Multipliers of a semigroup object in a monoidal category

TL;DR

This paper introduces a generalized notion of multipliers for semigroup objects within monoidal categories, extending classical concepts like translational hulls and multiplier algebras to an abstract categorical setting.

Contribution

It defines the multiplier monoid of a semigroup object in a monoidal category and relates it to classical translational hulls, establishing functorial properties and natural homomorphisms.

Findings

The multiplier monoid generalizes classical multiplier algebras.

A functor from semigroups to monoids is constructed.

Concretization homomorphisms are shown to be epimorphisms under certain conditions.

Abstract

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.