In mathematics, the free category or path category generated by a directed graph or quiver is the category that results from freely concatenating arrows together, whenever the target of one arrow is the source of the next.
is an edge of the quiver, and n ranges over the non-negative integers.
of the quiver, there is an "empty path" which constitutes the identity morphisms of the category.
Intuitively, U "[forgets] which arrows are composites and which are identities".
The free category on a quiver can be described up to isomorphism by a universal property.