Small object argument ($X\to Z_{\infty }\to Y$)

Attach cells for all lifting squares at each successive stage, then take a suitable colimit. Smallness of the generating domains makes every eventual lifting square occur at an earlier stage. This factors a map into a relative cell complex followed by a map with the prescribed right lifting property.

A composite built by successive pushouts of coproducts of prescribed generating maps. In nonnegative chain complexes, attaching a sphere-to-disk generator adds a free element with a specified existing cycle as boundary. The resulting inclusion has a degreewise free cokernel.

Mapping out of the object preserves sequential colimits. A map factors through a finite stage, and two finite-stage maps agreeing in the colimit agree at a later stage. Relative sequential smallness restricts the diagrams to a stated class of morphisms.