Bounded homotopy theory and the K-theory of weighted complexes

Given a bounding class B, we construct a bounded refinement BK(−) of Quillen’s K-theory functor from rings to spaces. As defined, BK(−) is a functor from weighted rings to spaces, and is equipped with a comparison map BK → K induced by “forgetting control.” In contrast to the situation with B-bounded cohomology, there is a functorial splitting BK(−) ≅ K(−)×BK ^rel(−) where BK ^rel(−) is the homotopy fiber of the comparison map.