nLab bipermutative category

Idea

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

Definition

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).

Properties

Relation to rig categories

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

Examples

• Ab, $R$Mod, Vect are symmetric rig categories under direct sum and tensor product of abelian groups/tensor product of modules. Thus, they have “strictifications” that are bipermutative.

Example

For $R$ 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 $R$, given by the Eilenberg-MacLane spectrum $H R$.

Example

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

$$Hom(n_1, n_2) = \left\{ \array{ \Sigma_{n_1} & | n_1 = n_2 \\ \emptyset & | n_1 \neq n_2 } \right. \,,$$

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

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

Related concepts

• K-theory of a bipermutative category

• BDR 2-vector bundle

• distributivity for monoidal structures

  • rig category

  • distributive monoidal category

  • distributive category

References

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

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

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

• Anthony Elmendorf, Michael Mandell, Rings, modules and algebras in infinite loop space theory, K-Theory 0680 (web, pdf)