Unbounded Weight Structures: (Re)construction and Completion

We develop a theory of completeness for weight structures on stable categories, dual to the theory of complete $t$-structures.

As in the bounded case, we show that complete weight structures are determined by their weight heart, giving rise to a universal construction $A \mapsto K(A)$ that assigns a complete weight category to an additive category and recovers classical examples such as homotopy categories of chain complexes.

We also give a general construction of weight structures on presentable stable categories generated by a small set of objects, generalizing a result of Bondarko and Pauksztello.

This recovers the standard weight structure on spectra and an exotic one related to Anderson duality.

We identify their completions with modules over the (spectral) integral Steenrod algebra.

To treat naturally occurring examples - such as derived categories of abelian categories and module categories over ring spectra - which are often only partially weight complete, we introduce the notion of weak $t$-structures.

Within this framework, we prove that any stable category equipped with compatible weight and weak $t$-structures, and satisfying left weight completeness and right $t$-completeness, can be reconstructed from its heart via a two-step completion process $A \mapsto \widehat{K}(A)$.