Let $mathcal{C}$ be a Waldhausen category. Suppose that $mathcal{B}$ is a subcategory of $mathcal{C}$, and that $mathcal{B}$ is closed under extensions. If $mathcal{B}$ is strictly cofinal in $mathcal{C}$ (in the sense that given any $Cin mathcal{C}$ there exists a $Bin mathcal{B}$ such that $Camalg Bin mathcal{B}$), can we say anything about $K(mathcal{B}) rightarrow K(mathcal{C})$?
In Waldhausen's paper "Algebraic K-theory of spaces" Waldhausen claims that the inclusion $mathcal{B}rightarrow mathcal{C}$ induces a weak equivalence $wS_bullet mathcal{B}rightarrow wS_bulletmathcal{C}$ (and thus an equivalence on K-theories), but I'm not sure that this is right, as $mathcal{B}$ does not need to be a full subcategory. In particular, if there are objects $C,C'$ which are in $mathcal{B}$ but are not isomorphic in $mathcal{B}$ they may well be isomorphic (or at least weakly equivalent) in $mathcal{C}$.
Consider the following example. Let $mathcal{C}$ be the category of pairs of pointed finite sets, whose morphisms $(A,B)rightarrow (A',B')$ are pointed maps $Avee Brightarrow A'vee B'$, and let $mathcal{B}$ be the category of pairs of pointed finite sets whose morphisms $(A,B)rightarrow (A',B')$ are pairs of pointed maps $Arightarrow B$ and $A'rightarrow B'$. We make $mathcal{C}$ a Waldhausen category by defining the weak equivalences to be the isomorphisms, and the cofibrations to be the injective maps. $mathcal{B}$ is clearly cofinal in $mathcal{C}$, but $K_0(mathcal{B}) = mathbf{Z}times mathbf{Z}$, while $K_0(mathcal{C}) = mathbf{Z}$. Going even further, the Barratt-Priddy-Quillen theorem should tell us that $K(mathcal{B}) = QS^0times QS^0$, while $K(mathcal{C}) = QS^0$.
If we add the condition that $mathcal{B}$ needs to be a full subcategory of $mathcal{C}$, then I believe that Waldhausen's paper is correct. But even without that, it is possible to say anything about the map $K(mathcal{B})rightarrow K(mathcal{C})$?
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