Groupoids

We recall the notion of groupoid:
a category in which each arrow is invertible.

This is common generalization of the notions of equivalence relation
(a groupoid which is “thin”, i.e., has at most one arrow between any two objects) and
group (a groupoid with only one object).

An important source of groupoids is:
Each group action has an associated groupoid.
If $X$ is a set acted on by a group $G$, with action \[\begin{array}{c} X \times G & \to & X\\ \langle x, g \rangle &\mapsto & xg , \end{array}\] then associated groupoid has objects the elements of $X$,
and arrows the set $X \times G$, with source and targets of each arrow as follows: $$ x \xrightarrow{\textstyle \langle x,g\rangle} xg .$$ When the source, $x$ in this case, is displayed, the arrow can be denoted simply as: $$ x \xrightarrow{\textstyle g} xg .$$

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Examples

Important examples are in combinatorics, algebra, and geometry.

In combinatorics, consider the group $S_3$
with its canonical action on $[3]$.
Of course $S_3$ is in fact the automorphism group of $[3]$,
so its six elements are normally viewed in the category of sets as loops on $[3]$,
so $S_3$ may be viewed as a subcategory of $\Set$ with one object $[3]$ and six arrows.

The groupoid associated to the action $[3]\times S_3 \to S_3$, in contrast,
has three objects, namely the elements $0,1,2$ of $[3]$,
and 18 arrows, one for each $\langle i,\sigma\rangle \in [3]\times S_3$,
with $\langle i,\sigma\rangle$ now being viewed an an arrow $$\langle i,\sigma\rangle : i \to i\sigma .$$ Thus, for example $0\in [3]$ has six arrows out of it: \[\begin{array}{} 1\\ \llap{(0,1),\,(0,1,2)}\Bigg\uparrow\\ \llap{\iota,\,(1,2) \circlearrowright\;}0 & \xrightarrow[\textstyle (0,2),\,(0,2,1)]{} & 2 ,\\ \end{array}\] and similarly for $1$ and $2$.

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In algebra, consider the subgroup $2\Z$ of $\Z$ acting on $\Z$ through addition.
A portion of the associated groupoid may be illustrated as follows: $\def\overtwo{\buildrel 2 \over \rightarrow}$ \[\begin{array}{c} & \dots & \overtwo & -2 & \overtwo & 0 & \overtwo & 2 & \overtwo & 4 & \overtwo & 6 & \overtwo & \dots\\ \dots & \overtwo & -3 & \overtwo & -1 & \overtwo & 1 & \overtwo & 3 & \overtwo & 5 & \overtwo & \dots \end{array}\] Note that the groupoid consists of two connected components.
In fact, these components may be viewed in several ways, as

cosets	of the subgroup $2\Z$,
orbits	of a (sub)group action,
equivalence classes	of the induced equivalence relation, or
connected componentsmmm	of a graph (the underlying graph of the groupoid).

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In geometry, consider a vector space $V$ acting on itself via addition,
\[\begin{array}{c} V\times V &\to &V\\ \langle x,v \rangle & \mapsto & x+v \end{array}\] If we apply the recipe above for defining the groupoid associated to this action,
we get precisely the picture normally drawn in elementary courses on vectors: \[ x \buildrel v \over \longrightarrow x+v .\] Of course, normally this picture is drawn with $x+v$ placed higher on the page,
with the arrow at an angle to the “$x$-axis”.
Sometimes $x$ will be called a “based vector” and $v$ a “free vector”.
But the idea is the same as in the groupoid case:
$v$ is an actor, taking $x$ to $x+v$.