Fiber bundles over Alexandroff spaces

Abstract

We introduce a topological variant of the Grothendieck construction which serves to represent every fiber bundle over an Alexandroff space. Using this result we give a classification theorem for fiber bundles over Alexandroff spaces with T${}_0$ fiber and we construct a universal bundle for bundles with T${}_0$ fiber over posets which are cofibrant objects of the category of small categories. Moreover, we prove that our construction induces an equivalence of categories between a suitable category of functors and the category of fiber bundles over a fixed Alexandroff space. In addition, we prove that any fiber bundle over an Alexandroff space is a fibration.

arXiv comment27 pages. Most of the results have been generalized and the overall presentation of the article has been bettered. In addition, several new results have been added

References in corpus (3)

• The homotopy theory of equivariant posets
• Cofibrant objects in the Thomason Model Structure
• Regular coverings and fundamental groupoids of Alexandroff spaces