Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-generated
How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Independent groupoid covering and universal-cover input

Statement

A covering p:(E,e)→(G,x) of nonempty connected small groupoids is a functor bijective on arrows with every fixed source. It is determined up to unique pointed isomorphism over G by its characteristic subgroup H=p(Ee)≤Gx; unpointed isomorphism classes correspond to conjugacy classes of subgroups of Gx. A normal covering means that its deck transformations act transitively on one, hence every, fibre. This is equivalent to H⊲Gx, and then Gx/H≅Deck(p). Here arrow products are left to right and deck transformations use ordinary function composition; the isomorphism sends aH to the deck transformation sending e to the endpoint of the lift of a. Every such G has a universal connected covering with trivial vertex groups, obtained by taking H={1x}.

Facts & Assumptions

[F1]

The groupoid conventions are Set-sized groupoids, functors and composition conversion and the quotient is The quotient group G/N and coset product (gN)(hN)=ghN. The five local prerequisites are Unique lifts, endpoint fibres and characteristic subgroups (unique lifts and characteristic subgroups), Explicit connected cover associated with a vertex subgroup (subgroup construction), Pointed classification of connected groupoid coverings (pointed classification), Normal coverings and the convention-correct deck quotient (normality and the deck quotient), and Universal groupoid covering from the trivial subgroup (universal covering).

Proof

Given: The data and hypotheses in the Statement.

Goal: Establish all assertions of the Statement.

1.1F1

The lift-fibre lemma gives the definition and injectivity on vertex groups. The subgroup construction realizes every subgroup, and pointed classification proves both its uniqueness with a chosen point and the conjugacy formulation without one. Thus the classification assertions hold with exactly the stated basepoint data.

2.1F1step 1.1∎

The normal-deck-quotient lemma proves the equivalence of normality with transitivity and the asserted isomorphism, using precisely the multiplication convention in the Statement. The universal-cover lemma gives existence from the trivial subgroup, trivial vertex groups and universality. These prove all remaining assertions.

Source notes

This is a local synthesis of five authored intermediate prerequisites, following Brown’s lifting/subgroup construction. The previous citation omitted the deck-group location: it is 10.6.3, printed p. 393, and regularity is 10.6.4, pp. 393–394. Its anti-isomorphism has been converted explicitly, rather than silently identified with an isomorphism. No outstanding imported theorem or source-proof reconstruction is claimed.

Depends on

Used by · 6

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