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.

Explicit connected cover associated with a vertex subgroup

Statement

Let $G$ be a connected small groupoid, $x$ an object, and $H \leq G_{x}$ a subgroup. Form objects $H a = {h a : h \in H}$ for all arrows $a : x \to y$ , recording $y$ . There is a connected covering $C_{H} \to G$ whose arrows are pairs $(H a, g)$ with $g : y \to z$ , from $H a$ to $H a g$ , projecting to $g$ . Its chosen object is $H 1_{x} = H$ and its characteristic subgroup there is $H$ .

Statement references

Set-sized groupoids, functors and composition conversion

Unique lifts, endpoint fibres and characteristic subgroups

Facts & Assumptions

[F1]

Use Set-sized groupoids, functors and composition conversion for small groupoids and left-to-right products, and Unique lifts, endpoint fibres and characteristic subgroups for coverings and characteristic subgroups; a subgroup contains the identity and is closed under multiplication and inverses.

Proof

Given: The data and hypotheses in the Statement.

Goal: Establish all assertions of the Statement.

1.1F1

Two sets $H a, H b$ with the same recorded target coincide precisely when $a = h b$ for some $h \in H$ , equivalently $a b^{- 1} \in H$ . Indeed a nonempty intersection yields $h_{1} a = h_{2} b$ and hence $a = h_{1}^{- 1} h_{2} b$ ; the converse follows by closure of $H$ . Consequently replacing $a$ by $h a$ leaves $H a g$ unchanged. These are sets because $G$ is small.

2.1F1step 1.1∎

Define $(H a, g) (H a g, k) = (H a, g k)$ . Associativity follows from that of $G$ ; the identity at $H a$ is $(H a, 1_{y})$ and the inverse of $(H a, g)$ is $(H a g, g^{- 1})$ . Thus this is a groupoid and the projection is a functor, bijective on each source star since its arrows with prescribed source are exactly the pairs with all possible labels $g$ . Between $H a$ and $H b$ the label $a^{- 1} b$ gives an arrow, so it is connected. The loops at $H$ have precisely labels $g \in G_{x}$ with $H g = H$ , exactly $H$ , which proves the characteristic-subgroup claim.

Depends on

Used by · 4

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