Geodesic
Bounded homotopy theory and the $K$-theory of weighted complexes
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Classical and modern mathematics in the wake of Boris Nikolaevich Delone, Tome 275 (2011), pp. 210-226

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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. 
@article{TM_2011_275_a13,
     author = {J. Fowler and C. Ogle},
     title = {Bounded homotopy theory and the $K$-theory of weighted complexes},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {210--226},
     publisher = {mathdoc},
     volume = {275},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2011_275_a13/}
}
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J. Fowler; C. Ogle. Bounded homotopy theory and the $K$-theory of weighted complexes. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Classical and modern mathematics in the wake of Boris Nikolaevich Delone, Tome 275 (2011), pp. 210-226. http://geodesic.mathdoc.fr/item/TM_2011_275_a13/