One can view the concept of a morphism or arrow in a category as an extreme abstraction of the concept of a spatial trajectory. It therefore comes with no surprise that (notions of) paths stemming from various conceptualizations of ‘space’ like e.g. quivers or topological spaces arrange themselves in appropriate categories thereby giving rise to several different notions of path category.
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 $a$ to $b$ in this free category are not merely the arrows from $a$ to $b$ in the quiver but instead are lists of the form $(a_n,f_n,a_{n-1},\ldots,a_{1},f_1,a_0)$ where $n \geq 0$ is a natural number, $a_0,a_1,\ldots,a_n$ are vertices of the graph, $a = a_0$, $b = a_n$, and for all $0 \lt i \leq n$, $f_i\colon a_{i-1} \to a_i$ is an edge from $a_{i-1}$ to $a_i$. The composition is given by the concatenation
whenever $a_0 = b_m$, and the target and source maps are given by $s(a_n,f_n,a_{n-1},\ldots,a_{1},f_1,a_0)=a_0$ and $t(a_n,f_n,a_{n-1},\ldots,a_{1},f_1,a_0) = a_n$. One informally writes $f$ for the morphism $(b,f,a)\colon a \to b$ in the free category and the identities of the free category are $id_a = (a,a)$; thus
With free objects, one is often primarily interested in taking quotients whence a free category over a graph is usually a somewhat auxiliary gadget its main interest lying in the role it plays in defining categories of fractions (cf. Gabriel-Zisman 1967 pp.1,6; probably the original source of the path category in this sense).
But similar techniques apply to various notions of graphs or more restricted classes of categories with forgetful functors to graphs permitting to syntactically generate various notions of free categories with extra structure and in some of these cases it occurs that the corresponding free category (of paths) is indeed interesting in itself e.g. the free topos arises this way and the syntactic derivations of context free grammars arrange themselves into such categories of paths as explored in Walters (1989a,1989b) or Latch (1991).
Given a topological space $X$ (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 $X$ and whose morphisms are paths in $X$. 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 $X$.
If $X$ is a directed space there is a notion of path category called the fundamental category of $X$.
When $X$ 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 $X$, the functor category $[I,X]$ for $I$ the interval category might be called a “directed path category of $X$” (similar to path space). However, this functor category is referred to instead as the arrow category of $X$ and sometimes even denoted $Arr(X)$.
The free category on a graph has a section in
or in
The following is another source for this, even an open source:
The path categories of context free grammars are explored in
Dana Latch, The connection between the fundamental groupoid and a unification algorithm for syntactil algebras (extended abstract) , Cah. Top. Géom. Diff. Cat. XXXII (1991). (link)
Bob Walters, The free category with products on a multigraph , JPAA 62 (1989). (link)
Bob Walters, A note on context-free languages , JPAA 62 (1989). (link)
Since the perspective on categories as ‘graphs with operations’ is closely related to their ‘logico-deductive’ side, it figures prominently in the work of Jim Lambek and Phil Scott (cf. the references at free topos).