DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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.

The dual of a morphism

Definition

Assume chosen left duals $X^{\lor}$ and $Y^{\lor}$ of objects $X$ and $Y$ . For a morphism $f : X \to Y$ , its left dual morphism

$f^{\lor} : Y^{\lor} \to X^{\lor}$

is the composite

$Y^{\lor} \overset{ρ^{- 1}}{\to} Y^{\lor} \otimes 1 \overset{1 \otimes {coev}_{X}}{\to} Y^{\lor} \otimes (X \otimes X^{\lor}) \overset{α^{- 1}}{\to} (Y^{\lor} \otimes X) \otimes X^{\lor} \overset{(1 \otimes f) \otimes 1}{\to} (Y^{\lor} \otimes Y) \otimes X^{\lor} \overset{{ev}_{Y} \otimes 1}{\to} 1 \otimes X^{\lor} \overset{λ}{\to} X^{\lor} .$

Equivalently, $f^{\lor}$ is the unique morphism making the transpose squares with evaluation and coevaluation commute. The right dual of a morphism is defined by the mirrored formula once right duals are chosen.

Depends on

Left dual and right dual object

Used by

Twist and ribbon structure Definition
Left duality is a contravariant antimonoidal functor Theorem

Dependency tree · two levels

2 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

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Section 2.10 (standard reference, not scraped)
nLab, rigid monoidal category (standard reference, not scraped)