On Completeness and Cocompleteness in an Around Small Categories

Abstract The simple connection of completeness and cocompleteness of lattices grows in categories into the Adjoint Functor Theorem. The connection of completeness and cocompleteness of Boolean algebras — even simpler — is similarly related to Pare's Theorem for toposes. We explain these relations, and then study the fibrational versions of both these theorems — for small complete categories. They can be interpreted as definability results in logic with proofs-as-constructions, and transferred to type theory.

[1]  John C. Reynolds,et al.  Towards a theory of type structure , 1974, Symposium on Programming.

[2]  Edmund Robinson,et al.  How complete is PER? , 1989, [1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science.

[3]  Dusko Pavlovic.  Predicates and fibrations : from type theoretical to category theoretical presentation of constructive logic , 1990 .

[4]  S. Maclane,et al.  Categories for the Working Mathematician , 1971 .

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

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

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

[8]  Robert Paré,et al.  Abstract families and the adjoint functor theorems , 1978 .

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

[10]  Thomas Streicher,et al.  Semantics of type theory - correctness, completeness and independence results , 1991, Progress in theoretical computer science.

[11]  de Ng Dick Bruijn,et al.  A survey of the project Automath , 1980 .

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

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

[14]  D. K. Harrison,et al.  Proceedings of the Conference on Categorical Algebra , 1966 .

[15]  Gordon D. Plotkin,et al.  Logical frameworks , 1991 .

[16]  Dusko Pavlovic,et al.  Categorical interpolation: descent and the Beck−Chevalley condition without direct images , 1991 .

[17]  Thierry Coquand.  Categories of Embeddings , 1988, LICS.

[18]  F. William Lawvere,et al.  Adjointness in Foundations , 1969 .

[19]  Jean Giraud,et al.  Cohomologie non abélienne , 1971 .

[20]  J. Roger Hindley,et al.  To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus, and Formalism , 1980 .

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

[22]  John C. Reynolds,et al.  Polymorphism is not Set-Theoretic , 1984, Semantics of Data Types.

[23]  Dusko Pavlovic,et al.  Constructions and Predicates , 1991, Category Theory and Computer Science.

[24]  Rohit Parikh.  Proceedings of the Fourth Annual Symposium on Logic in Computer Science (LICS '89), Pacific Grove, California, USA, June 5-8, 1989 , 1989, LICS.

[25]  Dusko Pavlovic,et al.  Maps I: relative to a factorisation system , 1995 .

[26]  Robert Paré,et al.  Colimits in Topoi , 1974 .

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

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

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

[30]  J. Hyland,et al.  The Discrete Objects in the Effective Topos , 1990 .