Generalized operads, also called generalized multicategories and T -monoids, are defined as monads within a Kleisli bicategory. With or without emphasizing their monoidal nature, generalized operads have been considered by numerous authors in different contexts, with examples including symmetric multicategories, topological spaces, globular operads and Lawvere theories. In this paper we study f...

During the last two decades, monads have become an indispensable tool for structuring functional programs with computational effects. In this setting, the mathematical notion of a monad is extended with operations that allow programmers to manipulate these effects. When several effects are involved, monad transformers can be used to build up the required monad one effect at a time. Although thi...

B. Courcelle studied algebraic trees as precisely the solutions of all recursive program schemes for a given signature in Set. He proved that the corresponding monad is iterative. We generalize this to recursive program schemes over a given finitary endofunctor H of a ”suitable” category. A monad is called second-order iterative if every guarded recursive program scheme (w.r.t. H) has a unique ...

Monads are used heavily in Haskell for supporting computational effects, and the language offers excellent support for defining monadic computations. Unfortunately, defining a monad remains a difficult challenge. There are no libraries that a programmer can use to define a monad that is not a composition of existing monad transformers; therefore every such effort must start from scratch despite...

Higher inductive types (HITs) in Homotopy Type Theory (HoTT) allow the definition of datatypes which have constructors for equalities over the defined type. HITs generalise quotient types, and allow to define types which are not sets in the sense of HoTT (i.e. do not satisfy uniqueness of equality proofs) such as spheres, suspensions and the torus. However, there are also interesting uses of HI...

Non-deterministic (also known as possibilistic) and probabilistic state based systems (or automata) have been studied for quite some time. Separately, they are reasonably well-understood. The combination however is difficult, both for conceptual and technical reasons. Here we study the combination from a coalgebraic perspective and identify a monad CM that captures the combination—following wor...

We give a category-theoretic formulation of Engeler-style models for the untyped λ-calculus. In order to do so, we exhibit an equivalence between distributive laws and extensions of one monad to the Kleisli category of another and explore the example of an arbitrary commutative monad together with the monad for commutative monoids. On Set as base category, the latter is the finite multiset mona...

Every algebraic theory gives rise to a monad, and monads allow a meta-language which is a basic programming language with sideeffects. Equations in the algebraic theory give rise to equations between programs in the meta-language. An interesting question is this: to what extent can we put equational reasoning for programs into the algebraic theory for the monad? In this paper I focus on local s...