nLab strict factorization system

Idea

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.

Definition

A strict factorization system on a category $C$ comprises wide subcategories $E$ and $M$ of $C$ such that every morphism in $C$ factors as $m e$ for $e \in E$ and $m \in M$.

Note that a strict factorization system is not necessarily an orthogonal factorization system, since $E$ and $M$ may not contain the isomorphisms in $C$. However, for every strict factorization system, there is a unique orthogonal factorization system containing it, given by closing $E$ under postcomposition by isomorphisms and closing $M$ 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:

• using a type of relaxed distributive law, as in Rosebrugh & Wood (2002);

• using wreaths as in Lack & Street (2002);

• by working in the bicategory of profunctors instead, as in Lack (2004) (see also factorization system over a subcategory); or by

• using weak distributive laws, as in Böhm (2012).

References

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:

• Robert Rosebrugh, Richard J. Wood, Distributive Laws and Factorization, Journal of Pure and Applied Algebra 175 1–3 (2002) 327-353 [doi:10.1016/S0022-4049(02)00140-8pdf]

• Stephen Lack, Ross Street, The formal theory of monads II, J. Pure Appl. Alg. 175 1–3 (2002) 243-265 [doi:10.1016/S0022-4049(02)00137-8]

• Stephen Lack, Composing PROPs, Theory and Applications of Categories, 13 9 (2004) 147-163 [tac:13-09]

• Gabriella Böhm, Factorization systems induced by weak distributive laws, Appl. Categ. Structures 20 3 (2012) 275-302 [doi:10.1007/s10485-010-9243-y, arXiv:1009.0732]