TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

Mutual factorisation is an equivalence relation on monomorphisms into an object and dually on epimorphisms out of it

Statement

For monomorphisms into a fixed object, mutual factorisation as defined in Subobject and quotient object as mutual-factorisation classes of monomorphisms and epimorphisms is an equivalence relation. If $m : A \to C$ and $n : B \to C$ mutually factor, the factor maps are unique inverse isomorphisms. Dually, mutual factorisation is an equivalence relation on epimorphisms out of a fixed object, and its factor maps are unique inverse isomorphisms.

Facts & Assumptions

Given: Monomorphisms $m : A \to C$ , $n : B \to C$ , and $p : D \to C$ , with the mutual-factorisation relation of Subobject and quotient object as mutual-factorisation classes of monomorphisms and epimorphisms.

[L1]

A morphism $f$ is monic when $f \circ g = f \circ h$ implies $g = h$ for every parallel pair $g, h$ , and epic when $g \circ f = h \circ f$ implies $g = h$ ; thus monomorphisms are left-cancellable and epimorphisms right-cancellable (Monomorphism and epimorphism by left and right cancellation).

[L2]

A morphism with a two-sided inverse is an isomorphism, and that inverse is unique (Isomorphism, groupoid, and connected category).

Proof

technique · direct

1.1givenL1

Identity factorisations give reflexivity, exchanging the two factor maps gives symmetry, and composing factor maps gives transitivity. The same three operations work dually for epimorphisms.

2.1step 1.1L1L2∎

Suppose $m = n \circ u$ and $n = m \circ v$ . Then $m = m \circ v \circ u$ , so monicity of $m$ gives $v \circ u = 1_{A}$ ; similarly monicity of $n$ gives $u \circ v = 1_{B}$ . Thus $u$ and $v$ are inverse isomorphisms by [L2], and monicity makes each factor map unique. For epimorphisms the same equations are cancelled on the right, proving the dual claim.

Depends on

Used by

Cited to discharge well-definedness by Subobject and quotient object as mutual-factorisation classes of monomorphisms and epimorphisms.

Dependency tree · two levels

5 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

E. Riehl, Category Theory in Context, section 4.5 (standard reference, not scraped)