nLab
path category

There are several concepts often called a path category.

Free category on a directed graph

There is a forgetful functor from small categories to quivers, (i.e. directed graphs allowing more than one edge from $a$ to $b$). This forgetful functor has a left adjoint, the free category or path category of a quiver, which has the same objects (vertices) but much more arrows. The morphisms from $a$ to $b$ in this free category are the sequences of the form $(a_n,f_n,a_{n-1},\dots ,a_2,f_1,a_0)$ where $n\ge 0$, $a_0,a_1,\dots ,a_n$ are vertices of the graph, $a=a_0$, $b=a_n$, for all $0<i\le n$, $f_i:a_{i-1}\to a_i$ is an edge from $a_{i-1}$ to $a_i$. The composition is given by the concatenation

$$(a_n,f_n,a_{n-1},\dots ,a_2,f_1,a_0)\circ (b_m,g_m,a_{m-1},\dots ,b_2,g_1,b_0):=(a_n,f_n,a_{n-1},\dots ,a_2,f_1,a_0=b_m,g_m,a_{m-1},\dots ,b_2,g_1,b_0)$$(a_n,f_n,a_{n-1},\ldots,a_{2},f_1,a_0)\circ (b_m,g_m,a_{m-1},\ldots,b_{2},g_1,b_0) := (a_n,f_n,a_{n-1},\ldots,a_{2},f_1,a_0= b_m,g_m,a_{m-1},\ldots,b_{2},g_1,b_0)

whenever $a_0=b_m$, and the target and source maps are given by $s(a_n,f_n,a_{n-1},\dots ,a_2,f_1,a_0)=a_0$ and $t(a_n,f_n,a_{n-1},\dots ,a_2,f_1,a_0)=a_n$. One informally writes $f$ for the morphism $(b,f,a):a\to b$ in the free category and the identities of the free category are ${\mathrm{id}}_a=(a,a)$; thus $f_n\circ f_{n-1}\circ \dots \circ f_1=(t(f_n),f_n,t(f_{n-1}),\dots ,t(f_1),f_1,s(f_1))$. The standard reference is Gabriel–Zisman.

Path category of a space

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 category whose collection of objects is $X$, and whose morphisms are images of intervals in $X$, i.e. 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 images of intervals 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.

Arrow 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 $\mathrm{Arr}(X)$.