nLab
bipermutative category

Idea

A bipermuatative category 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

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

Properties

Relation to bimonoidal categories

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

Examples

• Ab, $R$Mod, Vect are bipermutative categories under direct sum and tensor product of abelian groups/tensor product of modules.

Related concepts

• distributive category

• K-theory of a bipermutative category

• BDR 2-vector bundle

References

• Peter May, $E_{\mathrm{\infty }}$ Ring Spaces and $E_{\mathrm{\infty }}$ 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)