(1982). Fuzzy Sets and Systems — Theory and Applications. Journal of the Operational Research Society: Vol. 33, No. 2, pp. 198-198.

/pdf/fuzzy-sets-and-systems-theory-and-applications-a76qq6eile.pdf

Fuzzy Sets and Systems: Theory and Applications

Received by the editors 2004-10-30. Transmitted by Steve Lack, Ross Street and RJ Wood. Reprint published on 2005-04-23. Several typographical errors corrected 2012-05-13. 2000 Mathematics Subject Classification: 18-02, 18D10, 18D20.

/pdf/basic-concepts-of-enriched-category-theory-f320y2evfi.pdf

Basic concepts of enriched category theory

Conditional rewriting logic as a unified model of concurrency

There is a population explosion among the logical systems used in computing science. Examples include first-order logic, equational logic, Horn-clause logic, higher-order logic, infinitary logic, dynamic logic, intuitionistic logic, order-sorted logic, and temporal logic; moreover, there is a tendency for each theorem prover to have its own idiosyncratic logical system. The concept of institution is introduced to formalize the informal notion of “logical system.” The major requirement is that there is a satisfaction relation between models and sentences that is consistent under change of notation. Institutions enable abstracting away from syntactic and semantic detail when working on language structure “in-the-large”; for example, we can define language features for building large logical system. This applies to both specification languages and programming languages. Institutions also have applications to such areas as database theory and the semantics of artificial and natural languages. A first main result of this paper says that any institution such that signatures (which define notation) can be glued together, also allows gluing together theories (which are just collections of sentences over a fixed signature). A second main result considers when theory structuring is preserved by institution morphisms. A third main result gives conditions under which it is sound to use a theorem prover for one institution on theories from another. A fourth main result shows how to extend institutions so that their theories may include, in addition to the original sentences, various kinds of constraint that are useful for defining abstract data types, including both “data” and “hierarchy” constraints. Further results show how to define institutions that allow sentences and constraints from two or more institutions. All our general results apply to such “duplex” and “multiplex” institutions.

Institutions: abstract model theory for specification and programming

Although numerous contributions from divers authors, over the past fifteen years or so, have brought enriched category theory to a developed state, there is still no connected account of the theory, or even of a substantial part of it. As the applications of the theory continue to expand - some recent examples are given below - the lack of such an account is the more acutely felt. The present book is designed to supply the want in part, by giving a fairly complete treatment of the limited area to which the title refers. The basic concepts of category theory certainly include the notion of functor-category, of limit and colimit, of Kan extension, and of density; with their applications to completions, perhaps including those relative completions given by categories of algebras for limit-defined theories. If we read 'V-category' for 'category' here, this is essentially the list of our chapter-headings below, after the first chapter introducing V-categories.  In fact our scope is wider than this might suggest; for what we give is also a selfcontained account of basic category theory as described above, assuming as prior knowledge only the most elementary categorical concepts, and treating the ordinary and enriched cases together from Chapter 3 on.

/pdf/basic-concepts-of-enriched-category-theory-1mn867zj.pdf

The analogy between dist (a, b)+dist (b, c)≥dist (a, c) and hom (A, B) ⊗ hom (B, C)→hom (A, C) is rigorously developed to display many general results about metric spaces as consequences of a «generalized pure logic» whose «truth-values» are taken in an arbitrary closed category.

Metric spaces, generalized logic, and closed categories

In this article we see how already in 1967 category theory had made explicit a number of conceptual advances that were entering into the everyday practice of mathematics. For example, local Galois connections (in algebraic geometry, model theory, linear algebra, etc.) are globalized into functors, such as Spec, carrying much more information. Also, “theories” (even when presented symbolically) are viewed explicitly as categories; so are the background universes of sets that serve as the recipients for models. (Models themselves are functors, hence preserve the fundamental operation of substitution/composition in terms of which the other logical operations can be characterized as local adjoints.) My 1963 observation (referred to by Eilenberg and Kelly in La Jolla, 1965), that cartesian closed categories serve as a common abstraction of type theory and propositional logic, permits an invariant algebraic treatment of the essential problem of proof theory, though most of the later work by proof theorists still relies on presentation-dependent formulations. This article sums up a stage of the development of the relationship between category theory and proof theory. (For more details see Proceedings of the AMS Symposium on Pure Mathematics XVII (1970), pp. 1–14, and Marcel Dekker, Lecture Notes in Pure and Applied Mathematics, no. 180 (1996), pp. 181–189.) The main problem addressed by proof theory arises from the existential quantifier in “there exists a proof. . . ”. The strategy to interpret proofs themselves as structures had been discussed by Kreisel; however, the influential “realizers” of Kleene are not yet the usual mathematical sort of structures. Inspired by Lauchli’s 1967 success in finding a completeness theorem for Heyting predicate calculus lurking in the category of ordinary permutations, I presented, at the 1967 AMS Los Angeles Symposium on Set Theory, a common functorization of several geometrical structures, including such proof-theoretic structures. As Hyperdoctrines, those structures are described in the Proceedings of the

/pdf/adjointness-in-foundations-58jtb0r9m9.pdf

Adjointness in Foundations

In the last 60 years, the use of the notion of category has led to a remarkable unification and simplification of mathematics. Conceptual Mathematics, Second Edition, introduces the concept of 'category' for the learning, development, and use of mathematics, to both beginning students and general readers, and to practicing mathematical scientists. The treatment does not presuppose knowledge of specific fields, but rather develops, from basic definitions, such elementary categories as discrete dynamical systems and directed graphs; the fundamental ideas are then illuminated by examples in these categories.

Conceptual Mathematics: A First Introduction to Categories

Received by the editors 2005-04-01. Transmitted by M. Hyland, A. Kock, R. Rosebrugh. Reprint published on 2005-05-23. 2000 Mathematics Subject Classification: XXX.

https://www.pnas.org/content/pnas/52/6/1506.full.pdf

An elementary theory of the category of sets.

Foreword 1. Abstract sets and mappings 2. Sums, monomorphisms and parts 3. Finite inverse limits 4. Colimits, epimorphisms and the axiom of choice 5. Mapping sets and exponentials 6. Summary of the axioms and an example of variable sets 7. Consequences and uses of exponentials 8. More on power sets 9. Introduction to variable sets 10. Models of additional variation Appendices Bibliography.

F. William Lawvere

Papers

Metric spaces, generalized logic, and closed categories

Adjointness in Foundations

Conceptual Mathematics: A First Introduction to Categories

An elementary theory of the category of sets.

Sets for Mathematics