  Waldhausen categories were introduced to extend algebraic $K$-theory beyond
Quillen's exact categories. In this article, we modify Waldhausen's axioms so
that it matches better with the theory of extriangulated categories,
introducing a weak Waldhausen category and defining its Grothendieck group.
Examples of weak Waldhausen categories include any extriangulated category,
hence any exact or triangulated category, and any Waldhausen category. A key
feature of this structure is that it allows for "one-sided" extriangulated
localization theory, and thus enables us to extract right exact sequences of
Grothendieck groups that we cannot obtain from the theory currently available.
  To demonstrate the utility of our Weak Waldhausen Localization Theorem, we
give three applications. First, we give a new proof of the Extriangulated
Localization Theorem proven by Enomoto--Saito, which is a generalization at the
level of $K_0$ of Quillen's classical Localization Theorem for exact
categories. Second, we give a new proof that the index with respect to an
$n$-cluster tilting subcategory $\mathscr{X}$ of a triangulated category
$\mathscr{C}$ induces an isomorphism between $K_0^{\mathsf{sp}}(\mathscr{X})$
and the Grothendieck group of an extriangulated substructure of $\mathscr{C}$.
Last, we produce a weak Waldhausen $K_0$-generalization of a localization
construction due to Sarazola that involves cotorsion pairs but allows for
non-Serre localizations. We show that the right exact sequences of Grothendieck
groups obtained from our Sarazola construction and the Extriangulated
Localization Theorem agree under a common setup.
