DefinitionDefinition: AI-adaptedProof: 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.

Comember and the dual calculus

Definition

The dual member calculus in an abelian category is the member calculus of the opposite category, transported back along Exactness is self-dual.

Concretely, a comember of $A$ is a morphism with domain $A$ , $x : A \to X,$ and two comembers $x : A \to X$ and $y : A \to Y$ are equivalent when there exist one object $W$ and monomorphisms $u : X ↣ W, v : Y ↣ W$ with $u x = v y .$

So comembers compare after one common monomorphic enlargement, just as members compare after one common epimorphic cover.

Depends on

Used by

Two routes to every dual statement Remark

Dependency tree · two levels

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