Rig category (also known as a bimonoidal category) is an established concept in category theory. It refers to a category equipped with two monoidal structures — one playing the role of "addition" and the other of "multiplication" — such that the multiplicative structure distributes over the additive one, subject to a set of coherence conditions. The term is a categorification of the algebraic notion of a rig (a "ring without negatives").
Definition
A rig category is a category $ \mathbf{C} $ equipped with:
- A symmetric monoidal structure $ (\mathbf{C}, \oplus, 0) $ for addition.
- A monoidal structure $ (\mathbf{C}, \otimes, I) $ for multiplication.
- Distributivity natural isomorphisms: $$ d_\ell : x \otimes (y \oplus z) \longrightarrow (x \otimes y) \oplus (x \otimes z) $$ $$ d_r : (x \oplus y) \otimes z \longrightarrow (x \otimes z) \oplus (y \otimes z) $$
- Absorption (annihilation) natural isomorphisms: $$ a_\ell : x \otimes 0 \longrightarrow 0 $$ $$ a_r : 0 \otimes x \longrightarrow 0 $$
These structures must satisfy a set of coherence laws, first worked out by Laplaza (1972) and Kelly (1974).
Examples
- Set: The category of sets, with disjoint union as addition and Cartesian product as multiplication, forms a rig category (specifically, a distributive category).
- Vect: The category of vector spaces over a field, with direct sum as addition and tensor product as multiplication, forms a rig category.
- FinSet (the groupoid of finite sets and bijections): With disjoint union as addition and Cartesian product as multiplication, this rig category is a categorification of the natural numbers $ \mathbb{N} $.
- Ab (abelian groups), R-Mod (modules over a commutative ring), and Vect(X) (vector bundles over a space) are also rig categories under direct sum and tensor product.
Related Concepts
- Distributive category: A rig category where addition is the categorical coproduct and multiplication is the categorical product.
- Distributive monoidal category: A rig category where addition is the coproduct but multiplication is a possibly non-Cartesian monoidal structure.
- Bipermutative category: A strictification of a symmetric rig category. Every symmetric rig category is equivalent to a bipermutative category (May, 1977).
- 2-rig: A broader notion of a categorified rig, often defined as a cocomplete symmetric monoidal category where the monoidal product distributes over colimits.
Baez's Conjecture
John Baez conjectured that the groupoid of finite sets and bijections (FinSet$^\times$) is the initial symmetric rig category, analogous to how $ \mathbb{N} $ is the initial commutative rig. This conjecture was established by Elgueta (2021).
References
- Laplaza, M. L. (1972). "Coherence for distributivity". Lecture Notes in Mathematics 281, Springer, pp. 29–65.
- Kelly, G. M. (1974). "Coherence theorems for lax algebras and distributive laws". Lecture Notes in Mathematics 420, Springer, pp. 281–375.
- May, J. P. (1977). $E_\infty$ Ring Spaces and $E_\infty$ Ring Spectra. Springer Lecture Notes in Mathematics 577.
- Elgueta, J. (2021). "The groupoid of finite sets is biinitial in the 2-category of rig categories". Journal of Pure and Applied Algebra 225(11), 106738.
- nLab: Rig category
- Wikipedia: 2-rig
