TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Epimorphy is detected by members

Statement

For a morphism $g : B \to C$ in an abelian category, the following are equivalent:

$g$ is epic.
For every member $z : Z \to C$ , there exists a member $y : Y \to B$ with $g y \equiv z .$

Facts & Assumptions

Given: A morphism $g : B \to C$ .

[L1]

Members modulo equivalence correspond exactly to subobjects (Members modulo equivalence correspond to subobjects).

[L3]

The image is the least subobject through which a morphism factors (The image is the least subobject through which a morphism factors).

[L4]

Pullbacks exist, and pullbacks of epimorphisms are epimorphisms (Pullbacks and pushouts as limits and colimits of cospans and spans, The pullback of an epimorphism is an epimorphism).

Proof

technique · direct

1.1

Assume $g$ is epic. For a member $z : Z \to C$ , form the pullback of $z$ along $g$ with projections $y : P \to B$ and $α : P \to Z$ . By [L4], $α$ is epic, and the pullback equation $g y = z α$ shows $g y \equiv z$ .

L4assume-hypconstruct

1.2

Assume condition 2, and write $g$ as an epi-mono factorization $B \overset{e}{\to} I \overset{m}{\to} C$ . Apply condition 2 to the identity member $1_{C} : C \to C$ . Then there is $y : Y \to B$ with $g y \equiv 1_{C}$ , so [L1] says that $g y$ and $1_{C}$ determine the same whole subobject of $C$ . Since $g y$ factors through $g$ , [L3] gives $[im (g y)] \leq [im (g)] = [⁣ m ⁣]$ , hence $[1_{C}] \leq [⁣ m ⁣]$ , which forces $[⁣ m ⁣] = [1_{C}]$ . Thus $g$ is epic.

L1L3assume-hypalgebra

2.1

Therefore conditions 1 and 2 are equivalent.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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

Saunders Mac Lane, Categories for the Working Mathematician, Theorem VIII.4.3(iii) (standard reference, not scraped)