# nLab permutative category

Contents

### Context

#### Monoidal categories

monoidal categories

category theory

# Contents

## Idea

A permutative category is a symmetric monoidal category (possibly taken to be internal to Top) in which associativity (including unitality) holds strictly. Also known as a symmetric strict monoidal category.

## Properties

Every symmetric monoidal category is equivalent to a permutative one (Isbell).

The nerve of a permutative category is an E-infinity space, and therefore can be infinitely delooped to obtain an infinite loop space as its group completion.

## References

An original account is in

• John Isbell, On coherent algebras and strict algebras, J. Algebra 13 (1969)

Discussion in relation to symmetric spectra is in

Discussion in the context of K-theory of a permutative category is in

• Peter May, The spectra associated to permutative categories, Topology 17 (1978) (pdf)

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

• Anthony Elmendorf, Michael Mandell, Rings, modules and algebras in infinite loop space theory, K-Theory 0680 (web, pdf)

Discussion in the context of equivariant stable homotopy theory is in

• Bert Guillou, Peter May, Permutative $G$-categories in equivariant infinite loop space theory, Algebr. Geom. Topol. 17 (2017) 3259-3339 (arXiv:1207.3459)

Last revised on January 3, 2019 at 09:20:45. See the history of this page for a list of all contributions to it.