On perfectly generated weight structures and adjacent $t$-structures

TL;DR

This paper explores smashing weight structures and their relation to $t$-structures, proving that hearts of certain $t$-structures are Grothendieck abelian, and introduces the concept of perfect generation of weight structures.

Contribution

It introduces the notion of perfectly generated weight structures and shows their connection to adjacent $t$-structures, extending classification results to broader categories.

Findings

Hearts of compactly generated $t$-structures are Grothendieck abelian.

Any smashing weight structure on a well generated triangulated category is perfectly generated.

Classification of compactly generated torsion theories extended to arbitrary smashing categories.

Abstract

This paper is dedicated to the study of smashing weight structures (one may say that these are weight structures "coherent with arbitrary coproducts"), and the application of their properties to $t$-structures. In particular, we prove that hearts of compactly generated $t$-structures are Grothendieck abelian; this statement strengthens earlier results of several other authors. The central theorem of the paper is as follows: any perfect set of objects of a triangulated category generates a weight structure; we say that weight structures obtained this way are perfectly generated. An important family of perfectly generated weight structures are the ones adjacent to compactly generated $t$-structures; they give injective cogenerators for the hearts of the latter. We also establish the following not so explicit result: any smashing weight structure on a well generated triangulated category (this class of categories contains compactly generated ones) is perfectly generated; actually, we prove more than that. Moreover, we give a classification of compactly generated torsion theories (these generalize both weight structures and $t$-structures) that extends the corresponding result of D. Pospisil D. and J. \v{S}\v{t}ovi\v{c}ek to arbitrary smashing triangulated categories.