# Fundamental pro-groupoids and covering projections

Fundamenta Mathematicae (1998)

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

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topHernández-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 -

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