A categorical perspective on extended metric-topological spaces

TL;DR

This paper explores the category of extended metric-topological spaces, proving its bicompleteness and examining subcategories like compact spaces, to advance understanding of infinite-dimensional metric-measure structures.

Contribution

It establishes the bicompleteness of the category of extended metric-topological spaces and analyzes its subcategories, providing a foundational framework for infinite-dimensional geometry.

Findings

Proved bicompleteness of the category of extended metric-topological spaces.

Analyzed subcategories such as compact spaces.

Provided a categorical framework for infinite-dimensional metric-measure analysis.

Abstract

Motivated by the analysis and geometry of metric-measure structures in infinite dimensions, we study the category of extended metric-topological spaces, along with many of its distinguished subcategories (such as the one of compact spaces). One of the main achievements is the proof of the bicompleteness (i.e. of the existence of all small limits and colimits) of the aforementioned categories.