Approximating triangulated categories by spaces

TL;DR

This paper explores the structure of lattices of thick subcategories in triangulated categories, showing their properties and introducing two spectra that approximate these lattices by spaces.

Contribution

It systematically studies lattices of thick subcategories, characterizes when they are spatial frames, and constructs universal spectra for their approximation by spaces.

Findings

Distributive lattices of thick subcategories are spatial frames.

Two non-commutative spectra are constructed for universal approximations.

Examples illustrate various properties of these lattices.

Abstract

We initiate a systematic study of lattices of thick subcategories for arbitrary essentially small triangulated categories. To this end we give several examples illustrating the various properties these lattices may, or may not, have and show that as soon as a lattice of thick subcategories is distributive it is automatically a spatial frame. We then construct two non-commutative spectra, one functorial and one with restricted functoriality, that give universal approximations of these lattices by spaces.