Fundamental progroupoid and bundles with a structural category

  1. Ardanza-Trevijano, S. 2
  2. Hernández-Paricio, L.-J. 1
  1. 1 Universidad de Zaragoza
    info

    Universidad de Zaragoza

    Zaragoza, España

    ROR https://ror.org/012a91z28

  2. 2 Binghamton University
    info

    Binghamton University

    Binghamton, Estados Unidos

    ROR https://ror.org/008rmbt77

Revista:
Topology and its Applications

ISSN: 0166-8641

Año de publicación: 1999

Volumen: 92

Número: 2

Páginas: 85-99

Tipo: Artículo

DOI: HTTP://DX.DOI.ORG/10.1016/S0166-8641(97)00247-2 SCOPUS: 2-s2.0-0037920059 GOOGLE SCHOLAR

Otras publicaciones en: Topology and its Applications

Resumen

In this paper, for a given space X, a structural category C, and a faithful functor η from C to the category of spaces, we introduce a notion of (C,η)-bundle which contains as particular cases, the notions of covering space, of overlaying space (introduced by Fox), of suspension foliation and other well-known topological structures. The new notion allows us to use sheaf theory and category theory in order to obtain some classification theorems which appear in terms of equivalences of categories. We prove that the category (C,η)-bundle(X) of (C, η)-bundles over X is equivalent to the category pro(πC X, C), which is determined by the fundamental groupoid of X and the structural category C. As particular cases we obtain the standard classification of covering spaces, Fox's classification theorem for overlays with a finite number of leaves and the standard classification of suspension foliations. This paper illustrates the importance of the fundamental progroupoid of a space X, which plays in shape theory the role of the standard fundamental groupoid. If the space X satisfies some additional properties, the progroupoid πCX can be reduced to a surjective progroup, a groupoid or a group. In some cases a surjective progroupoid can be replaced by a topological prodiscrete group. In all these cases the category pro(π-CX,C) also reduces to well-known categories. © 1999 Elsevier Science B.V. All rights reserved.