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.

Equivalence of members

Definition

Let $x : X \to A$ and $y : Y \to A$ be members of the same object $A$ .

They are equivalent, written $x \equiv y,$ when there exist an object $W$ and epimorphisms $u : W ↠ X, v : W ↠ Y$ such that $x u = y v .$

So equivalence of members is equality after comparison on one common epimorphic cover of the two domains.

Depends on

Used by

Comember and the dual calculus Definition
A morphism carries members to members and preserves equivalence Proposition
Each object has a zero member and each member has a negative Proposition
Member equivalence is reflexive and symmetric Proposition
The covering criterion and the member calculus are the same tool Remark
Member equivalence is transitive Theorem
Members modulo equivalence correspond to subobjects Theorem
The covering criterion for exactness Theorem
The subtraction surrogate Theorem

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)