I am trying to understand quasi-isomorphisms in an exact category as defined via the mapping cylinder. I would like to know whether these form a category of weak equivalences in the sense of Waldhausen.
Some background.
Recall that in an exact category in the sense of Quillen we have a good notion of a long exact sequence that is simply a chain complex whose differentials factor through short exact sequences whose kernels and cokernels are also objects of the exact category. Call such sequences acyclic.
By the Gabriel-Quillen embedding theorem we can consider every exact category $\mathcal{E}$ as a subcategory of some abelian category $\mathcal{A}$, with short exact sequences inherited from $\mathcal{A}$. In general it is not the case that the acyclic sequences of $\mathcal{E}$ are exactly the long exact sequences of $\mathcal{A}$ whose objects are in $\mathcal{E}$ (consider the exact category of free $R$-modules sitting inside the abelian category of all $R$-modules, for example).
In an abelian category a morphism of chain complexes is a quasi-isomorphism iff its mapping cone is exact. Since we don't have homology in exact categories it is sensible to define quasi-isomorphisms via the mapping cone. But now there are two possible classes of quasi isomorphism I can choose, either $q_{1}$, the morphisms between chain complexes of exact categories whose mapping cones are acyclic in $\mathcal{E}$, or $q_{2}$, the morphisms whose mapping cone is a long exact sequence when viewed in the ambient abelian category.
Clearly $q_{1} \subseteq q_{2}$ but I see no reason for them to be equal. So, after all this set-up, my question is: is there a standard choice here? Do either form a category of weak equivalences in the sense of Waldhausen? I'd like to work with $q_{1}$ as it avoids extrinsic information about the exact category but I can pass to $q_{2}$ if necessary.
Any insights would be appreciated.
A bonus question, in case both form weak equivalences, do the corresponding $K$-theories differ? How?
