CorollaryStatement: 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 functor with small target is an equivalence exactly when it is fully faithful and essentially surjective

Statement

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

Facts & Assumptions

Given: A functor $F : C \to D$ with small $D$ , 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 · two levels

7 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)