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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 construction of algebraic (and topological due to Swan's theorem) $K$-theory or is the pseudo-abelian completion here based on a deeper 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 and Karoubian completeness of underlying category.
Now I would like to understand if and how the pseudo-abelian completion is involved in construction of $K_0$.

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  • $\begingroup$ You can also define the $K$-theory of $F_A$, it's just less interesting in general (as it is always a cyclic group). The higher $K$-groups agree with the usual one. So I would say the Karoubi completion is not conceptually important to define $K$-theory, it's conceptually important because we like projective modules $\endgroup$ Sep 9 at 19:29
  • $\begingroup$ hm, I'm still not 100 percent convinced that the procedure of Karoubi completion is "expendable" for constructions in $K$-theory in general setting. Do you have an idea which constructions from $K$-theory are referred to here: ncatlab.org/nlab/show/Karoubian+category#examples ? $\endgroup$ Sep 9 at 20:23
  • $\begingroup$ None of the notions of $K$-theory that I know require Karoubian completion ( I may be missing some, but I'm thinking of group completion $K$-theory, the $K$-theory of an exact ($\infty$-)category, and the $K$-theory of a stable $\infty$-category) . In fact $K$-theory detects dense subcategories (see Thomason, "classification of triangulated subcategories") and so is particularly well suited to study non Karoubian closed things $\endgroup$ Sep 9 at 20:29

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