This article is devoted to the investigation of the deformation (twisting) of monoidal structures, such as the associativity constraint of the monoidal category and the monoidal structure of monoidal functor. The sets of twistings have a (non-abelian) c ohomological nature. Using this fact the maps from the sets of twistings to some cohomology groups (Hochschild cohomology of K-theory) are cons...

Relative monads (Altenkirch, Chapman, and Uustalu 2010) are a recent generalisation of ordinary monads to cover similar structures where the underlying functor need not be an endofunctor. Our interest in this generalisation was triggered by some structures from programming theory that, in many ways, are strikingly similar to monads (even respecting the same laws), have similar programming appli...

Monads are of interest both in semantics and in higher dimensional algebra. It turns out that the idea behind usual notion finitary monads (whose values on all sets can be computed from their values on finite sets) extends to a more general class of monads called monads with arities, so that not only algebraic theories can be computed from a proper set of arities, but also more general structur...

We continue the study of Capretta’s delay monad as a means of introducing non-termination from iteration into Martin-Löf type theory. In particular, we explain in what sense this monad provides a canonical solution. We discuss a class of monads that we call ω-complete pointed classifying monads. These are monads whose Kleisli category is an ωcomplete pointed restriction category where pure maps...

For programmers, Monads are a well-known way to represent an abstract data structure, but without knowing its true nature. Admittedly, the definition of Monads in Category Theory is too subtle and obscure so makes people to avoid from trying to understand them. In this paper, we attempt to explore popular Monads in a purely functional programming language, Haskell, with a Category-theoretic min...

We revisit the connection between three notions of computation: Moggi’s monads, Hughes’s arrows and McBride and Paterson’s idioms (also called applicative functors). We show that idioms are equivalent to arrows that satisfy the type isomorphism A;B ' 1 ; (A→ B) and that monads are equivalent to arrows that satisfy the type isomorphism A; B ' A → (1 ; B). Further, idioms embed into arrows and ar...

We introduce a tensor product for symmetric monoidal categories with the following properties. Let SMC denote the 2-category with objects small symmetric monoidal categories, arrows symmetric monoidal functors and 2-cells monoidal natural transformations. Our tensor product together with a suitable unit is part of a structure on SMC that is a 2-categorical version of the symmetric monoidal clos...

Monads are a technique widely used in functional programming languages to address many different problems. This paper presents extensions, a functional-logic programming technique that constitutes an alternative to monads in several situations. Extensions permit the definition of easily reusable functions in the same way as monads, but are based on simpler concepts taken from logic programming,...

The category of all monads over many-sorted sets (and over other " set-like " categories) is proved to have coequalizers and strong coin-tersections. And a general diagram has a colimit whenever all the monads involved preserve monomorphisms and have arbitrarily large joint pre-fixpoints. In contrast, coequalizers fail to exist e.g. for monads over the (presheaf) category of graphs. For more ge...

1 ∞-Operads 4 1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Fibrations of ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Algebra Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Cartesian Monoidal Structures . . . . . . . . . . ....