# nLab bipermutative category

Contents

### Context

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

# Contents

## Idea

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

## Definition

Two nonequivalent definitions are given in (May, def. VI 3.3) and (Elmendorf-Mandell, def. 3.6).

May requires the left distributivity map to be an isomorphism and the right distributivity map to be an identity.

Elmendorf and Mandell allow both distributivity maps to be noninvertible.

A discussion of these two definitions is in (May2, Section 12).

## 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 $H R$.

###### Example

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

$Hom(n_1, n_2) = \left\{ \array{ \Sigma_{n_1} & | n_1 = n_2 \\ \emptyset & | n_1 \neq n_2 } \right. \,,$

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 $Core(FinSet)$, 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
• Peter May, The construction of $E_\infty$ ring spaces

from bipermutative categories_, Geometry and Topology Monographs, Vol. 16, (2009) (pdf) chaper VI

Last revised on January 1, 2019 at 01:02:03. See the history of this page for a list of all contributions to it.