DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-11

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.

Category with zero morphisms

Definition

A category with zero morphisms is a category $C$ (Category, object, morphism, domain, codomain, identity, composition, and hom-collection) equipped with a specified morphism $0_{A, B} : A \to B$ for every ordered pair of objects such that, for all $f : A^{'} \to A$ and $g : B \to B^{'}$ ,

$0_{A, B} \circ f = 0_{A^{'}, B}, g \circ 0_{A, B} = 0_{A, B^{'}} .$

The family is part of the structure. Merely naming an isolated morphism as zero does not supply zero morphisms.

Depends on

Category, object, morphism, domain, codomain, identity, composition, and hom-collection

Used by

Dependency tree · two levels

3 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

Haru Park, Category Theory and Homological Algebra, section 2.2 (standard reference, not scraped)