Free actions and quotient groupoids

PDF p. 6: §2.4, quotient-category construction

Let $N$ be a group and let $\mathcal{G}$ be a connected small groupoid: its objects and arrows form sets, every arrow is invertible, and there is an arrow between every ordered pair of objects. Composition is read left to right, so $ab$ means first $a$ and then $b$.

An automorphism of $\mathcal{G}$ is an invertible functor from $\mathcal{G}$ to itself. An action of $N$ on $\mathcal{G}$ is a homomorphism from $N$ to the group of these automorphisms, whose multiplication is composition of functors. Write $nx$ and $na$ for the images of an object $x$ and an arrow $a$ under the automorphism assigned to $n$; thus $(nm)x=n(mx)$ and $(nm)a=n(ma)$. The action is free on objects if $nx=x$ implies $n=1$.

Assume freeness on objects. The quotient groupoid $\mathcal{G}\mathrm{/}N$ has as objects the object orbits $[x]=\{nx:n\in N\}$. Its arrows are the arrow orbits $[a]=\{na:n\in N\}$, with source $[x]$ and target $[y]$ when $a:x\to y$. Equivalently, the arrows from $[x]$ to $[y]$ are precisely the orbits $[a]$ for which some representative has the form $a:gx\to g^{\mathrm{\prime }}y$, with $g,g^{\mathrm{\prime }}\in N$.

To compose $[a]:[x]\to [y]$ and $[b]:[y]\to [z]$, choose a representative $a$ and then the unique representative $b^{\mathrm{\prime }}$ of $[b]$ whose source is the target of $a$; define $[a][b]=[a{b}^{\mathrm{\prime }}]$. The identity at $[x]$ is $[1_x]$, and the inverse of $[a]$ is $[a^{-1}]$.

Covariant functor, identity functor, composite functor, and contravariant functor

Monoid homomorphism and group homomorphism

Connected small groupoids and vertex groups

Left group actions, transitive actions, and faithful actions

A free group action has no nonidentity element fixing a point

Equivalence relation, equivalence class, and the quotient set A/∼

Free object actions give orbit coverings

The homomorphism supplies compatible left actions on objects and arrows, and functors preserve sources, targets, identities and composition. Group-action orbits are equivalence classes, so the displayed object and arrow collections are well-defined sets. The source and target of an arrow orbit do not depend on the representative. An arrow orbit has source $[x]$ and target $[y]$ exactly when a representative has endpoints $gx$ and $g^{\mathrm{\prime }}y$ for suitable $g,g^{\mathrm{\prime }}\in N$.

By Free object actions give orbit coverings, freeness gives a unique representative of an arrow orbit with any prescribed source in its source orbit. Consequently the prescribed $b^{\mathrm{\prime }}$ exists uniquely; changing the representative $a$ changes the consecutive pair by a common translate, so the composite orbit is independent of the choice. The same proposition establishes associativity, the stated identity and inverse orbits, and connectedness of the resulting small groupoid.