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The History of Categorical Logic 1963–1977

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The History of Categorical Logic 1963–1977 THE HISTORY OF CATEGORICAL LOGIC 1963{1977 Jean-Pierre Marquis and Gonzalo E. Reyes Categorical logic, as its name indicates, is logic in the setting of category the- ory. But this description does not say much. Most readers would probably find more instructive to learn that categorical logic is algebraic logic, pure and simple. It is logic in an algebraic dressing. Just as algebraic logic encodes propositional logic in its different guises (classical, intuitionistic, etc.) by their Lindenbaum- Tarski algebras (Boolean algebras, Heyting algebras and so on), categorical logic encodes first-order and higher-order logics (classical, intuitionistic, etc.) by cate- gories with additional properties and structure (Boolean categories, Heyting cate- gories and so on). Thus, from the purely technical point of view, categorical logic constitutes a generalization of the algebraic encoding of propositional logic to first- order, higher-order and other logics. Furthermore, we shall present and discuss arguments (given by the main actors) to show that this encoding constitutes the correct generalization of the well-known algebraic encoding of propositional logics by the Lindenbaum-Tarski algebras. The proper algebraic structures are not only categories, but also morphisms between categories, mainly functors and more specially adjoint functors. A key example is provided by the striking fact that quantifiers, which were the stumbling block to the proper algebraic generalization of propositional logic, can be seen to be adjoint functors and thus entirely within the categorical framework. As is usually the case when algebraic techniques are imported and developed within a field, e.g. geometry and topology, vast generalizations and unification become possible. Furthermore, unexpected concepts and results show up along the way, often allowing a better understanding of known concepts and results. Categorical logic is not merely a convenient tool or a powerful framework. Again, as is usually the case when algebraic techniques are imported and used in a field, the very nature of the field has to be thought over. Furthermore, var- ious results shed a new light on what was assumed to be obvious or, what turns out to be often the same on careful analysis, totally obscure. Thus, categorical logic is philosophically relevant in more than one way. The way it encodes logical concepts and operations reveals important, even essential, aspects and properties of these concepts and operations. Again, as soon as quantifiers are seen as ad- joint functors, the traditional question of the nature of variables in logic receives a satisfactory analysis. Furthermore, many results obtained via categorical techniques have clear and essential philosophical implications. The systematic development of higher-order 2 Jean-Pierre Marquis and Gonzalo E. Reyes logic, type theory under a different name, and various completeness theorems are the most obvious candidates. But there is much more. Many important questions concerning the foundations of mathematics and the very nature of mathematical knowledge are inescapable. In particular, issues related to abstraction and the nature of mathematical objects emerges naturally from categorical logic. This paper covers the period that can be qualified as the birth and the consti- tution of categorical logic, that is the time span between 1963 and 1977. No one will deny that categorical logic started with Bill Lawvere's Ph.D. thesis written in 1963 under S. Eilenberg's supervision and widely circulated afterwards. (It is now available on-line on the TAC web-site.) In his thesis, Lawvere offered a categorical version of algebraic theories. He also suggested that the category of categories could be taken as a foundation for mathematics and that sets could be analyzed in a categorical manner. In the years that followed, Lawvere tried to extend his analysis and sketched a categorical version of first-order theories under the name of elementary theories. Then, in 1969, in collaboration with Myles Tierney, Law- vere introduced the notion of an elementary topos, making an explicit connection with higher-order logic and type theories. Both Lawvere and Tierney were aim- ing at an elementary, that is first-order, axiomatic presentation of what are now called Grothendieck toposes, a special type of categories introduced by Alexandre Grothendieck in the context of algebraic geometry and sheaf theory. Soon after, connections with intuitionistic analysis, recursive functions, completeness theo- rems for various logical systems, differential geometry, constructive mathematics were made. We have decided to end our coverage in 1977 for the following rea- sons. First, we had to stop somewhere, otherwise we would have to write a book. Second, and this is a more serious reason, three independent events in 1977 mark more or less a turning point in the history of categorical logic. First, the book First-Order Categorical