The set of paths in a space and its algebraic structure. A historical account

Abstract : The present paper provides a test case for the significance of the historical category “structuralism” in the history of modern mathematics. We recapitulate the various approaches to the fundamental group present in Poincaré’s work and study how they were developed by the next generations in more “structuralist” manners. By contrasting this development with the late introduction and comparatively marginal use of the notion of fundamental groupoid and the even later consideration of equivalence relations finer than homotopy of paths (their implicit presence from the outset in the proof of the group property of the fundamental group notwithstanding), we encounter “delay” phenomena which are explained by focussing on the actual uses of a concept in mathematical discourse.
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Ralf Krömer. The set of paths in a space and its algebraic structure. A historical account. Annales de la Faculté des Sciences de Toulouse. Mathématiques., Université Paul Sabatier _ Cellule Mathdoc 2013, 22 (5), pp.915 - 968. ⟨10.5802/afst.1392⟩. ⟨hal-01867403⟩

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