Day algebras

Abstract

In this paper we show that the Day monoidal product generalises in a straightforward way to other algebraic constructions and partial algebraic constructions on categories. This generalisation was motivated by its applications in logic, for example in hybrid and separation logic. We use the description of the Day monoidal product using profunctors to show that the definition generalises to an extension of an arbitrary algebraic structure on a category to a pseudo-algebraic structure on a functor category. We provide two further extensions. First we consider the case where some of the operations on the category are partial, and second we show that the resulting operations on the functor category have adjoints (they are residuated).