With duals for objects
With duals for morphisms
Special sorts of products
In higher 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, Ring Spaces and 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)