LemmaStatement: 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, essential surjectivity onto a small category admits a splitting

Statement

Assume the Axiom of Choice. If $F:\mathcal C\to\mathcal D$ is essentially surjective and $\mathcal D$ is small, then its essential surjectivity admits a splitting.

Facts & Assumptions

Given: An essentially surjective functor $F:\mathcal C\to\mathcal D$ with small target $\mathcal D$ .

[L1]

Essential surjectivity asserts an object and isomorphism witness over every target object, while split essential surjectivity records one such witness for each object (Faithful, full, fully faithful, essentially surjective, and split essentially surjective functors).

[L2]

A small category has a set of objects (Small, locally small, and large categories), and the Axiom of Choice selects from a set-indexed family of nonempty sets (The Axiom of Choice).

Proof

technique · direct

1.1

Since $\operatorname{Ob}\mathcal D$ is a set and each $D$ has some pair $(C,\varepsilon:FC\to D)$ , Collection bounds a witness for every $D$ inside one set of candidate pairs.

givenL1L2

2.1

For each $D$ , the candidates in that bounding set form a nonempty set, so [L2] chooses one pair $(C_D,\varepsilon_D)$ .

step 1.1L2choose

3.1

The selected family is exactly a splitting of essential surjectivity as defined in [L1].

step 2.1L1∎

Depends on

Used by

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

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 18 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)