Resolution theorem (algebraic K-theory)

Template:One source In algebraic K-theory, Quillen's resolution theorem states that if $𝒜\subset 𝒞$ is an exact subcategory where $𝒜$ is an extension-closed subcategory of $𝒞$, which is also closed under taking kernels of admissible surjections, and has a finite resolution by objects in $𝒜$; then the inclusion $𝒜\to 𝒞$ induces a homotopy equivalence of their K-theory spectra $K_0(𝒜)\simeq K_0(𝒞)$.