DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 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 $\mathcal C$ (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 $g:B\to B'$ ,

$0_{A,B}\circ f=0_{A',B},\qquad g\circ0_{A,B}=0_{A,B'}.$

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

A zero object supplies a unique compatible system of zero morphisms Proposition

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 11 results over 5 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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