Multipliers of a semigroup object in a monoidal category

Laurent Poinsot (1, 2)

1 CReA - Centre de Recherche de l'École de l'air
2 LIPN - Laboratoire d'Informatique de Paris-Nord

The monoid of multipliers of a semigroup object in a monoidal category is introduced, arising from an abstraction of the definition of the translational hull of an ordinary semigroup or of the multiplier algebra of a Banach algebra and dually, the monoid of comultipliers of a cosemigroup object is obtained. Its set-theoretic version, the classical translational hull, is shown to provide a functor from a subcategory of ordinary semigroups to monoids, similar to a left adjoint. The abstract multiplier monoid of a semigroup object is related to the concrete translational hull of its convolution semigroup by a ``concretization'' homomorphism. For semigroup objects for which this homomorphism is onto, the multiplier construction is functorial and the concretization homomorphisms form a natural epimorphism.

Mots clés: Semigroup object; multiplier; translational hull; convolution semigroup

Domaines: Catégories et ensembles [math.CT]

Laurent Poinsot. Multipliers of a semigroup object in a monoidal category. 2022. ⟨hal-03685334v2⟩