Iterative Algebras for a Base

Abstract

For algebras A whose type is given by an endofunctor, iterativity means that every flat equation morphism in A has a unique solution. In our previous work we proved that every object generates a free iterative algebra, and we provided a coalgebraic construction of those free algebras. Iterativity w.r.t. an endofunctor was generalized by Tarmo Uustalu to iterativity w.r.t. a "base", i.e., a functor of two variables yielding finitary monads in one variable. In the current paper we introduce iterative algebras in this general setting, and provide again a coalgebraic construction of free iterative algebras.

Complete Elgot Monads and Coalgebraic Resumptions

“…Remark 8. The order of arguments in X #Y is in agreement with [26] and differs from [4] where the notation Y X equivalent to the present X # Y is used. We chose the order of arguments to ensure agreement with the type profile of the iteration operator -- † , which is in turn in agreement with the expression (⋆).…”

“…Following [4] we will from now on denote the unit and monad multiplication of monads --#X by u X A : A → A # X and m X A : (A # X) # X → A # X, respectively.…”

“…We introduced the notion of complete Elgot algebra for a parametrized monad, based on the previous work [4,26]. We showed that the category of complete Elgot algebras for a parametrized monad # is isomorphic to the category of Eilenberg-Moore algebras for the monad νγ.…”

“…In order to study complete Elgot monads and their algebras it is helpful to make a further abstraction step and generalize from monads to parametrized monads [26] (finitary parametrized monads are also called bases [4]), which are of independent interest. Definition 7.…”

“…Following [4] we will from now on denote the unit and monad multiplication of monads --#X by u X A :…”