CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck 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, every epimorphism in $S e t$ is a split epimorphism

Statement

Assume the Axiom of Choice. Every epimorphism in $S e t$ is a split epimorphism.

Facts & Assumptions

Given: An epimorphism $p : X \to Y$ in $S e t$ .

[L1]

Epimorphisms in $S e t$ are precisely surjections (In $S e t$ , monomorphisms are exactly injections and epimorphisms are exactly surjections).

[L2]

The Axiom of Choice selects an element from every member of a set-indexed family of nonempty sets (The Axiom of Choice), and a split epimorphism has a right inverse (Split monomorphism, split epimorphism, retraction, and section).

Proof

technique · direct

1.1

By [L1], every fibre $p^{- 1} ({y})$ for $y \in Y$ is nonempty.

givenL1

2.1

By [L2], choose $s (y) \in p^{- 1} ({y})$ for every $y \in Y$ ; if $Y$ is empty this is the unique empty function, so the same construction covers that boundary case.

step 1.1L2choose

3.1

Then $p (s (y)) = y$ for every $y$ , so $p \circ s = 1_{Y}$ and $p$ is split epic.

step 2.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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)

Under the Axiom of Choice, every epimorphism in SetSet is a split epimorphism