From Grothendieck cofibrations to factorization systems: a formal 2-monadic account

Abstract

Grothendieck cofibrations describe transport in a category varying over a base, while factorization systems organize the arrows of a category into two complementary classes. We give a fully 2-categorical account of the passage from the former structure to the latter. The global comma 2-monad on the arrow 2-category encodes Grothendieck transport, whereas the squaring 2-monad encodes factorizations. We prove that split cofibrations are precisely the strict algebras for the comma 2-monad, including their 1-cells and 2-cells, and that normally cloven cofibrations are precisely its normal pseudoalgebras. A canonical colax morphism from the comma 2-monad to the squaring 2-monad then turns cocartesian transport into the cocartesian-vertical factorization of arrows in the total category. At the strict level, this yields the strict factorization system of designated cocartesian and vertical arrows; at the coherent level, it yields the orthogonal factorization system whose left class consists of all cocartesian arrows and whose right class consists of the arrows sent to isomorphisms in the base. We also separate unrestricted global pseudoalgebras, which retain a coherently trivial base action, from fixed-base pseudoalgebras, which correspond to arbitrary cleavages, and record the dual strict result for fibrations. This places the classical cofibration-factorization interaction, in all these variants, within a single change-of-2-monads construction and relates it directly to the existing fibrational and factorization literature.