Waldhausen's definition of a category with cofibrations includes the choice of a distinguished zero object. Probably this also means that an exact functor should preserve zero objects on the nose (Waldhausen writes "takes $\*$ to $\*$"). Isn't this rather evil from the general perspective of category theory? Even if we had fixed zero objects, we should only require that an exact functor preserves them up to isomorphism (which is unique anyway).
Remark that Weibel's K-Book doesn't demand this choice; there it is only required that some zero object exists. Isn't this more natural? On the other hand, the theories look quite the same.
So a question might be why Waldhausen has chosen his definition and if one definition of the two definitions has any real advantage in practice against the other.
This question has a topological analogue. If $(X,x)$ and $(Y,y)$ are pointed spaces, then why not defining a map $(X,x) \to (Y,y)$ to be a map $f : X \to Y$ together with a path $f(x) \to y$ (oplax) or $y \to f(x)$ (lax)? This has the advantage that we are more flexible as concerning the base points, we still have a well-defined induced map $\pi_n(X,x) \to \pi_n(Y,y)$ on homotopy groups, but on the other hand this will be more complicated than the usual notion of a pointed map. Any references are welcome.
$A_\infty$
category. If we're generalizing that far, why don't we replace points with contractible spaces? For many applications, the responsibility for maintaining the definition outweighs the added utility or increase in aesthetic value. $\endgroup$