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

Frobenius–Schur sign in a normalized anyonic vacuum channel Definition
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
The explicit Fibonacci model has a positive spherical pivotal structure Lemma

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)