# nLab bimonoidal category

### Context

#### Monoidal categories

monoidal categories

## With traces

• trace

• traced monoidal category?

# Contents

## Idea

A bimonoidal category is a category equipped with two monoidal category structure $\otimes$ (in the role of the tensor product) and $\oplus$ (in the role of the direct sum) such that $\otimes$ distributes over $\oplus$ up to coherent natural isomorphism.

This may be thought of as a natural categorification of the notion of rig in the way that the notion of monoidal category is the categorification of the notion of monoid.

## Definition

A bimonoidal category is a category $R$ with two structures $\left(R,\otimes ,1\right)$ and $\left(R.\oplus ,0\right)$ of a monoidal category, which satisfy a distributivity law up to natural isomorphism among each, paralleling that for addition and multiplication in a rig.

## Properties

### Strictification

The notion of a bipermutative category is a strictification of the notion of symmetric bimonidal categories.

Every symmetric bimonoidal category is equivalent to a bipermutative category (May, prop. VI 3.5).

Every bimonoidal category is equivalent to a strict bimonoidal category (Guillou, theorem 1.2).

## References

The coherence for the distributivity law in bimonoidal categories has been given in

• M. Laplaza, Coherence for distributivity, Lecture Notes in Mathematics 281, Springer Verlag, Berlin, 1972, pp. 29-72.

• G. Kelly, Coherence theorems for lax algebras and distributive laws, Lecture Notes in Mathematics 420, Springer Verlag, Berlin, 1974, pp. 281-375.

where these categories are called ring categories. Discussion with an eye towers the K-theory of a bipermutative category is in

• Peter May, ${E}_{\infty }$ Ring Spaces and ${E}_{\infty }$ Ring spectra, Springer lectures notes in mathematics, Vol. 533, (1977) (pdf) chaper VI
• Bertrand Guillou, Strictification of categories weakly enriched in symmetric monoidal categories, arXiv:0909.5270
• Angélica Osorno, Spectra associated to symmetric monoidal bicategories (arXiv)

Revised on September 11, 2012 01:08:31 by Urs Schreiber (82.169.65.155)