Bounded homotopy theory and the $K$-theory of weighted complexes

Given a bounding class $\mathcal B$, we construct a bounded refinement $\mathcal BK(-)$ of Quillen's $K$-theory functor from rings to spaces. As defined, $\mathcal BK(-)$ is a functor from weighted rings to spaces, and is equipped with a comparison map $\mathcal BK\to K$ induced by “forgetting control”. In contrast to the situation with $\mathcal B$-bounded cohomology, there is a functorial splitting $\mathcal BK(-)\simeq K(-)\times\mathcal BK^\mathrm{rel}(-)$ where $\mathcal BK^\mathrm{rel}(-)$ is the homotopy fiber of the comparison map.