# nLab path category

category theory

## Applications

There are several concepts often called a path category.

# Contents

## 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 $a$ 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 $\left({a}_{n},{f}_{n},{a}_{n-1},\dots ,{a}_{2},{f}_{1},{a}_{0}\right)$ where $n\ge 0$ is a natural number, ${a}_{0},{a}_{1},\dots ,{a}_{n}$ are vertices of the graph, $a={a}_{0}$, $b={a}_{n}$, and for all $0, ${f}_{i}:{a}_{i-1}\to {a}_{i}$ is an edge from ${a}_{i-1}$ to ${a}_{i}$. The composition is given by the concatenation

$\left({a}_{n},{f}_{n},{a}_{n-1},\dots ,{a}_{2},{f}_{1},{a}_{0}\right)\circ \left({b}_{m},{g}_{m},{a}_{m-1},\dots ,{b}_{2},{g}_{1},{b}_{0}\right):=\left({a}_{n},{f}_{n},{a}_{n-1},\dots ,{a}_{2},{f}_{1},{a}_{0}={b}_{m},{g}_{m},{a}_{m-1},\dots ,{b}_{2},{g}_{1},{b}_{0}\right)$(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 $s\left({a}_{n},{f}_{n},{a}_{n-1},\dots ,{a}_{2},{f}_{1},{a}_{0}\right)={a}_{0}$ and $t\left({a}_{n},{f}_{n},{a}_{n-1},\dots ,{a}_{2},{f}_{1},{a}_{0}\right)={a}_{n}$. One informally writes $f$ for the morphism $\left(b,f,a\right):a\to b$ in the free category and the identities of the free category are ${\mathrm{id}}_{a}=\left(a,a\right)$; thus ${f}_{n}\circ {f}_{n-1}\circ \dots \circ {f}_{1}=\left(t\left({f}_{n}\right),{f}_{n},t\left({f}_{n-1}\right),\dots ,t\left({f}_{1}\right),{f}_{1},s\left({f}_{1}\right)\right)$. 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 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.

## Arrow category

Given a category $X$, the functor category $\left[I,X\right]$ 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 $\mathrm{Arr}\left(X\right)$.

Revised on October 9, 2012 01:06:34 by Urs Schreiber (82.169.65.155)