TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28‡ 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.

A pullback is the kernel of the difference of the two legs, and dually for pushouts

Statement

Let $f : A \to C$ and $g : B \to C$ be morphisms in an abelian category. Then a pullback of the cospan $A \overset{f}{\to} C \overset{g}{\leftarrow} B$ is a kernel of the difference map

$f \circ p_{A} - g \circ p_{B} : A \oplus B \to C,$

where $p_{A}$ and $p_{B}$ are the biproduct projections. Dually, a pushout of $B \overset{s}{\leftarrow} A \overset{t}{\to} C$ is a cokernel of $i_{B} \circ s - i_{C} \circ t : A \to B \oplus C$ .

Facts & Assumptions

Given: An abelian category and a cospan $A \overset{f}{\to} C \overset{g}{\leftarrow} B$ .

[L1]

Abelian categories have finite limits and finite colimits (An abelian category has all finite limits and all finite colimits).

[L2]

On a biproduct the injections and projections satisfy the standard identity-sum relations (On a biproduct, the injections and projections satisfy the identity-sum relation).

[L3]

In a preadditive category, equalizers are kernels of differences (In a preadditive category, the equalizer of a parallel pair is the kernel of their difference).

[L4]

An abelian category is additive and therefore preadditive (Abelian category).

Proof

technique · direct

1.1

By [L1] there is a product of $A$ and $B$ , and by [L4] that product is the biproduct $A \oplus B$ . In the preadditive structure of [L4], a morphism $x : X \to A \oplus B$ satisfies $(f p_{A} - g p_{B}) x = 0$ exactly when $f p_{A} x = g p_{B} x$ . So by [L3], a kernel of $f p_{A} - g p_{B}$ is an equalizer of the parallel pair $f p_{A}, g p_{B}$ .

L1L3L4

2.1

Giving $x : X \to A \oplus B$ is the same as giving its two composites to $A$ and $B$ , and the equality in step 1.1 is exactly the pullback compatibility condition. Therefore the equalizer in step 1.1 is a pullback of $f$ and $g$ .

L2step 1.1

3.1

Reversing all arrows gives the pushout statement: in an abelian category a pushout is the cokernel of the corresponding difference map.

L1L3L4step 2.1∎

Depends on

Used by

Dependency tree · two levels

12 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

The Stacks Project, Section 12.5, Lemma 12.5.11 (standard reference, not scraped)