# nLab bipermutative category

### Context

#### Monoidal categories

monoidal categories

## With traces

• trace

• traced monoidal category?

# Contents

## Idea

A bipermuatative category is a semistrict rig category. More concretely, it is a permutative category $\left(C,\oplus \right)$ with a second symmetric monoidal category structure $\left(C,\otimes \right)$ that distributes over $\oplus$, with, again, some of the coherence laws required to hold strictly.

## Properties

### Relation to rig categories

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

## Examples

###### Example

For $R$ a plain ring, regarded as a discrete rig category, it is a bipermutative category. The corresponding K-theory of a bipermutative category is ordinary cohomology with coefficients in $R$, given by the Eilenberg-MacLane spectrum $HR$.

###### Example

Consider the category whose objects are the natural numbers and whose hom sets are

$\mathrm{Hom}\left({n}_{1},{n}_{2}\right)=\left\{\begin{array}{cc}{\Sigma }_{{n}_{1}}& \mid {n}_{1}={n}_{2}\\ \varnothing & \mid {n}_{1}\ne {n}_{2}\end{array}\phantom{\rule{thinmathspace}{0ex}},$Hom(n_1, n_2) = \left\{ \array{ \Sigma_{n_1} & | n_1 = n_2 \\ \emptyset & | n_1 \neq n_2 } \right. \,,

with ${\Sigma }_{n}$ being the symmetric group of permutations of $n$ elements. The two monoidal structures ar given by addition and multiplication of natural numbers. This is a bipermutative version of the $\mathrm{Core}\left(\mathrm{FinSet}\right)$, the core of the category FinSet of finite sets.

The corresponding K-theory of a bipermutative category is given by the sphere spectrum.

## References

• Peter May, ${E}_{\infty }$ Ring Spaces and ${E}_{\infty }$ Ring spectra, Springer lectures notes in mathematics, Vol. 533, (1977) (pdf) chaper VI

Revised on July 15, 2013 19:59:40 by Mike Shulman (107.201.30.59)