A parameterization process as a categorical construction

The parameterization process used in the symbolic computation systems Kenzo and EAT is studied here as a general construction in a categorical framework. This parameterization process starts from a given specification and builds a parameterized specification by transforming some operations into parameterized operations, which depend on one additional variable called the parameter. Given a model of the parameterized specification, each interpretation of the parameter, called an argument, provides a model of the given specification. Moreover, under some relevant terminality assumption, this correspondence between the arguments and the models of the given specification is a bijection. It is proved in this paper that the parameterization process is provided by a free functor and the subsequent parameter passing process by a natural transformation. Various categorical notions are used, mainly adjoint functors, pushouts and lax colimits

Diagrammatic logic applied to a parameterization process

This paper provides an abstract definition of some kinds of logics, called diagrammatic logics, together with a definition of morphisms and of 2-morphisms between diagrammatic logics. The definition of the 2-category of diagrammatic logics rely on category theory, mainly on adjunction, categories of fractions and limit sketches. This framework is applied to the formalization of a parameterization process. This process, which consists in adding a formal parameter to some operations in a given specification, is presented as a morphism of logics. Then the parameter passing process, for recovering a model of the given specification from a model of the parameterized specification and an actual parameter, is seen as a 2-morphism of logics

Formalization of a normalization theorem in simplicial topology

In this paper we present a complete formalization of the Normalization Theorem, a result in Algebraic Simplicial Topology stating that there exists a homotopy equivalence between the chain complex of a simplicial set, and a smaller chain complex for the same simplicial set, called the normalized chain complex. Even if the Normalization Theorem is usually stated as a higher-order result (with a Category Theory flavor) we manage to give a first-order proof of it. To this aim it is instrumental the introduction of an algebraic data structure called simplicial polynomial. As a demonstration of the validity of our techniques we developed a formal proof in the ACL2 theorem prover.Ministerio de Ciencia e Innovación MTM2009-13842European Commission FP7 STREP project ForMath n. 24384

An object-oriented interpretation of the EAT system

In a previous paper we characterized, in the Category Theory setting, a class of implementations of Abstract Data Types, which has been suggested by the way of programming in the EAT system. (EAT, Effective Algebraic Topology, is one of Sergeraert's systems for effective homology and homotopy computation.) This characterization was established using classical tools, in an unrelated way to the current mainstream topics in the field of Algebraic Specifications. Looking for a connection with these topics, we have found, rather unexpectedly, that our approach is related to some object-oriented formalisms, namely hidden specifications and the coalgebraic view. In this paper, we explore these relations making explicit the implicit object-oriented features of the EAT system and generalizing the data structure analysis we had previously done