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In the presence of symmetry, one hexagon implies the other

Statement

Let $C$ be a monoidal category equipped with a natural isomorphism $c_{X, Y} : X \otimes Y \to Y \otimes X$ that satisfies one of the two braiding hexagons. If moreover

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

for all $X, Y$ , then the other hexagon also holds. In particular, in a symmetric monoidal category either hexagon may be taken as the coherence axiom.

Facts & Assumptions

Given: A natural family $c_{X, Y}$ satisfying the symmetry equation and one braiding hexagon.

[L1]

In a symmetric monoidal category, the braiding satisfies $c_{Y, X} \circ c_{X, Y} = 1_{X \otimes Y}$ (Symmetric monoidal category).

Proof

technique · direct

1.1

Assume first that the given hexagon is $α_{Y, Z, X} \circ c_{X, Y \otimes Z} \circ α_{X, Y, Z} = (1_{Y} \otimes c_{X, Z}) \circ α_{Y, X, Z} \circ (c_{X, Y} \otimes 1_{Z}) .$ Inverting this equality gives $α_{X, Y, Z}^{- 1} \circ c_{X, Y \otimes Z}^{- 1} \circ α_{Y, Z, X}^{- 1} = (c_{X, Y}^{- 1} \otimes 1_{Z}) \circ α_{Y, X, Z}^{- 1} \circ (1_{Y} \otimes c_{X, Z}^{- 1}) .$

givenalgebra

2.1

Replace $(X, Y, Z)$ in step 1.1 by $(Z, X, Y)$ . By [L1], the symmetry equation implies $c_{U, V}^{- 1} = c_{V, U}$ for all objects $U, V$ . After that substitution, step 1.1 becomes $α_{Z, X, Y}^{- 1} \circ c_{X \otimes Y, Z} \circ α_{X, Y, Z}^{- 1} = (c_{X, Z} \otimes 1_{Y}) \circ α_{X, Z, Y}^{- 1} \circ (1_{X} \otimes c_{Y, Z}),$ which is exactly the second braiding hexagon.

L1step 1.1algebra

3.1

If instead the second hexagon is given, the same argument with the roles of the two hexagons reversed proves the first. Hence under the symmetry axiom either hexagon implies the other.

step 2.1algebra∎

Depends on

Used by

Symmetric coherence 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

Saunders Mac Lane, Natural Associativity and Commutativity, the hexagon (4.5) (standard reference, not scraped)
Michael Muger, Tensor Categories: A Selective Guided Tour, Version I of the coherence theorem (standard reference, not scraped)