A double monoid [Aguiar & Mahajan (2010) or duoid [Batanin & Markl (2012)] is an object of a duoidal category equipped with two compatible monoid structures.
Naively, a duoid is a set equipped with a pair of monoid structures and such that:
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).
See duoidal category.
Duoids were introduced as “double monoids” in the context of duoidal categories in :
The name “duoid”, following a suggestion by Ross Street, is due to:
.
Last revised on July 12, 2023 at 07:59:08. See the history of this page for a list of all contributions to it.