Smallness in Topology

TL;DR

This paper explores the complexity and variability of small objects in topological categories, revealing unexpected differences across various separation axioms and their implications for categorical topology.

Contribution

It provides full characterizations of small objects in different topological categories, highlighting surprising distinctions especially in T1 spaces and reflective subcategories.

Findings

Significant differences in small object characterizations across separation axioms

T1 spaces exhibit unique properties compared to other categories

Insights applicable to arbitrary reflective subcategories of topological spaces

Abstract

Quillen's notion of small object and the Gabriel-Ulmer notion of finitely presentable or generated object are fundamental in homotopy theory and categorical algebra. Do these notions always lead to rather uninteresting classes of objects in categories of topological spaces, such as all finite discrete spaces, or just the empty space, as the examples and remarks in the existing literature may suggest? This article demonstrates that the establishment of full characterizations of these notions (and some natural variations thereof) in many familiar categories of spaces can be quite challenging and may lead to unexpected surprises. In fact, we show that there are significant differences in this regard even amongst the categories defined by the standard separation axioms, with the T1-separation condition standing out. The findings about these specific categories lead us to insights also when considering rather arbitrary full reflective subcategories of the category of all topological spaces.