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Introduction to distributive categories

  • J. R. B. Cockett (a1)
Abstract

Distributive category theory is the study of categories with two monoidal structures, one of which “distributes” over the other in some manner. When these are the product and coproduct, this distribution is taken to be the law

which asserts that the obvious canonical map has an inverse. A distributive category is here taken to mean a category with finite products and binary coproducts such that this law is satisfied.

In any distributive category the coproduct of the final object with itself, 1 + 1, forms a boolean algebra. Thus, maps into 1 + 1 provide a boolean logic: if each such map recognizes a unique subobject, the category is a recognizable distributive category. If, furthermore, the category is such that these recognizers classify detachable subobjects (coproduct embeddings), it is an extensive distributive category.

Extensive distributive categories can be approached in various ways. For example, recognizable distributive categories, in which coproducts are disjoint or all preinitials are isomorphic, are extensive. Also, a category X having finite products and binary coproducts satisfying the slice equation (due to Schanuel and Lawvere) is extensive. This paper describes a series of embedding theorems. Any distributive category has a full faithful embedding into a recognizable distributive category. Any recognizable distributive category can be "solidified" faithfully to produce an extensive distributive category. Any extensive distributive category can be embedded into a topos.

A peculiar source of extensive distributive categories is the coproduct completion of categories with familial finite products. In particular, this includes the coproduct completion of cartesian categories, which is serendipitously, therefore, also the distributive completion. Familial distributive categories can be characterized as distributive categories for which every object has a finite decomposition into indecomposables.

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References
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Barr M. and Wells C. (1985) Toposes, triples and theories. Grundlehren der mathematischen Wis-senschaften 278, Springer-Verlag, Berlin, Heidelberg, New York.
Cockett J. R. B. (1989) Distributive logic, University of Tennessee, Department of Computer Science, Technical Report CS-89–01.
Cockett J. R. B. (1991) Conditional control is not quite categorical control. In: Birtwistle G. (ed.) IV Higher Order Workshop, Banff 1990, Workshops in Computing 190217.
Diers Y. (1985) Categories of boolean sheaves of simple algebras. Springer-Verlag Lecture Notes in Mathematics 1187.
Dress A. W. M. (1973) Contributions to the theory of induced representations. In: Algebraic K-Theory II. Springer-Verlag Lecture Notes in Mathematics 342 183240.
Gabriel P. and Zisman M. (1967) Calculus of fractions and homotopy theory. Ergebnisse der Mathematik und Ihre Grenzgebiete, New Series, 35, Springer-Verlag, Berlin, Heidelberg, New York.
Grothendiek A. and Verdier J. L. (1972) Théorie des topos, (SGA 4, exposé I-IV). Second edition. Springer- Verlag Lecture Notes in Mathematics 269270.
Johnstone P. T. (1977) Topos theory, Academic Press, London, New York, San Francisco.
Lintner H. (1976) A remark on Mackey-functors. Manuscripta math. 18 275278.
Mac Lane S. (1971) Categories for the working mathematician, Springer-Verlag, Berlin, Heidelberg, New York.
Schumacher D. and Street R. H. (1988) Some Parameterized Categorical Concepts. Communications in Algebra 16(11) 23132347.
Sydney Category Seminars (1988) Sydney category seminar abstracts.
Sydney Category Seminars (1990) Sydney category seminar abstracts.
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Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
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