nLab
permutative category

Contents

Context

Monoidal categories

monoidal categories

With symmetry

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

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.

Definition

(May, def. 1) (Elmendorf-Mandell, def. 3.1).

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 E_\infty Ring Spaces and E 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

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