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 uniquely (not just uniquely up to unique isomorphism) 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)).

Examples

• Every category has two “trivial” strict factorization systems in which $E$, respectively $M$, consists of the identity morphisms only. The corresponding orthogonal factorization systems are those with one class consisting of the isomorphisms.

• Part of the structure of a (non-generalized) Reedy category is a strict factorization system.

• The category Set, as defined in a material set theory, has a strict factorization system consisting of the surjections and the inclusions of subsets. Its associated orthogonal factorization system consists of surjections and injections. Other categories built out of structured sets have similar strict factorization systems.

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]