# Fibrations between finite topological spaces

## TL;DR

This paper explores Hurewicz fibrations in finite T0-spaces, establishing combinatorial conditions, linking them to Grothendieck bifibrations, and providing examples to illustrate the theory and necessity of assumptions.

## Contribution

It introduces combinatorial criteria for Hurewicz fibrations in finite T0-spaces and connects them to Grothendieck bifibrations, expanding understanding of their structure.

## Key findings

- Established strong conditions for Hurewicz fibrations in finite T0-spaces.
- Demonstrated the relationship between Hurewicz fibrations and Grothendieck bifibrations.
- Provided examples illustrating the theory and necessity of assumptions.

## Abstract

We study Hurewicz fibrations between finite T$_0$--spaces from a combinatorial viewpoint and give strong conditions that a continuous map between finite T$_0$--spaces must satisfy in order to be a Hurewicz fibration. We also show that there exists a strong relationship between Hurewicz fibrations between finite T$_0$--spaces and Grothendieck bifibrations. Finally we give several interesting examples that illustrate this theory and show that many of the assumptions of our results are necessary.