Classifying space of subcategories and its application

TL;DR

This paper introduces a topological space called the classifying space of subcategories, which helps classify prime subcategories via a bijective correspondence with closed subsets, unifying many existing classification results.

Contribution

It defines a new topological framework for classifying subcategories, extending the understanding of prime subcategories in triangulated categories and related structures.

Findings

Classifying space bijectively corresponds to prime subcategories.

Many known classification results are encompassed by this framework.

An example outside the scope of the framework is also provided.

Abstract

For a collection of subcategories satisfying a fixed set of conditions, for example thick subcategories of a triangulated category, we define a topological space called classifying space of subcategories. We show that this space classifies various prime subcategories in the sense that they bijectively correspond to the closed subsets of the classifying space. Many well-known results of subcategory classification fit into this framework. An example which cannot be classified by a topological space in the above sense is also given.