Operads were originally defined as V-operads, that is, enriched in a symmetric or braided monoidal category V. The symmetry or braiding in V is required in order to describe the associativity axiom the operads must obey, as well as the associativity that must be a property of the action of an operad on any of its algebras. A sequence of categorical types that filter the category of monoidal cat...

Completely iterative monads of Elgot et al. are the monads such that every guarded iterative equation has a unique solution. Free completely iterative monads are known to exist on every iteratable endofunctor H, i. e., one with final coalgebras of all functors H( ) + X. We show that conversely, if H generates a free completely iterative monad, then it is iteratable.

Structural synthesis creates a structural scheme according to the given characteristics - degree of abnormality structure S and its irrationality s . is created at first stage synthesis, containing only rotational kinematic pairs corresponds abnormality, which ensured by ratio units – plus monads, minus monads null monads. At second stage, independence makes it possible bring obtained downgradi...

We characterize the equational theories and Lawvere theories that correspond to the categories of analytic and polynomial monads on Set, and hence also the categories of the symmetric and rigid operads in Set. We show that the category of analytic monads is equivalent to the category of regular-linear theories. The category of polynomial monads is equivalent to the category of rigid theories, i...

The question of how to combine monads arises naturally in many areas with much recent interest focusing on the coproduct of two monads. In general, the coproduct of arbitrary monads does not always exist. Although a rather general construction was given by Kelly [15], its generality is reflected in its complexity which limits the applicability of this construction. Following our own research [1...

We give a new formulation of attribute grammars (AG for short) called monoidal AGs in traced symmetric monoidal categories. Monoidal AGs subsume existing domain-theoretic, graph-theoretic and relational formulations of AGs. Using a 2-categorical aspect of monoidal AGs, we also show that every monoidal AG is equivalent to a synthesised one when the underlying category is closed, and that there i...

This paper provides an introduction to the theory of monads. The main result of the paper is a folkloric proof of Beck’s monadicity theorem which gives an explicit construction of the equivalence involved. Several examples of monads are presented which illustrate the variety of guises in which monads can appear.

We consider sets of monad rules derived by focussing on the Kleisli category of a monad, and from these we derive some constructions for compound monads. Under certain conditions these constructions correspond to a distributive law connecting the monads. We also show how these relate to some constructions for compound monads described previously.

We investigate monads of partiality in Martin-Löf type theory, following Moggi’s general monad-based method for modelling effectful computations. These monads are often called lifting monads and appear in category theory with different but related definitions. In this paper, we unveil the relationship between containers and lifting monads. We show that the lifting monads usually employed in typ...

Monads are commonplace in computer science, and can be composed using Beck's distributive laws. Unfortunately, finding laws extremely difficult error-prone. The literature contains some general principles for constructing However, until now there have been no such techniques establishing when law exists. We present three families of theorems showing between two monads. first widely generalizes ...