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.

Braiding

Definition

Let $(C, \otimes, 1, α, λ, ρ)$ be a monoidal category (Monoidal category).

A braiding on $C$ is a natural isomorphism

$c_{X, Y} : X \otimes Y ⟶ Y \otimes X$

natural in both variables (Natural isomorphism) such that, for all objects $X, Y, Z$ ,

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

and

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

These are the two hexagon identities. The invertibility of each $c_{X, Y}$ is part of the data, not a consequence of the hexagons.

Depends on

Used by

Braided monoidal category Definition
The braiding is compatible with the unit constraints Theorem
The cartesian swap braiding is a symmetry Theorem
The inverse braiding is again a braiding Theorem

Dependency tree · two levels

8 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 8.1.1 (standard reference, not scraped)
Michael Muger, Tensor Categories: A Selective Guided Tour, Section 4 (standard reference, not scraped)