Monads are widely used in programming semantics and in functional programming to encapsulate notions of side-effect, such as state, exceptions, input/output, or continuations. One of their advantages is that they allow for a modular treatment of effects, using composition operators such as sum and tensor. Here, the sum represents the non-interacting combination of effects, while the tensor impo...

It has long been known that some of the most common uses of for and while-loops in imperative programs can easily be expressed using the standard higher-order functions fold and map. With this correspondence as a starting point, we derive parallel implementations of various iterative constructs, each having a better complexity than their sequential counterparts, and explore the use of monads to...

Monads provide a greatly useful capability to pure languages in simulating side-effects, but implementations such as the Monad Transformer Library [1] in Haskell prohibit reuse of those side-effects such as threading through two different states without some explicit workaround. Monad Factory provides a straightforward solution for opening the non-proper morphisms by indexing monads at both the...

We give an axiomatic category theoretic account of bisimu-lation in process algebras based on the idea of functional bisimulations as open maps. We work with 2-monads, T , on Cat. Operations on processes , such as nondeterministic sum, preexing and parallel composition are modelled using functors in the Kleisli category for the 2-monad T. We may deene the notion of open map for any such 2-monad...

We present an algorithm for deriving a spatial-behavioral type system from a formal presentation of a computational calculus. Given a 2-monad Calc : Cat → Cat for the free calculus on a category of terms and rewrites and a 2-monad BoolAlg for the free Boolean algebra on a category, we get a 2-monad Form = BoolAlg + Calc for the free category of formulae and proofs. We also get the 2-monad BoolA...

The definition of Azumaya algebras over commutative rings R requires the tensor product of modules over R and the twist map for the tensor product of any two R-modules. Similar constructions are available in braided monoidal categories, and Azumaya algebras were defined in these settings. Here, we introduce Azumaya monads on any category A by considering a monad (F,m, e) on A endowed with a dis...

The notion of commutative monad was denned by the author in [4]. The content of the present paper may briefly be stated: The category of algebras for a commutative monad can in a canonical way be made into a closed category, the two adjoint functors connecting the category of algebras with the base category are in a canonical way closed functors, and the frontand end-adjunctions are closed tran...

Many homotopy-coherent algebraic structures can be described by Segal-type limit conditions determined an “algebraic pattern”, which we mean ?-category equipped with a factorization system and collection of “elementary” objects. Examples that occur as such “Segal O-spaces” for pattern O include ?-categories, (?,n)-categories, ?-operads (including symmetric, non-symmetric, cyclic, modular ones),...

It is well known that, given an endofunctor H on a category C, the initial (A+H−)-algebras (if existing), i.e., the algebras of (wellfounded) H-terms over different variable supplies A, give rise to a monad with substitution as the extension operation (the free monad induced by the functor H). Moss [17] and Aczel, Adámek, Milius and Velebil [2] have shown that a similar monad, which even enjoys...

We present a coalgebraic generalisation of Fischer and Ladner’s Propositional Dynamic Logic (PDL) and Parikh’s Game Logic (GL). In earlier work, we proved a generic strong completeness result for coalgebraic dynamic logics without iteration. The coalgebraic semantics of such programs is given by a monad T , and modalities are interpreted via a predicate lifting λ whose transpose is a monad morp...