Topological and metric spaces are full subcategories of the category of simplicial objects of the category of filters

TL;DR

This paper reveals that topological spaces, uniform spaces, and simplicial sets can all be viewed as full subcategories within the category of simplicial filters, enabling categorical reformulations of key topological concepts.

Contribution

It demonstrates that these familiar categories are naturally embedded in the simplicial filters category and reformulates core topological notions categorically.

Findings

Categories of topological and uniform spaces are full subcategories of simplicial filters.

Reformulation of completeness, compactness, and related notions in categorical terms.

Open questions arising from these categorical perspectives.

Abstract

We observe that the category of topological space, uniform spaces, and simplicial sets are all, in a natural way, full subcategories of the same larger category, namely the simplicial category of filters; this is, moreover, implicit in the definitions of a topological and uniform space. We use these embeddings to rewrite the notions of completeness, precompactness, compactness, Cauchy sequence, and equicontinuity in the language of category theory, which we hope might be of use in formalisation of mathematics and tame topology. We formulate some arising open questions.