# Fiber bundles over Alexandroff spaces

## TL;DR

This paper develops a topological approach to classify fiber bundles over Alexandroff spaces, introducing a universal bundle and establishing categorical equivalences, thus advancing the understanding of bundle structures in this setting.

## Contribution

It introduces a topological variant of the Grothendieck construction for fiber bundles over Alexandroff spaces and establishes a classification theorem and categorical equivalences.

## Key findings

- Classification of fiber bundles over Alexandroff spaces with T$_0$ fiber.
- Construction of a universal bundle for T$_0$ fiber bundles over posets.
- Proof that all such fiber bundles are fibrations.

## 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.