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 square with monic legs is a pullback exactly when it identifies the source with the intersection subobject

Statement

Consider a commutative square in an abelian category

whose right and bottom legs are monomorphisms. Let $I \to D$ be the intersection of the subobjects represented by $m$ and $n$ . Then the square is a pullback if and only if the induced morphism $X \to I$ is an isomorphism.

Facts & Assumptions

Given: The displayed commutative square with monic right and bottom legs.

[L1]

In an abelian category, pullbacks of cospans exist and are computed by the construction of A pullback is the kernel of the difference of the two legs, and dually for pushouts.

[L2]

An intersection of two subobjects is their greatest lower bound in the subobject order (Intersection of a supplied family of subobjects as its greatest lower bound, Subobjects and quotient objects form oppositely oriented partially ordered collections).

Proof

technique · direct

1.1

By [L1], the pullback $P \to D$ of the two monics $m$ and $n$ exists. Because the square defining $P$ is a common lower bound of $[m]$ and $[n]$ , and every other common lower bound factors uniquely through that pullback, [L2] says that $P \to D$ represents their intersection subobject.

L1L2

2.1

If the displayed square is a pullback, then its source $X \to D$ is another representative of the same greatest lower bound from step 1.1. Therefore the induced morphism $X \to I$ is an isomorphism.

L2step 1.1

3.1

Conversely, if the induced map $X \to I$ is an isomorphism, then composing the pullback square representing $I$ from step 1.1 with that isomorphism yields the displayed square. Pullbackness is invariant under replacing the corner object by an isomorphic one, so the displayed square is a pullback.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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

Junhan Tan, The Freyd-Mitchell Embedding Theorem, Theorem 2.5 (standard reference, not scraped)