Fibrations, Logical Predicates and Indeterminates
<p>Within the framework of categorical logic/type theory, we provide a category-theoretic account of some logical concepts, i.e. first-order logical predicates for simply typed lambda-calculus, structural induction for inductive data types, and indeterminates for polymorphic calculi.</p><p> </p><p>The main concept which underlies the issues above is that of fibration, which gives an abstract presentation of the indexing present in all cases: predicates indexed by types/contexts in first-order logic and types indexed by kinds in polymorphic calculi.</p><p>The characterisation of the logical concepts in terms of fibrations relies on a fundamental property of adjunctions between fibrations, which in particular relates some structure in the total category of a fibration with that of the fibres. Suitable instances of this property reflect the above-mentioned logical concepts in an abstract way, independently of their syntactic presentation, thereby illuminating their main features.</p>