# Contents

## 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:

## 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.