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.

Equivalence, quasi-inverse, and adjoint equivalence of categories

Definition

An equivalence of categories from $C$ to $D$ consists of functors $F : C \to D$ and $G : D \to C$ , called quasi-inverses, together with natural isomorphisms

$η : 1_{C} \Rightarrow G F, ε : F G \Rightarrow 1_{D} .$

The categories are then called equivalent. The notions of functor and natural isomorphism are those of Covariant functor, identity functor, composite functor, and contravariant functor and Natural isomorphism.

An adjoint equivalence is such data satisfying the triangle identities

$G ε \circ η G = 1_{G}, ε F \circ F η = 1_{F} .$

Here $G ε$ , $η G$ , $ε F$ , and $F η$ are the whiskerings of Whiskering and horizontal composition of natural transformations. No triangle identity is required of a bare equivalence.

Equivalence, quasi-inverse, and adjoint equivalence of categories

Definition

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