# K-theory, monoidal vs. exact

My question is somewhat related to this one. However I think it adds something new to the table so I decided to post it sperately.

There is a construction of K-theory for symmetric monoidal categories (smc) and one for exact categories. From what I understand they are (a priori) two different pair of shoes.

For a symmetric monoidal category $S$ there is the classifying space $BS$. We define the K-theory $K(S)$ to be the group completion of this space. If we consider $BS$ to be a one-object topological category it is up to an equivalence of categories a strict monoidal category and a group completion is given by $BS\to\Omega BBS$.

If we now take our favorite category $P(R)$ of projective modules over a ring then the above construction isn't very interesting ($BS$ is contractible since $0\in S$ is an initial object). But we can take $iP(R)$ the isomorphism category and then $K(iP(R))$ magically coincides with the K-theory of $P(R)$ as an exact category (I write magically since I can read the proof but still don't really know why taking the isomorphism category is the sensible thing to do).

Something I like about the monoidal approach is Thomasons mapping cone construction. Given a functor of smc $A\to B$, such that $A=iA$ one can construct a smc $C$ such that there is a long exact sequence:

$$\cdots \to K_{i+1}(C)\to K_{i}(A)\to K_{i}(B)\to K_{i}(C)\to\cdots$$

Now I am in the situation that $A$ and $B$ are in fact exact categories. $A=P(R)$ and $B=P(R,\mathbb G_m)$ the category of projective modules equipped with an automorphism. So to apply Thomason's construction I may just take $A=iP(R)$, fair enough. May I also take $B=iP(R,\mathbb G_m)$ and get the K-theory of $P(R,\mathbb G_m)$ as an exact category (I have the feeling that we pass to direct sum K-theory here, or something like that)? If I now construct $C$ then I notice that $iC=iP(R,\mathbb G_m)$, so I definitely may not just take the isomorphism category here. Somehow I got the feeling that I am just jumping between the definitions and never really now whether it's justified.

So the question: For an exact category $D$, how are $K(D)$ (K-theory of $D$ as an exact category) $K(D)$ (as a symmetric monoidal category) and $K(iD)$ (again monoidal) related?

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