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. This recovers the standard weight structure on spectra and an exotic one related to Anderson duality. We identify their completions with Bousfield--Kan completions arising in Adams-type spectral sequences. 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 $A \mapsto \widehat{K}(A)$.