LemmaStatement: 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.

Unique lifts, endpoint fibres and characteristic subgroups

Statement

Let $p : E \to G$ be a functor of small groupoids, with $E, G$ connected and nonempty, bijective from the arrows with source $e$ to those with source $p (e)$ for every $e$ . Such a functor is called a covering. Put $x = p (e)$ and $H = p (E_{e}) \leq G_{x}$ (the characteristic subgroup at $e$ ). Products of arrows are read left to right. Lifts preserve products and inverses, $E_{e} \to G_{x}$ is injective, and two arrows $a, b : x \to y$ lifted from $e$ have equal endpoints exactly when $a b^{- 1} \in H$ .

Statement references

Set-sized groupoids, functors and composition conversion

Facts & Assumptions

[F1]

Use the small-groupoid, functor and vertex-group conventions of Set-sized groupoids, functors and composition conversion. A subgroup means a subset containing the identity and closed under multiplication and inverses. The assumed source-wise bijections are the definition of covering.

Proof

Given: The data and hypotheses in the Statement.

Goal: Establish all assertions of the Statement.

1.1F1

For composable arrows $a, b$ , lift $a$ from $u$ , then $b$ from its endpoint. Their composite projects to $a b$ and has source $u$ , hence is its unique lift. Identity arrows lift to identities, and the inverse of a lift projects to the inverse arrow and is its unique lift from its source. This uses exactly preservation of composition and the source-wise bijection.

2.1F1step 1.1∎

Two loops at $e$ with equal images are equal by injectivity on arrows with source $e$ . Their image $H$ contains the identity and is closed under products and inverses, so is a subgroup. If the lifts of $a, b$ have equal endpoints, their first composite with the inverse of the second is a loop over $a b^{- 1}$ , giving membership in $H$ . Conversely lift $a b^{- 1} \in H$ as a loop at $e$ , and then compose with the lift of $b$ from $e$ . This projects to $a$ and ends where that lift of $b$ ends; uniqueness gives the assertion.

Depends on

Set-sized groupoids, functors and composition conversion

Used by · 5

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

Ronald Brown, Topology and Groupoids, third edition, corrected electronic text · 10.3.1–10.3.3 and Corollary 1, printed pp. 371–374; 10.4.2–10.4.3 and Corollary 1, pp. 376–377; 10.6.3–10.6.4, pp. 393–394