Brief Historical Sketch

It is difficult to do justice to the short but intricate history of the field. In particular it is not possible to mention all those who have contributed to its rapid development. With this word of caution out of the way, we will look at some of the main historical threads.

Categories, functors, natural transformations, limits and colimits appeared almost out of nowhere in a paper by Eilenberg & Mac Lane (1945) entitled “General Theory of Natural Equivalences.” We say “almost,” because their earlier paper (1942) contains specific functors and natural transformations at work, limited to groups. A desire to clarify and abstract their 1942 results led Eilenberg & Mac Lane to devise category theory. The central notion at the time, as their title indicates, was that of natural transformation. In order to give a general definition of the latter, they defined functor, borrowing the term from Carnap, and in order to define functor, they borrowed the word ‘category’ from the philosophy of Aristotle, Kant, and C. S. Peirce, but redefining it mathematically.

After their 1945 paper, it was not clear that the concepts of category theory would amount to more than a convenient language; this indeed was the status quo for about fifteen years. Category theory was employed in this manner by Eilenberg & Steenrod (1952), in an influential book on the foundations of algebraic topology, and by Cartan & Eilenberg (1956), in a ground breaking book on homological algebra. (Curiously, although Eilenberg & Steenrod defined categories, Cartan & Eilenberg simply assumed them!) These books allowed new generations of mathematicians to learn algebraic topology and homological algebra directly in the categorical language, and to master the method of diagrams. Indeed, without the method of diagram chasing, many results in these two books seem inconceivable, or at the very least would have required a considerably more intricate presentation.

The situation changed radically with Grothendieck's (1957) landmark paper entitled “Sur quelques points d'algèbre homologique”, in which the author employed categories intrinsically to define and construct more general theories which he (Grothendieck 1957) then applied to specific fields, e.g., to algebraic geometry. Kan (1958) showed that adjoint functors subsume the important concepts of limits and colimits and could capture fundamental concepts in other areas (in his case, homotopy theory).

At this point, category theory became more than a convenient language, by virtue of two developments.

1. Employing the axiomatic method and the language of categories, Grothendieck (1957) defined in an abstract fashion types of categories, e.g., additive and Abelian categories, showed how to perform various constructions in these categories, and proved various results about them. In a nutshell, Grothendieck showed how to develop part of homological algebra in an abstract setting of this sort. From then on, a specific category of structures, e.g., a category of sheaves over a topological space X, could be seen as a token of an abstract category of a certain type, e.g., an Abelian category. One could therefore immediately see how the methods of, e.g., homological algebra could be applied to, for instance, algebraic geometry. Furthermore, it made sense to look for other types of abstract categories, ones that would encapsulate the fundamental and formal aspects of various mathematical fields in the same way that Abelian categories encapsulated fundamental aspects of homological algebra.
2. Thanks in large part to the efforts of Freyd and Lawvere, category theorists gradually came to see the pervasiveness of the concept of adjoint functors. Not only does the existence of adjoints to given functors permit definitions of abstract categories (and presumably those which are defined by such means have a privileged status) but as we mentioned earlier, many important theorems and even theories in various fields can be seen as equivalent to the existence of specific functors between particular categories. By the early 1970's, the concept of adjoint functors was seen as central to category theory.

With these developments, category theory became an autonomous field of research, and pure category theory could be developed. And indeed, it did grow rapidly as a discipline, but also in its applications, mainly in its source contexts, namely algebraic topology and homological algebra, but also in algebraic geometry and, after the appearance of Lawvere's Ph. D thesis, in universal algebra. This thesis also constitutes a landmark in this history of the field, for in it Lawvere proposed the category of categories as a foundation for category theory, set theory and, thus, the whole of mathematics, as well as using categories for the study of the logical aspects of mathematics.

Over the course of the 1960's, Lawvere outlined the basic framework for an entirely original approach to logic and the foundations of mathematics. He achieved the following:

• Axiomatized the category of sets (Lawvere 1964) and of categories (Lawvere 1966);
• Gave a categorical description of theories that was independent of syntactical choices and sketched how completeness theorems for logical systems could be obtained by categorical methods (Lawvere 1967);
• Characterized Cartesian closed categories and showed their connections to logical systems and various logical paradoxes (Lawvere 1969);
• Showed that the quantifiers and the comprehension schemes could be captured as adjoint functors to given elementary operations (Lawvere 1966, 1969, 1970, 1971);
• Argued that adjoint functors should generally play a major foundational role through the notion of “categorical doctrines” (Lawvere 1969).

Meanwhile, Lambek (1968, 1969, 1972) described categories in terms of deductive systems and employed categorical methods for proof-theoretical purposes.

All this work culminated in another notion, thanks to Grothendieck and his school: that of a topos. Even though toposes appeared in the 1960's, in the context of algebraic geometry, again from the mind of Grothendieck, it was certainly Lawvere and Tierney's (1972) elementary axiomatization of a topos which gave impetus to its attaining foundational status. Very roughly, an elementary topos is a category possessing a logical structure sufficiently rich to develop most of “ordinary mathematics”, that is, most of what is taught to mathematics undergraduates. As such, an elementary topos can be thought of as a categorical theory of sets. But it is also a generalized topological space, thus providing a direct connection between logic and geometry. (For more on the history of categorical logic, see Marquis & Reyes 2012, Bell 2005.)

The 1970s saw the development and application of the topos concept in many different directions. The very first applications outside algebraic geometry were in set theory, where various independence results were recast in terms of topos (Tierney 1972, Bunge 1974, but also Blass & Scedrov 1989, Blass & Scedrov 1992, Freyd 1980, Mac Lane & Moerdijk 1992, Scedrov 1984). Connections with intuitionistic and, more generally constructive mathematics were noted early on, and toposes are still used to investigate models of various aspects of intuitionism and constructivism (Lambek & Scott 1986, Mac Lane & Moerdijk 1992, Van der Hoeven & Moerdijk 1984a, 1984b, 1984c, Moerdijk 1984, Moerdijk 1995a, Moerdijk 1998, Moerdijk & Palmgren 1997, Moerdijk & Palmgren 2002), Palmgren 2012. For more on the history of topos theory, see McLarty (1992).

More recently, topos theory has been employed to investigate various forms of constructive mathematics or set theory (Joyal & Moerdijk 1995, Taylor 1996, Awodey 2008), recursiveness, and models of higher-order type theories generally. The introduction of the so-called “effective topos” and the search for axioms for synthetic domain theory are worth mentioning (Hyland 1982, Hyland 1988, 1991, Hyland et al. 1990, McLarty 1992, Jacobs 1999, Van Oosten 2008, Van Oosten 2002 and the references therein). Lawvere's early motivation was to provide a new foundation for differential geometry, a lively research area which is now called “synthetic differential geometry” (Lawvere 2000, 2002, Kock 2006, Bell 1988, 1995, 1998, 2001, Moerdijk & Reyes 1991). This is only the tip of the iceberg; toposes could prove to be for the 21st century what Lie groups were to the 20th century.

From the 1980s to the present, category theory has found new applications. In theoretical computer science, category theory is now firmly rooted, and contributes, among other things, to the development of new logical systems and to the semantics of programming. (Pitts 2000, Plotkin 2000, Scott 2000, and the references therein). Its applications to mathematics are becoming more diverse, even touching on theoretical physics, which employs higher-dimensional category theory — which is to category theory what higher-dimensional geometry is to plane geometry — to study the so-called “quantum groups” and quantum field theory (Majid 1995, Baez & Dolan 2001 and other publications by these authors).

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