Categories in geometry, algebra and logic. (English) Zbl 0831.18001

MacLane, Saunders

Math. Jap. 42, No. 1, 169-178 (1995).

The author, who created category theory together with S. Eilenberg, lists here some basic concepts of this topic and mentions some applications. The various notions considered are those of category, functor, natural transformation, limit, adjoint functors, sheaf, topos, cartesian closed category, monoidal category, 2-category, synthetic differential geometry, classifying topos. The length of the paper – only ten pages – forces the author to focus on the “flavour” of the notions, not on precise definitions or the statement of deep theorems. This tendency is sometimes exagerated: for example the way abelian categories are introduced almost forces the confusion with additive categories. On the other hand some classes of categories which played an important role in the development of the theory are not even mentioned: this is the case for regular, algebraic, monadic, locally presentable, accessible and filtered categories. The list of applications which are emphasized mentions homology, \(K\)-theory, free constructions on generators, Galois connections, Boolean algebras, logic, cohomology, \(\lambda\)-calculus, quantum fields, homotopy, geometry. In conclusion, this paper gives an excellent flavour of some basic achievements of category theory, inviting the reader to learn more about them via the detailed bibliography which is provided.

Reviewer: F.Borceux (Louvain-La-Neuve)

MSC:

18-03	History of category theory
01A60	History of mathematics in the 20th century

Keywords:

category theory; adjoint functors; sheaf; cartesian closed category; monoidal category; synthetic differential geometry; classifying topos; bibliography