Definition
Def Category on Graph Given a directed graph
, we define the category generated from by taking the vertices of as objects, and the paths in as arrows, we’ll write the arrows of in the form and define and define composition by concatenation: For each vertex , we have an empty path denoted , which is the identity arrow at .
Proposition
Prop If
has only one vertex, then is just the free monoid on the set of edges of .
Proposition
Prop If
has only vertices (no edges), then is the discrete category on the set of vertices of .
Def Universal Mapping Property