PropositionStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-11

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.

Under the Axiom of Choice, a connected small groupoid is equivalent to the automorphism group of any one of its objects

Statement

Assume the Axiom of Choice. If $G$ is a connected small groupoid and $A$ is any object, then $G$ is equivalent to the one-object category ${Aut}_{G} (A)$ .

Facts & Assumptions

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

[L1]

In a connected groupoid, every object is isomorphic to $A$ (Isomorphism, groupoid, and connected category), and smallness makes the object collection a set (Small, locally small, and large categories).

[L2]

Choice selects from set-indexed nonempty families (The Axiom of Choice); a group is a one-object category (A monoid is a one-object category, and a group is a one-object category in which every morphism is invertible), and the small-target equivalence criterion is Under the Axiom of Choice, a functor with small target is an equivalence exactly when it is fully faithful and essentially surjective.

Proof

technique · direct

1.1

By [L1] and [L2], choose for every object $X$ an isomorphism $p_{X} : A \to X$ , taking $p_{A} = 1_{A}$ .

givenL1L2choose

2.1

Define $F : G \to Aut (A)$ by sending every object to the sole object and $f : X \to Y$ to $p_{Y}^{- 1} f p_{X}$ ; identities and composites are preserved by cancellation of $p_{Y} p_{Y}^{- 1}$ .

step 1.1L2

3.1

For each $X, Y$ , the inverse hom-map sends $u \in Aut (A)$ to $p_{Y} u p_{X}^{- 1}$ , so $F$ is fully faithful; it is split essentially surjective because its target has one object, and [L2] makes it an equivalence.

step 2.1L2∎

Depends on

Used by

The orbit-set and fixed-point constructions as Kan extensions Example

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)
Ahrens, Kapulkin and Shulman, Univalent categories and the Rezk completion, section 6 (standard reference, not scraped)