Multiple monoidal categories

Most categories have more than one monoidal product, which can interact with each other in interesting ways, such as:

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.