A convenient category of locally preordered spaces

TL;DR

This paper introduces a new category of locally preordered spaces that is Cartesian closed and supports all limits and colimits, providing a practical foundation for homotopy theory of spacetime.

Contribution

It extends a category of compact partially ordered spaces to a convenient, Cartesian closed category of locally preordered spaces, facilitating homotopy theory applications.

Findings

The new category is Cartesian closed.

The forgetful functor creates all limits and colimits.

Supports a homotopy theory of abstract spacetime.

Abstract

As a practical foundation for a homotopy theory of abstract spacetime, we extend a category of certain compact partially ordered spaces to a convenient category of locally preordered spaces. In particular, we show that our new category is Cartesian closed and that the forgetful functor to the category of compactly generated spaces creates all limits and colimits.