In algebraic K-theory, Quillen's resolution theorem states that if A ⊂ C {\displaystyle {\mathcal {A}}\subset {\mathcal {C}}} is an exact subcategory where A {\displaystyle {\mathcal {A}}} is an extension-closed subcategory of C {\displaystyle {\mathcal {C}}} , which is also closed under taking kernels of admissible surjections, and has a finite resolution by objects in A {\displaystyle {\mathcal {A}}} ; then the inclusion A → C {\displaystyle {\mathcal {A}}\rightarrow {\mathcal {C}}} induces a homotopy equivalence of their K-theory spectra K 0 ( A ) ≃ K 0 ( C ) {\displaystyle K_{0}({\mathcal {A}})\simeq ~K_{0}({\mathcal {C}})} .
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