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Abstract. A semantic formulation of Lindley and Stark’s ⊤⊤-lifting is given. We first illustrate our semantic formulation of the ⊤⊤-lifting in Set with several examples, and apply it to the logical predicates for Moggi’s computational metalanguage. We then abstract the semantic ⊤⊤-lifting as the lifting of strong monads across bifibrations with lifted symmetric monoidal closed structures. 1

ABSTRACT. New techniques for constructing a distributive law of a monad over another are studied using submonads, quotient monads, product monads, recursively-defined distributive laws, and linear equations. Sequel papers will consider distributive laws in closed categories and will construct monad approximations for compositions which fail to be a monad. 1.

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We take some first steps in providing a synthetic theory of distributions. In particular, we are interested in the use of distribution theory as foundation, not just as tool, in the study of the wave equation. 1.

Abstract. The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop a theory of free constructions for such equational systems; and to illustrate the use of equational systems as needed in modern applications, specifically to the theory of substitution in the presence of variable binding and to models of name-passing process calculi. 1

Generalised Distributive laws in Computer Science are rules governing the transformation of one programming structure into another. In programming, they are programs satisfying certain formal conditions. Their importance has been to date documented in several isolated cases by diverse formal approaches. These applications have always meant leaps in understanding the nature of the subject. However, distributive laws have not yet been given the attention they deserve. One of the reasons for this omission is certainly the lack of a formal notion of distributive laws in their full generality. This hinders the discovery and formal description of occurrences of distributive laws, which is the precursor of any formal manipulation. In this thesis, an approach to formalisation of distributive laws is presented based on the functorial approach to formal Category Theory pioneered by Lawvere and others, notably Gray. The proposed formalism discloses a rather simple nature of distributive laws of the kind found in programming structures based on lax 2-naturality and Gray’s tensor product of 2-categories. It generalises the existing more specific notions of distributive

Abstract—Although increasingly popular, Model Driven Architecture (MDA) still lacks suitable formal foundations on top of which rigorous methodologies for the description, analysis and transformation of models could be built. This paper aims to contribute in this direction: building on previous work by the authors on coalgebraic refinement for software components and architectures, it discusses refactoring of models within a coalgebraic semantic framework. Architectures are defined through aggregation based on a coalgebraic semantics for (subsets of) UML. On the other hand, such aggregations, no matter how large and complex they are, can always be dealt with as coalgebras themselves. This paves the way to a discipline of models’ transformations which, being invariant under either behavioural equivalence or refinement, are able to formally capture a large number of refactoring patterns. The main ideas underlying this research are presented through a detailed example in the context of refactoring of UML class diagrams. I.

Abstract. It is shown how the theory of commutative monads provides an axiomatic framework for several aspects of distribution theory in a broad sense, including probability distributions, physical extensive quantities, and Schwartz distributions of compact support. Among the particular aspects considered here are the notions of convolution, density, expectation, and conditional probability.

calculus with extensive quantities: wave equation