permutative category


Monoidal categories

Category theory



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.


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


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.


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)

Revised on December 15, 2016 18:22:10 by Anonymous Coward (