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

Ralf Krömer

Annales de la faculté des sciences de Toulouse Mathématiques (2013)

  • Volume: 22, Issue: 5, page 915-968
  • ISSN: 0240-2963

Abstract

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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.

How to cite

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Krömer, Ralf. "The set of paths in a space and its algebraic structure. A historical account." Annales de la faculté des sciences de Toulouse Mathématiques 22.5 (2013): 915-968. <http://eudml.org/doc/275292>.

@article{Krömer2013,
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.},
author = {Krömer, Ralf},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
language = {eng},
month = {12},
number = {5},
pages = {915-968},
publisher = {Université Paul Sabatier, Toulouse},
title = {The set of paths in a space and its algebraic structure. A historical account},
url = {http://eudml.org/doc/275292},
volume = {22},
year = {2013},
}

TY - JOUR
AU - Krömer, Ralf
TI - The set of paths in a space and its algebraic structure. A historical account
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2013/12//
PB - Université Paul Sabatier, Toulouse
VL - 22
IS - 5
SP - 915
EP - 968
AB - 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.
LA - eng
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ER -

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