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The localization theorem generalizes the [[localization theorem for an abelian categories]].
{{math_theorem|name=Localization theorem|Let <math>A</math> be the category with cofibrations, equipped with two categories of weak equivalences, <math>v(A) \subset w(A)</math>, such that <math>(A, v)</math> and <math>(A, w)</math> are both Waldhausen categories. Assume <math>(A, w)</math> has a [[cylinder functor]] satisfying the Cylinder Axiom, and that <math>w(A)</math> satisfies the Saturation and Extension Axioms. Then
:<math>K(A^w) \to K(A, v) \to K(A, w)</math>
is a [[homotopy fibration]].
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