to \(i\) — if every commutative square

admits a diagonal lift.