Logic, by Makkai and Reyes appears, a book that more or less codifies the work done by the Montreal school in the period 1970-1974 and now constitutes the core of categorical first-order logic. Second, the same year witnesses the publication of Johnstone's Topos Theory, the first systematic and comprehensive presentation of topos theory as it was known in 1974-75. (The reader should compare this edition with Johnstone's recent Sketches of an Ele- phant, a comprehensive reference on topos theory in three volumes.) Third, 1977 was also the year of the Durham meeting on applications of sheaf theory to logic, algebra and analysis, whose proceedings were published in 1979. We submit that around the end of the nineteen seventies, categorical logic was on firm ground and could be developed in various directions, which is precisely what happened, from theoretical computer science, modal logic and other areas. The usual warnings, caveat and apologies are now necessary. It is impossible to cover, even in a long article such as this one and for such a short time period, all events involved in the history of categorical logic. This paper is but a first attempt at a more precise and detailed history of a complicated and fascinating period in the history of ideas. We hope that it will stimulate more work on the topic. We hasten to add that it also reflects our interests and (hopefully not too limited) The History of Categorical Logic: 1963{1977 3 knowledge of the field. It is our hope that it will nonetheless be useful to logicians and philosophers alike. We sincerely apologize to mathematicians, logicians and philosophers whose names ought to have appeared in this history but have not because of our ignorance. 1 THE BIRTH OF CATEGORY THEORY AND ITS EARLY DEVELOPMENTS Category theory as a discipline in itself and was born in the context of algebraic topology in the nineteen forties. We will briefly sketch the history of category theory before the advent of categorical logic and rehearse the fundamental notions of the theory required for the exposition of the following sections. 1.1 Category theory: its origins We will here only rehearse the ingredients required for the history of categorical logic. The reader is referred to [Landry and Marquis, 2005], [Marquis, 2006] and [Kr¨omer, 2007] for more details. Category theory made its official public appearance in 1945 in the paper enti- tled \General Theory of Natural Equivalences" written by Samuel Eilenberg and Saunders Mac Lane. This “off beat" and \far out" paper, as Mac Lane came to qualify it later [Mac Lane, 2002, 130], was meant to provide an autonomous framework for the concept of natural transformation, a concept whose generality, pervasiveness and usefulness had become clear to both of them during their col- laboration on the clarification of an unsuspected link between group extensions and homology groups. Such a general, pervasive and conceptually useful notion seemed to deserve a precise, rigorous, systematic and abstract treatment. Eilenberg and Mac Lane decided to devise an axiomatic framework in which the notion of natural transformation would receive an entirely general and autonomous definition. This is where categories came in. Informally, a natural transformation is a family of maps that provides a systematic \translation" or a \deformation" between two systems of interrelated entities within a given framework. But in order to give a precise definition of natural transformations, one needs to clarify the systematic nature of these deformations, that is, one has to specify what these deformations depend upon and how they depend upon it. Eilenberg and Mac Lane introduced what they called functors | the term was borrowed from Carnap | so that one could say between what the natural transformations were acting: a natural transformation is a family of maps between functors. Clearly, one has to define the notion of a functor: the concept of category was tailored for that purpose. The systematic nature of natural transformations was also made clear by categories themselves. Thus categories were introduced in 1945 and, as Mac Lane reported (see [Mac Lane, 2002]), Eilenberg believed that their paper would be the only paper written on \pure" category theory. 4 Jean-Pierre Marquis and Gonzalo E. Reyes As we have already mentioned, Eilenberg and Mac Lane gave a purely axiomatic definition of category in their original paper. It is worth mentioning that they explicitly avoided using a set-theoretical terminology and notation in the axioms themselves. Here is how their definition unfolds (only with a slightly different notation): A category C is an aggregate of abstract elements X, called the objects of the category, and abstract elements f, called mappings of the category. Certain pairs of mappings f; g of C determine uniquely a product mapping g ◦ f, satisfying the axioms C1, C2, C3 below. Corresponding to each object X of C, there is a unique mapping, denoted by 1X satisfying the axioms C4 and C5. The axioms are: C1 The triple product h ◦ (g ◦ f) is defined if and only if (h ◦ g) ◦ f is defined.
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