In category theory, the phrase ‘to factor through’ essentially means that a morphism (or arrow) can be expressed as a composition of two other morphisms. More formally, if we have three objects A, B, and C in a category, and two morphisms f: A → B and g: B → C, we say that f factors through g if there exists some morphism h: A → C such that the following diagram commutes:
A ---f---> B | | | | g | | v v C <--h----
This means that if you first apply h to A, it should be equivalent to first applying f to A to get to B, and then applying g to get to C. In other words, the morphism h can be viewed as a 'shortcut' from A to C that passes through B.
Factoring through is a useful concept as it allows us to break down complex relationships in categories into more manageable pieces. It shows how compositions of morphisms relate to each other and can help to illustrate properties such as equivalence and isomorphism between objects.