Hearts of set-generated t-structures have a set of generators

TL;DR

This paper proves that in certain well-generated triangulated categories, t-structures generated by a set of objects have hearts that are locally presentable abelian categories, extending understanding of their structure.

Contribution

It establishes that hearts of t-structures generated by a set of objects are locally lpha-presentable abelian categories in lpha-compactly generated triangulated categories.

Findings

Hearts of such t-structures are locally lpha-presentable.

Any t-structure generated by a set of objects has a heart with a set of generators.

The result applies to well-generated triangulated categories.

Abstract

We show that if $\alpha$ is a regular cardinal, $\mathcal{D}$ is an $\alpha$-compactly generated triangulated category, in the sense of Neeman \cite{N}, and $\tau$ is a t-structure in $\mathcal{D}$ generated by a set of $\alpha$-compact objects, then the heart of $\tau$ is a locally $\alpha$-presentable (not necessarily Ab5) abelian category. As a consequence, in a well-generated triangulated category any t-structure generated by a set of objects has a heart with a set of generators.