TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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In a strict braided monoidal category the braiding satisfies the Yang-Baxter equation

Statement

Let $C$ be a strict braided monoidal category. Then for all objects $X, Y, Z$ ,

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

In particular, when $X = Y = Z$ , both sides are endomorphisms of $X \otimes X \otimes X$ , and the relation becomes

$(c \otimes 1) (1 \otimes c) (c \otimes 1) = (1 \otimes c) (c \otimes 1) (1 \otimes c) .$

Facts & Assumptions

Given: A strict braided monoidal category.

[L1]

A braided monoidal category carries a braiding satisfying the two hexagon identities (Braided monoidal category).

[L2]

In a strict monoidal category, the associator and both unitors are identity morphisms (Strict monoidal category).

Proof

technique · direct

1.1

By [L2], both hexagon identities from [L1] lose all associators. They become $c_{X, Y \otimes Z} = (1_{Y} \otimes c_{X, Z}) \circ (c_{X, Y} \otimes 1_{Z})$ and $c_{X \otimes Y, Z} = (c_{X, Z} \otimes 1_{Y}) \circ (1_{X} \otimes c_{Y, Z})$ .

givenL1L2algebra

2.1

Naturality of $c_{X, -}$ with respect to the morphism $c_{Y, Z} : Y \otimes Z \to Z \otimes Y$ gives $(c_{Y, Z} \otimes 1_{X}) \circ c_{X, Y \otimes Z} = c_{X, Z \otimes Y} \circ (1_{X} \otimes c_{Y, Z}) .$

L1step 1.1algebra

3.1

Expand the left-hand occurrence of $c_{X, Y \otimes Z}$ in step 2.1 by the first formula of step 1.1, and expand $c_{X, Z \otimes Y}$ by the same formula with $Y$ and $Z$ interchanged. This yields $(c_{Y, Z} \otimes 1_{X}) \circ (1_{Y} \otimes c_{X, Z}) \circ (c_{X, Y} \otimes 1_{Z}) = (1_{Z} \otimes c_{X, Y}) \circ (c_{X, Z} \otimes 1_{Y}) \circ (1_{X} \otimes c_{Y, Z}),$ which is the stated Yang-Baxter relation. The special case $X = Y = Z$ is the displayed braid equation.

step 1.1step 2.1algebra∎

Depends on

Used by

The braid category is the free strict braided monoidal category on one generator Theorem

Dependency tree · two levels

4 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, Proposition 8.1.10 (standard reference, not scraped)