Resolution theorem (algebraic K-theory)

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