Multiple monoidal categories
Most categories have more than one monoidal product, which can interact with each other in interesting ways, such as:
- Distributivity , the classical relation between products and sums:
\[
X \otimes (Y \oplus Z) \cong (X \otimes Y) \oplus (X \otimes Z) \\
(X \oplus Y) \otimes Z \cong (X \otimes Z) \oplus (Y \otimes Z)
\]
Here are several possible definitions, in increasing order of generality:
- Distributive category
- Distributive monoidal category
- Rig category (or 2-rig ), a categorification of a rig
- Linear distributivity , a weak associativity relation appearing in the multiplicative fragment of linear logic: \[ X \otimes (Y\ ⅋\ Z) \to (X \otimes Y)\ ⅋\ Z \\ (X\ ⅋\ Y) \otimes Z \to X\ ⅋\ (Y \otimes Z) \]
- Interchangeability, in the lax form of duoidal categories : \[ (X \star Y) \diamond (Z \star W) \to (X \diamond Z) \star (Y \diamond W) \]
Literature
I do not know of any systematic survey of multiple monoidal categories, comparable to Selinger’s survey of monoidal categories and their graphical languages. Someone should write one.
Distributivity
- Carboni, Lack, Walters, 1993: Introduction to extensive and distributive categories (doi)
- Jay, 1993: Tail recursion through universal invariants (doi, tech report ), Sec. 3.2: Distributive monoidal categories
Linear distributivity
See page on linear logic and its categorical semantics.
Interchangeability
Duoidal categories (aka 2-monoidal categories) and intercategories both involve monoidal products that interchange laxly.
- Aguiar & Mahajan, 2010: Monoidal functors, species and Hopf algebras (doi, pdf), Ch. 6: 2-monoidal categories
- Booker & Street, 2013: Tannaka duality and convolution for duoidal categories (pdf, arxiv)
For intercategories, see internal category theory.