CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-09-01

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Monoid objects in a symmetric monoidal category form a symmetric monoidal category

Statement

If $C$ is a symmetric monoidal category, then $Mon (C)$ is symmetric monoidal under the tensor product of Monoid objects in a braided monoidal category form a monoidal category.

Facts & Assumptions

Given: A symmetric monoidal category $C$ .

[L1]

A symmetric monoidal category is, in particular, braided (Symmetric monoidal category).

[L2]

In a braided monoidal category, monoid objects form a monoidal category under the braided tensor product (Monoid objects in a braided monoidal category form a monoidal category).

Proof

technique · direct

1.1

By [L1] and [L2], $Mon (C)$ is already a monoidal category.

L1L2given

2.1

The ambient symmetry $c_{A, B} : A \otimes B \to B \otimes A$ is a monoid morphism between the tensor-product monoids because the symmetry is involutive and natural, so it commutes with the braided interchange formula defining multiplication. Its square is the identity because it already is in $C$ . Therefore these maps provide a symmetric braiding on $Mon (C)$ .

L1L2step 1.1algebra∎

Depends on

Used by

Commutative monoid objects in sets are ordinary commutative monoids Example

Dependency tree · two levels

5 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 8.8.2(iv) (standard reference, not scraped)