Fundamental pro-groupoids and covering projections

Luis Hernández-Paricio

Fundamenta Mathematicae (1998)

  • Volume: 156, Issue: 1, page 1-31
  • ISSN: 0016-2736

Abstract

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We introduce a new notion of covering projection E → X of a topological space X which reduces to the usual notion if X is locally connected. We use locally constant presheaves and covering reduced sieves to find a pro-groupoid π crs (X) and an induced category pro (π crs (X), Sets) such that for any topological space X the category of covering projections and transformations of X is equivalent to the category pro (π crs (X), Sets). We also prove that the latter category is equivalent to pro (π CX, Sets), where π CX is the Čech fundamental pro-groupoid of X. If X is locally path-connected and semilocally 1-connected, we show that π crs (X) is weakly equivalent to π X, the standard fundamental groupoid of X, and in this case pro (π crs (X), Sets) is equivalent to the functor category S e t s π X . If (X,*) is a pointed connected compact metrisable space and if (X,*) is 1-movable, then the category of covering projections of X is equivalent to the category of continuous π ˇ 1 ( X , * ) -sets, where π ˇ 1 ( X , * ) is the Čech fundamental group provided with the inverse limit topology.

How to cite

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Hernández-Paricio, Luis. "Fundamental pro-groupoids and covering projections." Fundamenta Mathematicae 156.1 (1998): 1-31. <http://eudml.org/doc/212259>.

@article{Hernández1998,
abstract = {We introduce a new notion of covering projection E → X of a topological space X which reduces to the usual notion if X is locally connected. We use locally constant presheaves and covering reduced sieves to find a pro-groupoid π crs (X) and an induced category pro (π crs (X), Sets) such that for any topological space X the category of covering projections and transformations of X is equivalent to the category pro (π crs (X), Sets). We also prove that the latter category is equivalent to pro (π CX, Sets), where π CX is the Čech fundamental pro-groupoid of X. If X is locally path-connected and semilocally 1-connected, we show that π crs (X) is weakly equivalent to π X, the standard fundamental groupoid of X, and in this case pro (π crs (X), Sets) is equivalent to the functor category $Sets^\{π X\}$. If (X,*) is a pointed connected compact metrisable space and if (X,*) is 1-movable, then the category of covering projections of X is equivalent to the category of continuous $\check\{π\}_1 (X,*)$-sets, where $\check\{π\}_1 (X,*)$ is the Čech fundamental group provided with the inverse limit topology.},
author = {Hernández-Paricio, Luis},
journal = {Fundamenta Mathematicae},
keywords = {covering projection; covering transformation; pro-groupoid, Čech fundamental pro-groupoid; covering reduced sieve; locally constant presheaf; category of fractions; subdivision; fundamental groupoid; Čech fundamental group; G-sets; continuous G-sets; pro-groupoid; Čech fundamental pro-groupoid},
language = {eng},
number = {1},
pages = {1-31},
title = {Fundamental pro-groupoids and covering projections},
url = {http://eudml.org/doc/212259},
volume = {156},
year = {1998},
}

TY - JOUR
AU - Hernández-Paricio, Luis
TI - Fundamental pro-groupoids and covering projections
JO - Fundamenta Mathematicae
PY - 1998
VL - 156
IS - 1
SP - 1
EP - 31
AB - We introduce a new notion of covering projection E → X of a topological space X which reduces to the usual notion if X is locally connected. We use locally constant presheaves and covering reduced sieves to find a pro-groupoid π crs (X) and an induced category pro (π crs (X), Sets) such that for any topological space X the category of covering projections and transformations of X is equivalent to the category pro (π crs (X), Sets). We also prove that the latter category is equivalent to pro (π CX, Sets), where π CX is the Čech fundamental pro-groupoid of X. If X is locally path-connected and semilocally 1-connected, we show that π crs (X) is weakly equivalent to π X, the standard fundamental groupoid of X, and in this case pro (π crs (X), Sets) is equivalent to the functor category $Sets^{π X}$. If (X,*) is a pointed connected compact metrisable space and if (X,*) is 1-movable, then the category of covering projections of X is equivalent to the category of continuous $\check{π}_1 (X,*)$-sets, where $\check{π}_1 (X,*)$ is the Čech fundamental group provided with the inverse limit topology.
LA - eng
KW - covering projection; covering transformation; pro-groupoid, Čech fundamental pro-groupoid; covering reduced sieve; locally constant presheaf; category of fractions; subdivision; fundamental groupoid; Čech fundamental group; G-sets; continuous G-sets; pro-groupoid; Čech fundamental pro-groupoid
UR - http://eudml.org/doc/212259
ER -

References

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