DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-29

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.

Member of an object

Definition

Let $A$ be an object of a category. A member of $A$ is simply a morphism with codomain $A$ .

Thus a member is written $x : X \to A,$ and one may read this as " $x$ is a member of $A$ with domain $X$ ". The point of the terminology is that an abelian category need not have honest elements, but it always has arrows into an object.

Depends on

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

Used by

Comember and the dual calculus Definition
Equivalence of members Definition
An ordinary element as the member from the integers Example

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

Saunders Mac Lane, Categories for the Working Mathematician, VIII.4 (standard reference, not scraped)