  We study the Grothendieck monoid (a monoid version of the Grothendieck group)
of an extriangulated category, and give some results which are new even for
abelian categories. First, we classify Serre subcategories and dense 2-out-of-3
subcategories using the Grothendieck monoid. Second, in good situations, we
show that the Grothendieck monoid of the localization of an extriangulated
category is isomorphic to the natural quotient monoid of the original
Grothendieck monoid. This includes the cases of the Serre quotient of an
abelian category and the Verdier quotient of a triangulated category. As a
concrete example, we introduce an intermediate subcategory of the derived
category of an abelian category, which lies between the abelian category and
its one shift. We show that intermediate subcategories bijectively correspond
to torsionfree classes in the abelian category, and then compute the
Grothendieck monoid of an intermediate subcategory.
