With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
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.
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).
Every symmetric rig category is equivalent to a bipermutative category ([May, prop. VI 3.5]).
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 $H R$.
Consider the category whose objects are the natural numbers and whose hom sets are
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.
from bipermutative categories_, Geometry and Topology Monographs, Vol. 16, (2009) (pdf) chaper VI
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