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Quotient groupoids give normal covers

PDF p. 6: §2.4, last paragraph

Statement

Let a group $N$ act by automorphisms on a connected small groupoid $G$ , freely on objects. Thus $G$ has sets of objects and arrows, every arrow is invertible, and an arrow exists between every ordered pair of objects; freeness means $n x = x$ only for $n = 1$ . Use the left action and left-to-right arrow composition conventions of Free actions and quotient groupoids.

Every arrow orbit whose source is the object orbit $[x]$ has a unique representative with source $x$ . The object and arrow orbits form a groupoid $G / N$ , with composition obtained by aligning consecutive representatives. The quotient functor $Q : G \to G / N$ is a normal covering, and $n \mapsto (a \mapsto n a)$ is an isomorphism $N ≅ Deck (Q)$ , where deck transformations are automorphisms over $Q$ . For every object $x$ there is an exact sequence $1 \to G_{x} \overset{Q}{\to} (G / N)_{[x]} \overset{δ}{\to} N \to 1$ , where the vertex groups consist of arrows from the indicated object to itself. If the unique representative of a loop orbit with source $x$ ends at $n x$ , then $δ$ sends that orbit to $n$ .

Statement references

Free actions and quotient groupoids

Groupoid coverings, universal and normal covers

Free object actions give orbit coverings

Facts & Assumptions

[F1]

For a left action by automorphisms on a connected small groupoid, free on objects, the orbit construction has unique representatives with prescribed source and produces a connected groupoid and a normal covering. Its deck group is the acting group via the action, and the vertex-group sequence defined by endpoints of lifted loops is exact. Free object actions give orbit coverings

[F2]

In the quotient convention, objects and arrows are their respective $N$ -orbits, and composition aligns the source of the second representative with the target of the first. A normal covering is a covering whose deck group acts transitively on a fibre. Free actions and quotient groupoids Groupoid coverings, universal and normal covers

Proof

Given: The groupoid and free object action in the Statement.

Proof technique: direct.

Goal: Establish the quotient construction, normal covering, deck-group identification and exact sequence.

1.1F1F2given

Apply the orbit construction. The hypotheses of [F1] hold by the Given. Its quotient has the orbit and composition conventions recorded in [F2], so it is precisely $G / N$ as defined here. Hence the prescribed-source representative is unique, the composition is well defined, and $Q$ is a normal covering.

2.1F1step 1.1∎

Identify the group maps. By [F1], the action homomorphism is an isomorphism from $N$ to $Deck (Q)$ . The same result gives the exact vertex-group sequence, with $δ$ defined by the endpoint $n x$ of the unique lift from $x$ . These are the maps in the Statement, completing the proof.

Depends on

Used by · 2

Dependency tree · 2 levels

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 6, §2.4, last paragraph