PropositionStatement: AI-adaptedProof: AI-generated

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Vertex-group exact sequence of a normal groupoid cover

PDF p. 6: §2.4, (2.10) and following sentence; PDF p. 2: §1.2.1 before Theorem 1.3: universal-cover deck group

Statement

Let $F : \tilde{G} \to G$ be a normal covering of nonempty connected small groupoids, and let $F (\tilde{x}) = x$ . A vertex group consists of all arrows from the specified object to itself. Use left-to-right arrow products and ordinary composition of deck functors. Then there is an exact sequence of groups

$1 ⟶ {\tilde{G}}_{\tilde{x}} \overset{F}{\to} G_{x} \overset{ψ}{\to} Deck (F) ⟶ 1.$

Here $1$ denotes the trivial group. For group homomorphisms $A \overset{f}{\to} B \overset{g}{\to} C$ , exactness at $B$ means $im f = \ker g$ ; the displayed sequence requires this equality at each of its three intervening groups. The maps from and to $1$ are the unique homomorphisms.

For $α \in G_{x}$ , the deck transformation $ψ (α)$ sends $\tilde{x}$ to the target of the unique lift of $α$ with source $\tilde{x}$ . In particular, if $F$ is universal, then $Deck (F) ≅ G_{x}$ .

Statement references

Connected small groupoids and vertex groups

Groupoid coverings, universal and normal covers

Independent groupoid covering and universal-cover input

The quotient group G/N and coset product (gN)(hN)=ghN

The kernel and image of a group homomorphism

Facts & Assumptions

Given: The covering, base objects, and conventions in the Statement.

[F1]

A covering has a unique lift of each base arrow from each prescribed object over its source. A universal covering has trivial vertex groups upstairs. Groupoid coverings, universal and normal covers

[F2]

For a connected normal covering, the induced map on the chosen vertex group is injective, its image $H = F ({\tilde{G}}_{\tilde{x}})$ is normal in $G_{x}$ , and the quotient $G_{x} / H$ is isomorphic to the deck group by the map sending $α H$ to the deck transformation that takes $\tilde{x}$ to the target of the lift of $α$ . These statements use the multiplication conventions in the Statement. Independent groupoid covering and universal-cover input

[F3]

For a normal subgroup $H$ of a group $G$ , the quotient map $q : G \to G / H$ , $q (g) = g H$ , is surjective and has kernel $H$ . Indeed, the coset product gives $q (g g^{'}) = q (g) q (g^{'})$ , every coset has a representative, the quotient identity is $H$ , and $q (g) = H$ exactly when $g \in H$ . The quotient group G/N and coset product (gN)(hN)=ghN

[F4]

For a group homomorphism $f : A \to B$ , its kernel is ${a \in A : f (a) = 1_{B}}$ and its image is ${f (a) : a \in A}$ . Use exactness as defined in the Statement, including the unique homomorphisms from and to the trivial group. The kernel and image of a group homomorphism

Proof

Given: The data in the Statement.

Goal: Prove the exact sequence, the description of $ψ$ , and the universal-cover consequence.

1.1F1F2F3F4

By [F2], $F$ embeds the upstairs vertex group in $G_{x}$ with normal image $H$ . Compose the quotient map $G_{x} \to G_{x} / H$ with the isomorphism in [F2] and call the composite $ψ$ . By [F3] it is surjective with kernel $H$ , while the first map is injective. Thus the displayed sequence is exact. The stated endpoint description of $ψ$ is the defining description of the isomorphism in [F2], with the unique lift supplied by [F1].

2.1F1F2F3F4step 1.1∎

If $F$ is universal, [F1] makes ${\tilde{G}}_{\tilde{x}}$ trivial, so $H$ is trivial. The normality criterion in the groupoid-covering input applies because the trivial subgroup is normal; hence the preceding construction applies to a universal connected covering. The quotient by the trivial subgroup is canonically isomorphic to $G_{x}$ , and [F2] identifies it with $Deck (F)$ . This proves the final assertion.

Source notes

The article citation records attribution of the original assertion. The proof above uses the independently authored groupoid-covering input and the quotient-group and kernel/image definitions; exactness is defined inline.

Depends on

Used by · 2

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. 6, §2.4, (2.10) and following sentence (normal-cover exact sequence and endpoint map); PDF p. 2, §1.2.1 before Theorem 1.3 (universal-cover deck-group assertion)