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## Fundamenta Mathematicae

1998 | 156 | 1 | 1-31
Tytuł artykułu

### Fundamental pro-groupoids and covering projections

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
1-31
Opis fizyczny
Daty
wydano
1998
otrzymano
1995-05-30
poprawiono
1997-05-14
Twórcy
Bibliografia
• [A-M] M. Artin and B. Mazur, Etale Homotopy, Lecture Notes in Math. 100, Springer, Berlin, 1967.
• [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.
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• [G-Z] P. Gabriel and M. Zisman, Calculus of Fractions and Homotopy Theory, Springer, Heidelberg, 1967.
• [God] C. Godbillon, Éléments de Topologie Algébrique, Hermann, Paris, 1971.
• [Go] M. Golasiński, Homotopies of small categories, Fund. Math. 114 (1981), 209-217.
• [Gro] A. Grothendieck, Revêtements Etales et Groupe Fondamental (SGA 1), Lecture Notes in Math. 222, Springer, Berlin, 1971.
• [H] L. J. Hernández, Applications of simplicial M-sets to proper and strong shape theories, Trans. Amer. Math. Soc. 347 (1995), 363-409.
• [J] P. T. Johnstone, Topos Theory, Academic Press, New York, 1977.
• [M-M] S. MacLane and I. Moerdijk, Sheaves in Geometry and Logic, Springer, 1992.
• [M-S] S. Mardešić and J. Segal, Shape Theory. The Inverse Systems Approach, North-Holland, 1982.
• [M] I. Moerdijk, Prodiscrete groups and Galois toposes, Proc. Konink. Nederl. Akad. Wetensch. Ser. A 92 (1988), 219-234.
• [P] T. Porter, Abstract homotopy theory in procategories, Cahiers Topologie Géom. Différentielle 17 (1976), 113-124.
• [S] E. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.
Typ dokumentu
Bibliografia
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