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
category theory

Concepts
Universal constructions
Theorems
Extensions
Applications
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 , $E_\infty$ -spaces, group completions, and permutative categories , London Math. Soc. Lecture Notes No. 11, 1974, 61-94 (doi:10.1017/CBO9780511662607.008 , 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

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