Free category

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.
More precisely, the objects of the category are the vertices of the quiver, and the morphisms are paths between objects. Here, a path is defined as a finite sequence
where formula_2 is a vertex of the quiver, formula_3 is an edge of the quiver, and "n" ranges over the non-negative integers. For every vertex formula_4 of the quiver, there is an "empty path" which constitutes the identity morphisms of the category.
The composition operation is concatenation of paths. Given paths
their composition is
Note that the result of the composition starts with the right operand of the composition, and ends with its left operand.
Properties.
The category of small categories Cat has a forgetful functor into the quiver category Quiv:
which takes objects to vertices and morphisms to arrows. Intuitively, "[forgets] which arrows are composites and which are identities". This forgetful functor is right adjoint to the functor sending a quiver to the corresponding free category.
Universal property.
The free category on a quiver can be described up to isomorphism by a universal property. Let : Quiv → Cat be the functor that takes a quiver to the free category on that quiver (as described above), let be the forgetful functor defined above, and let be any quiver. Then there is a graph homomorphism : → (()) and given any category D and any graph homomorphism : → , there is a unique functor : () → D such that ()∘=, i.e. the following diagram commutes:
The functor is left adjoint to the forgetful functor .