Charles Ehresmann. Introduction of Categories

Why have categories not been discovered earlier?

It is an interesting question. To answer it, we have to immerse to ourselves in the atmosphere of the first half of the 20th century.  I am not a historian, but I have some ideas of what it was through my discussions with Charles Ehresmann. Mathematics was much more ‘concrete’, tackling problems without the introduction of many new concepts. If you look at the works of most mathematicians of that time (such as Poincare, Lebesgue, Elie Cartan, Herman Weyl?), their preoccupations and style are very different from that of modern works.

Formalism in mathematics began to develop in Germany with Hilbert’s program in 1920, whose aim was to give a solid conceptual foundation to mathematics. The mathematicians of Charles’ generation, in particular those who created the Bourbaki group in the late 30’s, were much inspired by this approach, for instance by the book “Moderne Algebra” (2 volumes, Springer 1930) by van der Waerden, written in a formal style.  The idea of Bourbaki was to develop a new kind of mathematics based on a general notion of ‘mathematical structure’. It is not surprising that Eilenberg was a member of Bourbaki.

It has been difficult for more ‘classical’ mathematicians to accept new abstract notions of which they did not see the interest in their own work. I recall Bouligand in the 50’s mocking the need for new notions such as fiber bundles or foliated manifold. Thus it took a long time for the interest of categories to be seen outside homological algebra. It is essentially in the late fifties that 3 main papers (by Kan on adjoint functors, by Grothendieck on abelian categories, and by Ehresmann on species of structures and differentiable categories) paved the way for applications in other mathematical domains, allowing for a larger diffusion of categories. However up to now, there has been a fierce resistance against them by some people looking at them as “abstract nonsense”.

To introduce categories through an axiomatisation of the properties of functions, as proposed by Posina, seems a good approach.  It is not so different from the approach of Charles who arrived to categories through groupoids, considered as a generalization of pseudogroups of transformations, themselves generalizing groups of transformations (cf. the site http://vbm-ehr.pagesperso-orange.fr/ChEh/ and my 2 general conferences in it).  Another approach is through graph theory, a category being defined as a directed (multi-)graph equipped with a composition associating to a path of the graph a unique composite arrow with the same extremities.

Andree Ehresman