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

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The braiding is compatible with the unit constraints

Statement

Let $c$ be a braiding on a monoidal category $C$ . Then for every object $X$ ,

$ρ_{X} \circ c_{1, X} = λ_{X}, λ_{X} \circ c_{X, 1} = ρ_{X},$

and consequently

$c_{1, X} = c_{X, 1}^{- 1} .$

Facts & Assumptions

Given: A braided monoidal category with braiding $c$ .

[F1]

In any braided monoidal category, Exercise 8.1.6 of EGNO derives the identities $ρ_{X} \circ c_{1, X} = λ_{X}, λ_{X} \circ c_{X, 1} = ρ_{X} .$

[L1]

A braiding is, in particular, a family of isomorphisms satisfying the hexagons (Braiding).

Proof

technique · direct

1.1

The present hypotheses are exactly those of [F1], so the two displayed unit-compatibility identities hold for every object $X$ .

givenF1

2.1

By [L1], the morphisms $c_{1, X}$ and $c_{X, 1}$ are isomorphisms. Step 1.1 therefore gives $c_{1, X} = ρ_{X}^{- 1} λ_{X}, c_{X, 1} = λ_{X}^{- 1} ρ_{X} .$ These two formulas are inverse to each other, so $c_{1, X} = c_{X, 1}^{- 1}$ .

L1step 1.1algebra∎

Depends on

Used by

The double-braiding center is a symmetric monoidal subcategory Theorem

Dependency tree · two levels

6 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, Exercise 8.1.6 (standard reference, not scraped)