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 dimension of an object relative to a pivotal structure

Definition

Let $a$ be a pivotal structure on a rigid monoidal category. The dimension of an object $X$ relative to $a$ is

$\dim_{a} (X) : = {Tr}_{L} (a_{X}) \in {End}_{C} (1) .$

The subscript is part of the notation: changing the pivotal structure changes the morphism whose trace is being taken, so the resulting dimension can change as well.

Depends on

Used by

Spherical structure Definition
The categorical dimension of a vector space is the scalar image of its linear dimension Example
FALSE: the dimension of an object is independent of the pivotal structure False statement

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

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Definition 4.7.11 (standard reference, not scraped)