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 . 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 -sets, where is the Čech fundamental group provided with the inverse limit topology.
@article{bwmeta1.element.bwnjournal-article-fmv156i1p1bwm,
author = {Luis Hern\'andez-Paricio},
title = {Fundamental pro-groupoids and covering projections},
journal = {Fundamenta Mathematicae},
volume = {158},
year = {1998},
pages = {1-31},
zbl = {0906.55008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv156i1p1bwm}
}
Hernández-Paricio, Luis. Fundamental pro-groupoids and covering projections. Fundamenta Mathematicae, Tome 158 (1998) pp. 1-31. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv156i1p1bwm/
[00000] [A-M] M. Artin and B. Mazur, Etale Homotopy, Lecture Notes in Math. 100, Springer, Berlin, 1967.
[00001] [E-H] D. A. Edwards and H. M. Hastings, Čech and Steenrod Homotopy Theories with Applications to Geometric Topology, Lecture Notes in Math. 542, Springer, 1970. | Zbl 0334.55001
[00002] [F1] R. H. Fox, On shape, Fund. Math. 74 (1972), 47-71.
[00003] [F2] R. H. Fox, Shape theory and covering spaces, in: Lecture Notes in Math. 375, Springer, 1974, 71-90.
[00004] [G-Z] P. Gabriel and M. Zisman, Calculus of Fractions and Homotopy Theory, Springer, Heidelberg, 1967. | Zbl 0186.56802
[00005] [God] C. Godbillon, Éléments de Topologie Algébrique, Hermann, Paris, 1971.
[00006] [Go] M. Golasiński, Homotopies of small categories, Fund. Math. 114 (1981), 209-217. | Zbl 0485.55014
[00007] [Gro] A. Grothendieck, Revêtements Etales et Groupe Fondamental (SGA 1), Lecture Notes in Math. 222, Springer, Berlin, 1971.
[00008] [H] L. J. Hernández, Applications of simplicial M-sets to proper and strong shape theories, Trans. Amer. Math. Soc. 347 (1995), 363-409. | Zbl 0855.55007
[00009] [J] P. T. Johnstone, Topos Theory, Academic Press, New York, 1977.
[00010] [M-M] S. MacLane and I. Moerdijk, Sheaves in Geometry and Logic, Springer, 1992.
[00011] [M-S] S. Mardešić and J. Segal, Shape Theory. The Inverse Systems Approach, North-Holland, 1982.
[00012] [M] I. Moerdijk, Prodiscrete groups and Galois toposes, Proc. Konink. Nederl. Akad. Wetensch. Ser. A 92 (1988), 219-234. | Zbl 0687.18004
[00013] [P] T. Porter, Abstract homotopy theory in procategories, Cahiers Topologie Géom. Différentielle 17 (1976), 113-124. | Zbl 0349.18012
[00014] [S] E. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.