Resolution theorem (algebraic K-theory)

In algebraic K-theory, Quillen's resolution theorem states that if ${\displaystyle {\mathcal {A}}\subset {\mathcal {C}}}$ is an exact subcategory where ${\displaystyle {\mathcal {A}}}$ is an extension-closed subcategory of ${\displaystyle {\mathcal {C}}}$, which is also closed under taking kernels of admissible surjections, and has a finite resolution by objects in ${\displaystyle {\mathcal {A}}}$; then the inclusion ${\displaystyle {\mathcal {A}}\rightarrow {\mathcal {C}}}$ induces a homotopy equivalence of their K-theory spectra ${\displaystyle K_{0}({\mathcal {A}})\simeq ~K_{0}({\mathcal {C}})}$.^[1]