nLab strict factorization system



A strict factorization system is like an orthogonal factorization system, but the factorizations are specified uniquely on the nose, rather than merely up to isomorphism.


A strict factorization system on a category CC comprises wide subcategories EE and MM of CC such that every morphism in CC factors as mem e for eEe \in E and mMm \in M.

Note that a strict factorization system is not necessarily an orthogonal factorization system, since EE and MM may not contain the isomorphisms in CC. However, for every strict factorization system, there is a unique orthogonal factorization system containing it, given by closing EE under postcomposition by isomorphisms and closing MM under precomposition by isomorphisms (see §2.1 of Grandis (2000)).

As distributive laws in a bicategory

One reason strict factorization systems are of interest is that they can be identified with distributive laws in the bicategory of spans, as shown by Rosebrugh & Wood (2002).

Ordinary orthogonal factorization systems can be similarly characterized by:


Strict factorization systems were defined in:

  • Marco Grandis. Weak subobjects and the epi-monic completion of a category. Journal of Pure and Applied Algebra 154.1-3 (2000): 193-212.

See also:

Last revised on December 16, 2022 at 10:39:19. See the history of this page for a list of all contributions to it.