nLab bipermutative category



Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



A bipermutative 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.


Two nonequivalent definitions are given in (May, def. VI 3.3) and (Elmendorf-Mandell, def. 3.6).

May requires the left distributivity map to be an isomorphism and the right distributivity map to be an identity.

Elmendorf and Mandell allow both distributivity maps to be noninvertible.

A discussion of these two definitions is in (May2, Section 12).


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
  • Peter May, The construction of E E_\infty ring spaces

    from bipermutative categories_, Geometry and Topology Monographs, Vol. 16, (2009) (pdf) chaper VI

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