Type theory via exact categories

Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a category-theoretic explanation of why the construction of a category of partial equivalence relations often produces a cartesian closed category. We show how several familiar examples of categories of partial equivalence relations fit into the general framework.

[1]  Giuseppe Longo,et al.  On church's formal theory of functions and functionals: The λ-calculus: connections to higher type recursion theory, proof theory, category theory , 1988, Ann. Pure Appl. Log..

[2]  Roberto M. Amadio,et al.  Recursion over Realizability Structures , 1991, Inf. Comput..

[3]  Andre Scedrov,et al.  Functorial Polymorphism , 1990, Theor. Comput. Sci..

[4]  A. Carboni,et al.  The free exact category on a left exact one , 1982 .

[5]  Edmund Robinson,et al.  Colimit completions and the effective topos , 1990, Journal of Symbolic Logic.

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

[7]  Robert Harper,et al.  Constructing Type Systems over an Operational Semantics , 1992, J. Symb. Comput..

[8]  Eugenio Moggi,et al.  Notions of Computation and Monads , 1991, Inf. Comput..

[9]  A. Carboni,et al.  Some free constructions in realizability and proof theory , 1995 .

[10]  Giuseppe Longo,et al.  A Category-Theoretic Characterization of Functional Completeness , 1990, Theor. Comput. Sci..

[11]  Dana S. Scott,et al.  Lambda Calculus: Some Models, Some Philosophy , 1980 .

[12]  Roberto M. Amadio.  On the Adequacy of Per Models , 1993, MFCS.

[13]  Luca Cardelli,et al.  A Semantic Basis for Quest , 1991, J. Funct. Program..

[14]  Giuseppe Rosolini,et al.  Extensional PERs , 1990, [1990] Proceedings. Fifth Annual IEEE Symposium on Logic in Computer Science.

[15]  Stefano Berardi,et al.  An application of PER models to program extraction , 1993, Mathematical Structures in Computer Science.

[16]  Martin Hofmann,et al.  Elimination of Extensionality in Martin-Löf Type Theory , 1994, TYPES.

[17]  Giuseppe Rosolini,et al.  Categories and Effective Computations , 1987, Category Theory and Computer Science.

[18]  Martin Hofmann,et al.  A Simple Model for Quotient Types , 1995, TLCA.

[19]  Martín Abadi,et al.  A PER model of polymorphism and recursive types , 1990, [1990] Proceedings. Fifth Annual IEEE Symposium on Logic in Computer Science.

[20]  Thomas Ehrhard.  Hypercoherences: A Strongly Stable Model of Linear Logic , 1993, Math. Struct. Comput. Sci..

[21]  Peter W. O'Hearn,et al.  Parametricity and local variables , 1995, JACM.

[22]  Eugenio Moggi,et al.  Constructive Natural Deduction and its 'Omega-Set' Interpretation , 1991, Math. Struct. Comput. Sci..

[23]  Giuseppe Rosolini,et al.  Locally cartesian closed exact completions ( , 2000 .

[24]  Giuseppe Rosolini.  About Modest Sets , 1990, Int. J. Found. Comput. Sci..

[25]  E. Moggi.  The partial lambda calculus , 1988 .

[26]  J. Hyland.  The Effective Topos , 1982 .

[27]  G. M. Kelly,et al.  Categories of continuous functors, I , 1972 .

[28]  Maria Emilia Maietti,et al.  The Internal Type Theory of a Heyting Pretopos , 1996, TYPES.

[29]  P. Aczel.  The Type Theoretic Interpretation of Constructive Set Theory: Choice Principles , 1982 .

[30]  Manfred Droste.  Concurrent Automata and Domains , 1992, Int. J. Found. Comput. Sci..

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

[32]  Martín Abadi,et al.  A Theory of Objects , 1996, Monographs in Computer Science.

[33]  Andre Scedrov,et al.  Categories, allegories , 1990, North-Holland mathematical library.

[34]  Dana S. Scott,et al.  Data Types as Lattices , 1976, SIAM J. Comput..

[35]  Andre Scedrov,et al.  A Categorical Approach to Realizability and Polymorphic Types , 1987, MFPS.

[36]  Rance Cleaveland,et al.  Implementing mathematics with the Nuprl proof development system , 1986 .