# homotopy theory – construction of \$K_0\$-group and Karoubian completion

Let $$A$$ be a ring. The $$K_0$$ group of $$A$$ can be defined in most
old fashioned way as the Grothendieck group of the set of isomorphism classes
of its finitely generated projective $$R$$ modules, regarded as a monoid under direct sum.

One can show that this $$K_0(R)$$ coincides with Quillen-construction
$$K_0(P_A) := pi_1(BQ(P_A),0)$$ if $$P_A$$ is the category of finitely generated projective A-modules. The important issue of Q-construction is that it associates $$K_i$$ groups to every
exact category. In more general setting applying S-construction
one can associate $$K$$-theory groups to categories with cofibrations (called
Waldhausen categories).

If we go back to the first construction of $$K_0(R)$$ we observe that the
auxilary category $$P_A$$ of finitely generated projective A-modules is formally
the Karoubi or preudo-abelian completion of the category $$F_A$$ of finitely generated free $$A$$-modules.
What we see is that the preudo-abelian completion or preudo-abelian categories play
somehow a central role for the definition of algebraic $$K_0(R)$$ group.

My question is how concretely is the pseudo-abelian completion involved in the
construction of $$K_0$$ group in more general setting like in Quillen’s construction
or S-construction if we start with an arbitrary exact or Waldhausen category $$mathcal{C}$$.

One could also ask if this definition of $$K_0(A)$$ in algebraic $$K$$-theory
involving pseudo-abelian completion of fin.gen. free $$A$$-modules is something
special what happens only in algebraic (and topological) $$K$$-theory or
is the pseudo-abelian completion here based on a more general principle
used in construction of $$K_0$$ in $$K$$-theories in general setting?

The question is closely related to this discussion that mostly uncovers the
obstructional relation between existence of nonzero negative $$K$$-groups
Karoubian completeness.
Now I would like to understand if and how the pseudo-abelian completion is
involved in construction of $$K_0$$.