CorollaryStatement: 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.
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Under the Axiom of Choice, a functor with small target is an equivalence exactly when it is fully faithful and essentially surjective

Statement

Assume the Axiom of Choice and let $F:\mathcal C\to\mathcal D$ have small target. Then $F$ is an equivalence exactly when it is fully faithful and essentially surjective.

Facts & Assumptions

Given: A functor $F:\mathcal C\to\mathcal D$ with small $\mathcal D$ , under the Axiom of Choice.

[L1]

An equivalence is fully faithful and split essentially surjective, and the converse holds (A functor is an equivalence exactly when it is fully faithful and split essentially surjective, without Choice).

[L2]

Essential surjectivity over a small target can be split under Choice (Under the Axiom of Choice, essential surjectivity onto a small category admits a splitting).

Proof

technique · direct

1.1

If $F$ is an equivalence, [L1] makes it fully faithful and split essentially surjective, hence essentially surjective.

givenL1

2.1

If $F$ is fully faithful and essentially surjective, [L2] supplies a splitting.

step 1.1L2

3.1

The converse direction of [L1] now makes $F$ an equivalence, proving the biconditional.

step 2.1L1∎

Depends on

Used by

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

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 6 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)