We define a notion of grading monoid T in monoidal category C, relative to class morphisms M (which provide M-subobject). show that, under reasonable conditions (including that forms factorization system), there is canonical T. Our application graded monads and models computational effects. demonstrate our results by characterizing the gradings number monads, for which C endofunctors with compo...

We give a framework for combining n monads on the same category via distributive laws satisfying Yang-Baxter equations, extending the classical result of Barr and Wells which combines two monads via one distributive law. We show that this corresponds to iterating n-times the process of taking the 2-category of monads in a 2-category, extending the result of Street characterising distributive la...

Whilst the relationship between initial algebras and monads is well understood, the relationship between final coalgebras and comonads is less well explored. This paper shows that the problem is more subtle than might appear at first glance: final coalgebras can form monads just as easily as comonads, and, dually, initial algebras form both monads and comonads. In developing these theories we s...

Around twenty years ago two important developments happened in the areas of modularity and reuse in programming languages. On the one hand, Moggi showed how computational effects found in impure languages could be simulated using the notion of monads from category theory. Inspired by Moggi’s work, Wadler showed how monads could be used to structure (purely functional) programs. On the other han...

An action ∗ : V × A−→ A of a monoidal category V on a category A corresponds to a strong monoidal functor F : V−→ [A,A] into the monoidal category of endofunctors of A. In many practical cases, the ordinary functor f : V−→ [A,A] underlying the monoidal F has a right adjoint g; and when this is so, F itself has a right adjoint G as a monoidal functor—so that, passing to the categories of monoids...

We introduce a method to lift monads on the base category of a fibration to its total category using codensity monads. This method, called codensity lifting, is applicable to various fibrations which were not supported by the categorical >>-lifting. After introducing the codensity lifting, we illustrate some examples of codensity liftings of monads along the fibrations from the category of preo...

The simplicial category ∆ plays many roles all over mathematics. In one of these roles it is a monad freely generated by a single object (see [9], Section 3, [2], Section 4, and references therein). That ∆ is isomorphic to this free monad may be understood as a coherence result connecting the syntax brought by the equational presentation of monads and the semantics given by the order preserving...