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.

Exactness is detected by members

Statement

For a composable pair $A \overset{f}{\to} B \overset{g}{\to} C$ in an abelian category, the following are equivalent:

the pair is exact at $B$ ;
$g f = 0$ , and for every member $y : Y \to B$ with $g y \equiv 0$ there exists a member $x : X \to A$ with $f x \equiv y .$

Facts & Assumptions

Given: The composable pair $A \overset{f}{\to} B \overset{g}{\to} C$ .

[L1]

Exactness at $B$ is the equality $[im (f)] = [\ker (g)]$ (Exactness at a node).

[L2]

The subobject inequalities underlying exactness are exactly the two factorization statements for morphisms killed by $g$ (The subobject inequalities underlying exactness).

[L3]

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

[L4]

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

[L5]

Every morphism admits an image factorization (Every morphism factors as an epimorphism followed by a monomorphism, uniquely up to unique isomorphism).

Proof

technique · direct

1.1

Assume the pair is exact at $B$ , and let $y : Y \to B$ satisfy $g y \equiv 0$ . Choose an epic $u : W \to Y$ with $g y u = 0$ . If $A \overset{e}{\to} I \overset{m}{\to} B$ is an image factorization of $f$ , then exactness and [L2] give a map $w : W \to I$ with $m w = y u$ .

L1L2L5assume-hypchoose

1.2

Assume condition 2. For any $u : U \to B$ with $g u = 0$ , we have $g u \equiv 0$ , so condition 2 gives a member $x : X \to A$ with $f x \equiv u$ . By [L3], the members $f x$ and $u$ determine the same subobject of $B$ , and since $f x$ factors through $f$ , that subobject lies below the image of $f$ . Thus every morphism killed by $g$ factors through the image of $f$ . Together with $g f = 0$ , [L2] yields exactness at $B$ .

L2L3L5assume-hyp

2.1

Pull back the epic $e : A \to I$ along $w$ , obtaining $x : P \to A$ and an epic $α : P \to W$ by [L4]. Then $f x = m e x = m w α = y u α$ , so $f x \equiv y$ .

L4L5step 1.1constructalgebra

3.1

Thus conditions 1 and 2 are equivalent.

step 2.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

24 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(v) (standard reference, not scraped)