DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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.

Symmetric monoidal category

Definition

A symmetric monoidal category is a braided monoidal category (Braided monoidal category) whose braiding $c$ satisfies

$c_{Y, X} \circ c_{X, Y} = 1_{X \otimes Y}$

for all objects $X, Y$ .

Depends on

Braided monoidal category

Used by

Monoid objects in a symmetric monoidal category form a symmetric monoidal category Corollary
Supervector spaces with the sign braiding Example
In the presence of symmetry, one hexagon implies the other Theorem
Symmetric coherence Theorem
The cartesian swap braiding is a symmetry Theorem
The double-braiding center is a symmetric monoidal subcategory 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, Definition 8.1.12 (standard reference, not scraped)
Saunders Mac Lane, Natural Associativity and Commutativity, Section 4 (standard reference, not scraped)