Rational Homotopy in Pseudotopological Spaces

TL;DR

This paper develops a rational homotopy theory for pseudotopological spaces, establishing their homotopy equivalences with topological spaces and constructing a model category framework.

Contribution

It introduces a model category for pseudotopological spaces, proves their homotopy equivalence to CW complexes, and constructs rational homotopy theory within this setting.

Findings

Pseudotopological spaces are Quillen equivalent to simplicial sets.

Every pseudotopological space is weak homotopy equivalent to a CW complex.

The Cartesian product of pseudotopological CW complexes is a CW complex.

Abstract

Pseudotopological spaces are the Cartesian closed hull of the category of \v{C}ech closure spaces. In this paper, we give a direct proof that the model category of the pseudotopological spaces constructed by Rieser is Quillen equivalent to the category of simplicial sets. In addition to noting that every pseudotopological space is weak homotopy equivalent to a topological CW complex, we prove that any weak equivalence of pseudotopological spaces can be converted to a weak equivalence of topological spaces. Finally, combining these ingredients, we construct rational homotopy for simply connected pseudotopological spaces. In this paper, we also prove that the cartesian product of two pseudotopological CW complexes is a CW complex.