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The fact that categorial concepts are used despite the difficulties in giving them satisfactory set-theoretical foundations leads to the idea of studying first of all the of these concepts, their pragmatic aspect. More specifically, workers in the field seem not to ask whether the concepts legitimate in the sense that they refer to some objects which exist but whether they are . We have to analyze, hence, what it means for a use to be legitimate.

The concepts category, functor, and natural transformation were introduced (in reverse order) during the early 1940s by Samuel Eilenberg and Saunders Mac Lane, aiming at resolving certain conceptual problems in algebraic topology. Before explaining in detail the points concerned, it might be useful to develop some hypotheses. In view of the intention of the category concept, the idea comes to mind that category theory should have emerged from some study of mappings. In algebraic topology, there was indeed a strong tendency beginning in the 1920s to study mappings, as exemplified in the Lefschetz fixed point theorem and the study of homotopy classes of mappings initiated by Brouwer, Hopf and others.

Before around 1955, CT was almost exclusively used in algebraic topology and served there, at least up to Eilenberg and Steenrod, mainly as a conceptual (or linguistic) framework for the organization of a knowledge system. Arrows and arrow composition played an important role there, and the new framework emphasizing these aspects changed considerably the organization of topology as a whole (compare [Volkert 2002] chapter 6), but this change was rather a shift of emphasis from problem solving to conceptual clarification than direct progress in solving the problems formerly considered as central in the discipline (as, for instance, the classification of 3-manifolds). In the domain of algebraic topology, it was Kan who entered first a level of conceptual innovation on which CT came to serve also as a means of . This means that results in the topological context have been obtained by the application of results established on the categorial level—results deeper than those available using solely the base concepts of category theory, ., results the proof (and already the formulation) of which used new, more involved concepts like adjoint functors and the general limit concept.

In the sequel to his work on sheaf cohomology, Grothendieck in the period 1958–1970 undertakes a complete renewal of the conceptual bases of algebraic geometry. CT intervenes at every stage of this conceptual work, for instance in the introducion of the fundamental concepts of scheme and topos and in important characteristics of Grothendieck’s methodology (descent, relativization). All these innovations are tested, for instance, in the case of the so-called Weil conjectures, but in this case, Grothendieck’s approach yielded only partial results.

There has been considerable development of CT from the beginning to the end of the period under consideration, often in interaction with the applications. While particular conceptual achievements often are mentioned in the context of the original applications in chapters 2–4, it is desirable to present also some diachronical, organized overview of these developments. This will be done in the present chapter. Some parts of this chapter have the character of a commented subject index ordered according to systematic criteria and hence are more appropriate for reference purposes than for direct reading; but others contain important bricks in the wall of my overall interpretation.

The possibilities and problems attendant on the construction of a set-theoretical foundation for CT and the relevance of such foundations have been subject to extensive debates for many years. In this chapter, I will consider the historical development of these debates. So far, a detailed discussion of this subject matter is absent from the historical writing on CT; I do not know whether this lack of interest is but one more expression of the profound indifference exhibited by most mainstream mathematicians towards set-theoretical foundations of mathematics in general and of category theory in particular, or whether it indicates merely that the problem is an open one and hence in a trivial sense does not yet admit a conclusive historical treatment. Anyway, in a historical and philosophical analysis of a theory, one is not supposed to parrot uncritically the prejudices of the workers in the field. To the contrary, such prejudices are to be analyzed with priority; questions like: What are the motives underlying them? What basic convictions of the people active in the field do they reveal? What have been their consequences for the development of the theory and of the debates concerning it? The answers to these questions are most important both for an understanding of the theory’s history and for its philosophical interpretation.

Mathematics in the 20th century was marked by an extensive discussion of its foundations. The subdisciplines set theory, model theory and proof theory emerged at least partly as scientific methods for foundational research. The task of giving mathematics a foundation was taken up by the mathematicians themselves as well as by philosophers.

In the introduction, I said that the way mathematicians work with categories reveals interesting insights into their implicit philosophy (how they interpret mathematical objects, methods, and the fact that these methods work). On the grounds of the evidence presented, we can now observe that the history of CT shows a switch in this interpretation: at first, objects of categories were always interpreted as sets (as in the case of the representations of Eilenberg and Mac Lane; see section 5.4.4.2); the purely formal character of categorial concepts was acknowledged but not consequently stressed. What was stressed positively is that concerning the categories themselves, the “all” is to be taken seriously <#20 p.237>. One was not aware of the fact that the difference between set theory and formal CT allowed for an interpretation of CT beyond sets (as far as the objects are concerned). This changed with Grothendieck on the one hand and Buchsbaum on the other. Grothendieck was interested in infinitistic argumentation and tried to extend the scope of the (formal) concept of set. Buchsbaum was interested in formal purity. The result of this development is a new technical intuition. This paradigm change took a different shape in the American and the French community, respectively.