Use this URL to cite or link to this record in EThOS: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.652380
Title: Fibrations, logical predicates and indeterminates
Author: Hermida, Claudio Alberto
Awarding Body: University of Edinburgh
Current Institution: University of Edinburgh
Date of Award: 1993
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Abstract:
Within the framework of categorical logic or categorical type theory, predicate logics and type theories are understood as fibrations with structure. Fibrations, or fibred categories, provide an abstract account of the notions of indexing and substitution. These notions are central to the interpretations of predicate logics and type theories with dependent types or polymorphism. In these systems, predicates/dependent types are indexed by the contexts which declare the types of their free variables, and there is an operation of substitution of terms for free variables. With this setting it is natural to give a category-theoretic account of certain logical issues in terms of fibrations. In this thesis we explore logical predicates for simply typed theories, induction principles for inductive data types, and indeterminate elements for fibrations in relation to polymorphic λ-calculi. The notions of logical predicate is a useful tool in the study of type theories like simply typed λ-calculus. For a categorical account of this concept, we are led to study certain structure of fibred categories. In particular, the kind of structure involved in the interpretation of simply typed λ-calculus, namely cartesian closure, is expressed in terms of adjunctions. Hence we are led to consider adjunctions between fibred categories. We give a characterisation of these adjunctions which allows us to provide categorical structure, given by adjunctions, to a fibred category using similar structure on its base and its fibres. By expressing the abovementioned categorical construction logically, in the internal language of a fibration, we can then account for the notion of logical predicate for a cartesian closed category. With a similar argument, we provide a categorical interpretation of the induction principle for inductive data types, given by initial algebras for endofunctors on a distributive category. We also consider the problem of adjoining indeterminate elements to fibrations.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.652380  DOI: Not available
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