What is the correct notion of homomorphism for interpolative semigroups?
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.