There are several concepts often called a path category.
Free category on a directed graph
There is a forgetful functor from small strict categories to quivers. This forgetful functor has a left adjoint, giving the free category or path category of a quiver, whose objects are the vertices of the quiver. The morphisms from to in this free category are not merely the arrows from to in the quiver but instead are lists of the form where is a natural number, are vertices of the graph, , , and for all , is an edge from to . The composition is given by the concatenation
whenever , and the target and source maps are given by and . One informally writes for the morphism in the free category and the identities of the free category are ; thus . The standard reference is Gabriel–Zisman.
Path category of a space
Given a topological space (or some other kind of space with a notion of maps from intervals into it), there are various ways to obtain a small strict category whose objects are the points of and whose morphisms are paths in . This is also often called a path category.
In particular, for every topological space there is its fundamental groupoid whose morphisms are homotopy classes of paths in .
If is a directed space there is a notion of path category called the fundamental category of .
When is a smooth space, there is a notion of path category where less than homotopy of paths is divided out: just thin homotopy. This yields a notion of path groupoid.
If parameterized paths are used, there is a way to get a category of paths without dividing out any equivalence relation: this is the Moore path category.
Given a category , the functor category for the interval category might be called a “directed path category of ” (similar to path space). However, this functor category is referred to instead as the arrow category of and sometimes even denoted .