# category of paths on a graph

A nice class of illustrative examples of some notions of category theory^{}
is provided by categories^{} of paths on a graph.

Let $G$ be an undirected graph. Denote the set of vertices of $G$ by “$V$” and denote the set of edges of $G$ by “$E$”.

A path of the graph $G$ is an ordered tuplet of vertices $({x}_{1},{x}_{2},\mathrm{\dots}{x}_{n})$
such that, for all $i$ between $1$ and $n-1$, there exists an edge connecting
${x}_{i}$ and ${x}_{i+i}$. As a special case, we allow trivial paths which consist
of a single vertex — soon we will see that these in fact play an important
role as identity elements^{} in our category.

In our category, the vertices of the graph will be the objects and the
morphisms^{} will be paths; given two of these objects $a$ and $b$, we set
$\mathrm{Hom}(a,b)$ to be the set of all paths $({x}_{1},{x}_{2},\mathrm{\dots}{x}_{n})$
such that ${x}_{1}=a$ and ${x}_{n}=b$. Given an object $a$, we set ${1}_{a}=(a)$,
the trivial path mentioned above.

To finish specifying our category, we need to specify the composition operation.
This operation^{} will be the concatenation of paths, which is defined as follows:
Given a path $({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n})\in \mathrm{Hom}(a,b)$ and a path
$({y}_{1},{y}_{2},\mathrm{\dots},{y}_{m})\in \mathrm{Hom}(a,b)$, we set

$$a\circ b=({x}_{1},{x}_{2},\mathrm{\dots}{x}_{n},{y}_{2},\mathrm{\dots},{y}_{m}).$$ |

(Remember that ${x}_{n}={y}_{1}=b$.) To have a bona fide category, we need to check that this choice satisfies the defining properties (A1 - A3 in the entry http://planetmath.org/node/965category). This is rather easily verified.

A1: Given a morphism $({x}_{1},{x}_{2},\mathrm{\dots}{x}_{n})$, it can only belong to $\mathrm{Hom}(a,b)$ if ${x}_{1}=a$ and ${x}_{n}=b$, hence $\mathrm{Hom}(a,b)\cup \mathrm{Hom}(c,d)=\mathrm{\varnothing}$ unless $a=c$ and $b=d$.

A2: Suppose that we have four objects $a,b,c,d$ and three morphisms, $({x}_{1},{x}_{2},\mathrm{\dots}{x}_{n})\in \mathrm{Hom}(a,b)$, $({y}_{1},{y}_{2},\mathrm{\dots}{y}_{m})\in \mathrm{Hom}(b,c)$, and $({z}_{1},{z}_{2},\mathrm{\dots}{z}_{k})\in \mathrm{Hom}(c,d)$. Then, by the definition of the operation $\circ $ given above,

$(({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n})\circ ($ | ${y}_{1},{y}_{2},\mathrm{\dots},{y}_{m}))\circ ({z}_{1},{z}_{2},\mathrm{\dots},{z}_{k})$ | ||

$=({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n},{y}_{2},\mathrm{\dots},{y}_{m})\circ ({z}_{1},{z}_{2},\mathrm{\dots},{z}_{k})$ | |||

$=({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n},{y}_{2},\mathrm{\dots},{y}_{m},{z}_{2},\mathrm{\dots},{z}_{k})$ | |||

$({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n})\circ ($ | $({y}_{1},{y}_{2},\mathrm{\dots},{y}_{m})\circ ({z}_{1},{z}_{2},\mathrm{\dots},{z}_{k}))$ | ||

$=({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n})\circ ({y}_{1},{y}_{2},\mathrm{\dots},{y}_{m},{z}_{2},\mathrm{\dots},{z}_{k})$ | |||

$=({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n},{y}_{2},\mathrm{\dots},{y}_{m},{z}_{2},\mathrm{\dots},{z}_{k}).$ |

Since these two quantities are equal, the operation is associative.

A3: It is easy enough to check that paths with a single vertex act as identity elements:

$({x}_{1})\circ ({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n})$ | $=({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n})$ | ||

$({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n})\circ ({x}_{n})$ | $=({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n})$ |

It is also possible to consider the equivalence class^{} of paths
modulo retracing. To introduce this category, we first define
a binary relation^{} $\approx $ on the class of paths as follows:
Let $A$ and $B$ be any two paths such that the right endpoint^{}
of $A$ is the same as the left endpoint of $B$, i.e. $A\in \mathrm{Hom}(a,b)$ and $B\in \mathrm{Hom}(b,c)$
for some vertices $a,b,c$ of our graph. Let $d$ be any vertex
which shares an edge with $d$. Then we set $A\circ B\approx A\circ (c,d,c)\circ B$.

Let $\sim $ be the smallest equivalence relations which contains
$\approx $. We will call this equivalence relation *retracing*.

As defined above, it may not intuitively obvious what this
equivalence amounts to. To this end, we may consider a different
description. Define the *reversal* of a path to be the
path obtained by reversing the order of the vertices traversed:

$${({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n-1},{x}_{n})}^{-1}=({x}_{n},{x}_{n-1},\mathrm{\dots},{x}_{2},{x}_{1})$$ |

Then we may show that two paths are equivalent^{} under retracing
if they may both be obtained from a third path by inserting
terms of the form $X{X}^{-1}$. In symbols, we claim that $A\sim B$
if there exists an integer $n\xc3\mathrm{\xaf}\xc2\mathrm{\xbc}\AA \mathrm{\xbe}0$ and paths ${X}_{1},\mathrm{\dots}{X}_{n+1},{Y}_{1},\mathrm{\dots}{Y}_{n-1},{Z}_{1},\mathrm{\dots}{Z}_{n}$ such that

$$A={X}_{1}\circ {X}_{1}^{-1}\circ {Z}_{1}\circ {X}_{2}\circ {X}_{2}^{-1}\circ \mathrm{\cdots}\circ {X}_{n-1}\circ {X}_{n-1}^{-1}\circ {Z}_{n}\circ {X}_{n}\circ {X}_{n}^{-1}\circ {Z}_{n}\circ {X}_{n+1}\circ {X}_{n+1}^{-1}$$ |

and

$$B={Y}_{1}\circ {Y}_{1}^{-1}\circ {Z}_{1}\circ {Y}_{2}\circ {Y}_{2}^{-1}\circ \mathrm{\cdots}\circ {Y}_{n-1}\circ {Y}_{n-1}^{-1}\circ {Z}_{n}\circ {Y}_{n}\circ {Y}_{n}^{-1}\circ {Z}_{n}\circ {Y}_{n+1}\circ {Y}_{n+1}^{-1}$$ |

This characterization explains the choice of the term “retracing” — we do not change the equivalence class of the path if we happen to wander off somewhere in the course of following the path but then backtrack and pick the path up again where we left off on our digression.

Rather than presenting a detailed formal proof, we will sketch how the two definitions may be shown to be equivalent.

Title | category of paths on a graph |
---|---|

Canonical name | CategoryOfPathsOnAGraph |

Date of creation | 2013-03-22 16:45:54 |

Last modified on | 2013-03-22 16:45:54 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 18 |

Author | rspuzio (6075) |

Entry type | Example |

Classification | msc 20L05 |

Classification | msc 18B40 |

Related topic | IndexOfCategories |