DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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.

Monoidal category

A monoidal category is a category $C$ equipped with:

a bifunctor $\otimes : C \times C \to C$ (Product category and its projection functors, Covariant functor, identity functor, composite functor, and contravariant functor);
an object $1$ of $C$ ;
natural isomorphisms (Natural isomorphism) $α_{X, Y, Z} : ((X \otimes Y) \otimes Z) ⟶ X \otimes (Y \otimes Z),$ $λ_{X} : (1 \otimes X) ⟶ X, ρ_{X} : (X \otimes 1) ⟶ X,$

such that the following two equations hold for all objects $W, X, Y, Z \in C$ .

The pentagon axiom is

$α_{W, X, Y \otimes Z} \circ α_{W \otimes X, Y, Z} = (1_{W} \otimes α_{X, Y, Z}) \circ α_{W, X \otimes Y, Z} \circ (α_{W, X, Y} \otimes 1_{Z}) .$

The triangle axiom is

$(1_{X} \otimes λ_{Y}) \circ α_{X, 1, Y} = ρ_{X} \otimes 1_{Y} .$

This page imposes no further axiom. In particular, the equality $λ_{1} = ρ_{1}$ is not built into the definition; it is derived later on this page.

Depends on