A commutative diagram is a diagram in which composition is path-independent?.
For our purposes, a diagram in a category consists of a quiver and a functor to from the free category on :
Then this diagram commutes if this functor factors (up to natural isomorphism) through a poset :
or equivalently (treating as a strict category) if the functor factors up to equality through a proset :
In the above, we are identifying quivers, posets, and prosets with certain categories in the usual ways.
Recall that a quiver consists of a set of vertices, a set of edges, and two functions . Given a category , a diagram of shape in a category is consists of a map from to the objects of and a map from to the morphisms of , both denoted , such that and for each edge , where are the source and target maps in .
Recall that a path in consists of a list of vertices and a list of edges such that and for each , where is any natural number (possibly zero). We say that is the source of the path and that is its target. Given a path and a diagram , the composite of under is the composite in .
A diagram commutes if, given any two vertices in and any two paths with source and target , the composites of and under are equal in .