Taking Categories Seriously

The relation between teaching and research is partly embodied in simple general concepts which can guide the elaboration of examples in both. Notions and constructions, such as the spectral analysis of dynamical systems, have important aspects that can be understood and pursued without the complication of limiting the models to specific classical categories. Pursuing that idea leads to a dynamical objectification of Dedekind reals which is particularly suited to the simple identification of metric spaces as enriched categories over a special closed category. Rejecting the complacent description of that identification as a mere analogy or amusement, its relentless pursuit [8] is continued, revealing convexity and geodesics as concepts having a definite meaning over any closed category. Along the way various hopefully enlightening exercises for students (and possible directions for research) are inevitably encountered: (1) an explicit treatment of the contrast between multiplication and divisibility that, in inexorable functorial fashion, mutates into the adjoint relation between autonomous and non-autonomous dynamical systems; (2) the role of commutation relations in the contrast between equilibria and orbits, as well as in qualitative distinctions between extensions of Heyting logic; (3) the functorial contrast between translations and rotations (as appropriately defined) in an arbitrary non symmetric metric space. The theory of categories originated [1] with the need to guide complicated calculations involving passage to the limit in the study of the qualitative leap from spaces to homotopical/homological objects. Since then it is still actively used for those problems but also in algebraic geometry [2], logic and set theory [3], model theory [4], functional analysis [5], continuum physics [6], combinatorics [7], etc. In all these the categorical concept of adjoint functor has come to play a key role. Such a universal instrument for guiding the learning, development, and use of advanced mathematics does not fail to have its indications also in areas of school and college mathematics, in the most basic relationships of space and quantity and the calculations based on those relationships. In saying “take categories seriously”, I advocate noticing, cultivating, After the original manuscript was stolen, I was able to reconstruct, in time for inclusion in the proceedings of the 1983 Bogotá Workshop, this“sketch” (as Saunders Mac Lane’s review was to describe it). Interested teachers can elaborate this and similar material into texts for beginning students. For the present opportunity, I am grateful to Xavier Caicedo who gave permission to reprint, to Christoph Schubert for his expert transcription into TEX, and to the editors of TAC. Received by the editors 2004-11-02. Transmitted by M. Barr, R. Rosebrugh and R. F. C. Walters. Reprint published on 2005-03-23. 2000 Mathematics Subject Classification: 06D20, 18D23, 37B55, 34L05, 52A99, 53C22, 54E25, 54E40.