nLab duoid

Idea

A double monoid [Aguiar & Mahajan (2010) or duoid [Batanin & Markl (2012)] is an object of a duoidal category equipped with two compatible monoid structures.

Definitions

With a binary operation and an element

Naively, a duoid is a set $D$ equipped with a pair of monoid structures $(\otimes, I)$ and $(\odot, J)$ such that:

$$(v \otimes x) \odot (y \otimes z) \;=\; (v \odot y) \otimes (x \odot z) \,.$$

In this form, a duoid can be viewed as a strict double category with a single object, one horizontal morphism, and one vertical morphism.

However, due to the Eckmann–Hilton argument, this is equivalent to a commutative monoid. In fact, this is true for a duoid in any braided monoidal category (viewed as a duoidal category in which both tensor products coincide).

In a duoidal category

See duoidal category.

References

Duoids were introduced as “double monoids” in the context of duoidal categories in :

• Marcelo Aguiar, Swapneel Arvind Mahajan, Definition 6.28 of: Monoidal functors, species and Hopf algebras 29 Providence, RI: American Mathematical Society (2010) [pdf, ams:crmm-29]

The name “duoid”, following a suggestion by Ross Street, is due to:

• Michael Batanin, Martin Markl, Centers and homotopy centers in enriched monoidal categories, Advances in Mathematics 230 4-6 (2012) 1811-1858 [doi:10.1016/j.aim.2012.04.011]

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