Fibrations, logical predicates and indeterminates

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. 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. 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.

[1]  Peter M. W. Knijnenburg,et al.  Partial Hyperdoctrines: Categorical Models for Partial Function Logic and Hoare Logic , 1994, Math. Struct. Comput. Sci..

[2]  Bart Jacobs,et al.  Categorical type theory , 1991 .

[3]  R. Cockett,et al.  Strong categorical datatypes I , 1991 .

[4]  John L. Bell,et al.  Toposes and local set theories - an introduction , 1988 .

[5]  G. M. Kelly.  Elementary observations on 2-categorical limits , 1989, Bulletin of the Australian Mathematical Society.

[6]  Martin Hyland.  A small complete category , 1988, Ann. Pure Appl. Log..

[7]  Peter T. Johnstone,et al.  Fibrations and partial products in a 2-category , 1993, Appl. Categorical Struct..

[8]  R. Goldblatt.  Topoi, the Categorial Analysis of Logic , 1979 .

[9]  Robert A. G. Seely,et al.  Hyperdoctrines, Natural Deduction and the Beck Condition , 1983, Math. Log. Q..

[10]  Samson Abramsky,et al.  Abstract Interpretation, Logical Relations and Kan Extensions , 1990, J. Log. Comput..

[11]  R. Street,et al.  Review of the elements of 2-categories , 1974 .

[12]  Joseph A. Goguen,et al.  Some Fundamental Algebraic Tools for the Semantics of Computation: Part 3: Indexed Categories , 1991, Theor. Comput. Sci..

[13]  Robin Milner,et al.  Definition of standard ML , 1990 .

[14]  J. Lambek,et al.  Introduction to higher order categorical logic , 1986 .

[15]  Peter W. O'Hearn,et al.  Relational parametricity and local variables , 1993, POPL '93.

[16]  G. M. Kelly,et al.  A $2$-categorical approach to change of base and geometric morphisms I , 1991 .

[17]  Benjamin C. Pierce,et al.  Programming in higher-order typed lambda-calculi , 1989 .

[18]  Rod M. Burstall,et al.  Deliverables: A Categorial Approach to Program Development in Type Theory , 1993, MFCS.

[19]  John C. Mitchell,et al.  Type Systems for Programming Languages , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[20]  Jean-Yves Girard,et al.  The System F of Variable Types, Fifteen Years Later , 1986, Theor. Comput. Sci..

[21]  M. Raynaud,et al.  Categories Fibrees et Descente , 1971 .

[22]  Giuseppe Longo,et al.  Categories, types and structures - an introduction to category theory for the working computer scientist , 1991, Foundations of computing.

[23]  Glynn Winskel,et al.  A Compositional Proof System on a Category of Labelled Transition Systems , 1989, Inf. Comput..

[24]  Bart Jacobs.  Comprehension Categories and the Semantics of Type Dependency , 1993, Theor. Comput. Sci..

[25]  Simon Ambler.  First order linear logic in symmetric monoidal closed categories , 1991 .

[26]  Bart Jacobs,et al.  Fibrations with Indeterminates: Contextual and Functional Completeness for Polymorphic Lambda Calculi , 1995, Math. Struct. Comput. Sci..

[27]  R. A. G. Seely,et al.  Linear Logic, -Autonomous Categories and Cofree Coalgebras , 1989 .

[28]  John C. Mitchell,et al.  The Semantics of Second-Order Lambda Calculus , 1990, Inf. Comput..

[29]  William A. Howard,et al.  The formulae-as-types notion of construction , 1969 .

[30]  Michael Barr,et al.  Category theory for computing science , 1995, Prentice Hall International Series in Computer Science.

[31]  Simone Martini.  Categorical Models for Non-Extensional lambda-Calculi and Combinatory Logic , 1992, Math. Struct. Comput. Sci..

[32]  David E. Rydeheard,et al.  Foundations of Equational Deduction: A Categorical Treatment of Equational Proofs and Unification Algorithms , 1987, Category Theory and Computer Science.

[33]  Andrew M. Pitts,et al.  Polymorphism is Set Theoretic, Constructively , 1987, Category Theory and Computer Science.

[34]  Dusko Pavlovic,et al.  Predicates and Fibrations , 1990 .

[35]  J. Gray.  Fibred and Cofibred Categories , 1966 .

[36]  Edmund Robinson,et al.  Categories of Partial Maps , 1988, Inf. Comput..

[37]  M. Barr,et al.  Toposes, Triples and Theories , 1984 .

[38]  Michael P. Fourman,et al.  A Proposed Categorial Semantics for Pure ML , 1992, ICALP.

[39]  John C. Mitchell,et al.  Kripke-Style Models for Typed lambda Calculus , 1991, Ann. Pure Appl. Log..

[40]  Jean Benabou,et al.  Fibered categories and the foundations of naive category theory , 1985, Journal of Symbolic Logic.

[41]  John C. Reynolds,et al.  Types, Abstraction and Parametric Polymorphism , 1983, IFIP Congress.

[42]  John C. Mitchell,et al.  Notes on Sconing and Relators , 1992, CSL.

[43]  Ross Street,et al.  Fibrations and Yoneda's lemma in a 2-category , 1974 .

[44]  Thierry Coquand,et al.  The Calculus of Constructions , 1988, Inf. Comput..