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

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.