Some recherche monoidal structures

Richard Garner – 17 May 2023

It is well known that commutative monoids in (Set, ×, 1) are commutative monoids. We describe other monoidal structures on Set for which the commutative monoids are: commutative semigroups; non-commutative semigroups; and magmas. In a similar spirit, we describe a monoidal structure on presheaves over a Boolean algebra for which the idempotent commutative monoids are sheaves.