What is the correct notion of homomorphism for interpolative semigroups?

Jürgen Koslowski

Abstract

Such semigrups are defined by requiring every element to be a product. Clearly, monoids have this property, and it is well-known that semigroup morphisms between monoids need not preserve the neutral element. So the question is how to strengthen the notion of semigroup homomorphism in the interpolative case in order to guarantee the preservation of neutral elements, if these are present. This problem is a special case of the question how to generalize the notion of functor to taxonomies, ie, categories without identities where the associativity diagram for the composition is in fact a coequalizer.

Along the way, for suitable bicategories we construct first a general multicategory of modules and then another multicategory with idempotent modules as objects. Isomorphism classes of these are shown to have canonical representatives, interpolads, which specialize to interpolative semigroups over the suspension of set and to taxonomies over spn. When restricting to these objects, the obvious 1-cells, called profunctors, turn out to be bimodules subject to two coequalizer requirements. As Richard Wood has shown, profunctors between monads are compatible with the monad identities. By exploiting the connection between functors and profunctors in case of monads, we are then able us to extend the notion of functor to interpolads.