category with duals (list of them)
dualizable object (what they have)
Recall that a rig is a ‘ring without negatives’: a monoid object in the monoidal category of commutative monoids with the usual tensor product. Categorifying this notion, we obtain various notions of 2-rig. One of these, in which both “addition” and “multiplication” are represented by abstract monoidal structures, is the notion of rig category, also known as a bimonoidal category.
A typical example would be the groupoid of finite sets and bijections, with disjoint union playing the role of addition and cartesian product playing the role of multiplication. This rig category can be thought of as a categorification of the set of natural numbers. Note that in this example, disjoint union is not the categorical coproduct, and product of sets is not the categorical product (because we are working in the groupoid of finite sets).
A rig category, or bimonoidal category, is a category with a symmetric monoidal structure for addition and a monoidal structure for multiplication, together with left and right distributivity natural isomorphisms
d_\ell : x \otimes (y \oplus z) \to (x \otimes y) \oplus (x \otimes z)
d_r : (x \oplus y) \otimes z \to (x \otimes z) \oplus (y \otimes z)
and absorption/annihilation isomorphisms
a_\ell : x \otimes 0 \to 0
a_r : 0 \otimes x \to 0
Note that these authors used the term ‘ring category’. We take the liberty of switching to ‘rig category’ because it is typical for these to lack additive inverses.
While a rig can have the extra property of being commutative (i.e. of its multiplication being commutative), a rig category can have the extra structure of (its monoidal structure ) being braided (compatibly with the distributive laws) and may then have the further property of being symmetric.
Rig categories are part of the hierarchy of distributivity for monoidal structures. If is the categorical coproduct and is the categorical product, then we have the notion of a distributive category, which is a special case of a rig category. For example, (or any topos) is a distributive category, hence a rig category with and .
Using the correct definition of the 2-category of symmetric rig categories, the groupoid of finite sets and bijections is the initial symmetric rig category, just as is the initial commutative rig. Note that a suitably weakened concept of ‘initial’ is needed here; see 2-limit. In other words, given any symmetric rig category , there is a unique symmetric rig morphism , up to an equivalence which is itself unique up to an isomorphism which is actually unique (up to equality).
The notion of a bipermutative category is a strictification of the notion of symmetric rig category. Every symmetric rig category is equivalent to a bipermutative category (May, prop. VI 3.5). Similarly, every rig category is equivalent to a strict rig category (Guillou, theorem 1.2).
where these categories are called ring categories. Discussion with an eye towers the K-theory of a bipermutative category is in