PropositionStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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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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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 $\mathcal G$ is a connected small groupoid and $A$ is any object, then $\mathcal G$ is equivalent to the one-object category $\operatorname{Aut}_{\mathcal G}(A)$ .

Facts & Assumptions

Given: A connected small groupoid $\mathcal G$ 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 $F:\mathcal G\to\operatorname{Aut}(A)$ by sending every object to the sole object and $f:X\to Y$ to $p_Y^{-1}fp_X$ ; identities and composites are preserved by cancellation of $p_Yp_Y^{-1}$ .

step 1.1L2

3.1

For each $X,Y$ , the inverse hom-map sends $u\in\operatorname{Aut}(A)$ to $p_Yup_X^{-1}$ , 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

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 27 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)