bipermutative category


Monoidal categories



A bipermuatative category is a semistrict rig category. More concretely, it is a permutative category (C,)(C, \oplus) with a second symmetric monoidal category structure (C,)(C, \otimes) that distributes over \oplus, with, again, some of the coherence laws required to hold strictly.


(May, def. VI 3.3) (Elmendorf-Mandell, def. 3.6)


Relation to rig categories

Every symmetric rig category is equivalent to a bipermutative category ([May, prop. VI 3.5]).



For RR a plain ring, regarded as a discrete rig category, it is a bipermutative category. The corresponding K-theory of a bipermutative category is ordinary cohomology with coefficients in RR, given by the Eilenberg-MacLane spectrum HRH R.


Consider the category whose objects are the natural numbers and whose hom sets are

Hom(n 1,n 2)={Σ n 1 |n 1=n 2 |n 1n 2, Hom(n_1, n_2) = \left\{ \array{ \Sigma_{n_1} & | n_1 = n_2 \\ \emptyset & | n_1 \neq n_2 } \right. \,,

with Σ n\Sigma_n being the symmetric group of permutations of nn elements. The two monoidal structures ar given by addition and multiplication of natural numbers. This is a bipermutative version of the Core(FinSet)Core(FinSet), the core of the category FinSet of finite sets.

The corresponding K-theory of a bipermutative category is given by the sphere spectrum.


  • Peter May, E E_\infty Ring Spaces and E E_\infty Ring spectra, Springer lectures notes in mathematics, Vol. 533, (1977) (pdf) chaper VI

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