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Three themes in the work of Charles Ehresmann: local-to-global; groupoids; higher dimensions. (English) Zbl 1123.01015

Kubarski, Jan (ed.) et al., Geometry and topology of manifolds. The mathematical legacy of Charles Ehresmann. On the occasion of the hundredth anniversary of his birthday. Proceedings of the conference, Bȩdlewo, Poland, May 8–15, 2005. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 76, 51-63 (2007).
The author reports on interactions of his own work with Charles Ehresmann’s achievements, mainly the latter’s book on Catégories et Structures, published in 1965 (see Zbl 0192.09803). Ehresmann was one of the first members of the Bourbaki group, but actually stepped further than Bourbaki himself: To find the structure of everything! A local-to-global theme! Brown considers as one of the major triumphs of category theory during the 20th century the fact of making progress in unifying mathematics along various areas of mathematics.
Among new concepts studied by Ehresmann range differentiable categories and groupoids as well as multiple categories. Brown stresses the necessity of considering higher dimensions.
The article is quite comprehensive, whereas initially Brown found Ehresmann’s book difficult to understand.
Bearing in mind that the notion of structure is closely related to the notion of analogy, the author expresses his personal conviction:
A description of a new structure is in some sense a development of part of a new language. The aim of doing this contrasts with that of many, who feel that the development of mathematics is mainly guided by the solution of famous problems.
For the entire collection see [Zbl 1112.57300].

MSC:

01A60 History of mathematics in the 20th century
18-03 History of category theory
55-03 History of algebraic topology
22-03 History of topological groups

Citations:

Zbl 0192.09803
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