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Existence of universal connected groupoid covers

PDF p. 2: §1.2.1 before Theorem 1.3

Statement

Every nonempty connected small groupoid $G$ has a universal connected groupoid covering $F : \tilde{G} \to G$ . For every object $x$ of $G$ , the deck transformation group $Deck (F)$ is isomorphic to the vertex group $G_{x}$ .

Here a groupoid is a category in which every morphism is invertible; connected means that any two objects are joined by a morphism. The vertex group at $x$ consists of the morphisms from $x$ to itself. A covering has a unique lift of each base morphism from each prescribed object over its source. It is universal when all upstairs vertex groups are trivial. A deck transformation is an automorphism of the upstairs groupoid whose composite with $F$ equals $F$ .

Statement references

Connected small groupoids and vertex groups

Groupoid coverings, universal and normal covers

Independent groupoid covering and universal-cover input

Vertex-group exact sequence of a normal groupoid cover

Facts & Assumptions

Given: A nonempty connected small groupoid $G$ and an object $x$ of $G$ .

[F1]

A connected covering of $G$ with trivial characteristic subgroup at $x$ exists and is universal; its upstairs vertex groups are trivial. Such a covering is normal because the trivial subgroup is normal. Independent groupoid covering and universal-cover input

[F2]

For a normal covering of nonempty connected small groupoids, the vertex-group exact sequence identifies the deck group with the base vertex group when the covering is universal. Vertex-group exact sequence of a normal groupoid cover

[F3]

The covering, universal-cover and deck-transformation conventions are those stated above. Groupoid coverings, universal and normal covers The groupoid and vertex-group conventions are as stated above. Connected small groupoids and vertex groups

Proof

technique · direct

Goal: Construct a universal connected covering and identify its deck group with $G_{x}$ .

1.1F1F3given

By [F1], the trivial subgroup of $G_{x}$ gives a connected covering $F : \tilde{G} \to G$ whose upstairs vertex groups are trivial. By the convention in [F3], this covering is universal. The same fact gives normality.

2.1F2step 1.1given∎

Apply [F2] to this normal universal covering: its deck transformation group is isomorphic to $G_{x}$ . The construction works for any chosen object $x$ , proving the statement.

Source notes

The article citation attributes the assertion. The proof uses the independent groupoid-covering input and the stated exact-sequence consequence.

Depends on

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 2, §1.2.1 before Theorem 1.3; p. 6, §2.4