nLab
permutative category

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.

Related concepts

• K-theory of a permutative category

• bipermutative category

References

An original account is in

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

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, $E_{\mathrm{\infty }}$ Ring Spaces and $E_{\mathrm{\infty }}$ 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)