M Theory Lesson 222

A useful notion in categorical Galois theory is the idea of precategory. This is a collection of objects $O$ and morphisms $A$ (ie. a graph) together with source and target maps ($s$ and $t$) and suitable maps $m$ and $e$ expressing composition and identities. But in a category we can define an internal precategory using the pullback diagram where certain pieces through $O$ commute. That is, the lower and right hand triangles express the fact that taking a source or target of an identity arrow defines the identity on objects. The equation $s m = s p_1$ says that taking the source of a composition is the same as taking the source of the first arrow in the pair. Similarly, taking the target of the second arrow is the same as taking the target of the pair. The pullback square expresses the fact that one can only compose arrows when the target of the first matches the source of the second.

Observe that not all pieces of the diagram commute. For example, it is not true that $sm = tm$, unless the composition forms a loop. But this is always true for a one object category, such as a group, in which case one is permitted to draw in an arrow $1_{A}$ diagonally across the pullback square, and then basically everything commutes.