Small object argument

Remarks

The proof builds $X = X_0 \to X_1 \to X_2 \to \cdots$ over $Y$ by transfinite induction. At each step one takes all commutative squares from a morphism of $I$ to $X_\beta \to Y$. One pushout of a coproduct then glues in a solution for every square at once. At limit stages one takes the colimit. One stops at an ordinal $\kappa$ so large that every map from a domain of $I$ into $X_\kappa$ factors through some earlier $X_\beta$. The smallness hypothesis guarantees such a $\kappa$; then any lifting problem against $X_\kappa \to Y$ is solved at stage $\beta + 1$. Without smallness the transfinite construction need not stop at any stage, and the argument does not apply.

A consequence is that every morphism with the left lifting property with respect to $I\text{-inj}$ is a retract of a morphism of $I\text{-cell}$. In a locally presentable category every object is small, so the hypothesis holds for every set $I$.

Example: in topological spaces let $I$ be the set of boundary inclusions $S^{n-1} \to D^n$ for $n \ge 0$. The theorem factors every map into a relative cell complex followed by a map with the right lifting property against these inclusions. The second map is then a weak homotopy equivalence and a Serre fibration. This is how the cofibration and trivial fibration factorization of the Quillen model structure is obtained.

Going deeper

For a student

The theorem is the main tool to build cofibrantly generated model structures. One gives a set $I$ of generating cofibrations and a set $J$ of generating trivial cofibrations, and applies the argument to both. The cofibrations are then the retracts of $I$-cell maps, and the fibrations are the $J$-injective maps. This is how the standard model structures on topological spaces, simplicial sets and chain complexes are constructed. The functoriality of the factorization comes from the fact that the gluing step uses all lifting problems, not a chosen one.