[Waldhausen](http://www.math.uni-bielefeld.de/~fw/LNM1126_318-419.pdf)'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](http://ncatlab.org/nlab/show/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](http://www.math.rutgers.edu/~weibel/Kbook.html) 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.



Choice of base point in a Waldhausen category
When Waldhausen defines $K$-theory, he considers sequences of the form


$\* \rightarrow A\_1 \rightarrow A\_2 \rightarrow \cdots \rightarrow A\_n$


for varying $n$, with all maps cofibrations. For any fixed $n$, these sequences together with the obvious morphisms again form a category, and these categories nearly assemble to a simplicial object in categories. Degeneracy maps are defined by inserting identities and the face maps except $d\_0$ are defined by composition. $d\_0$ of the above sequence is obtained by ignoring the $\*$ and dividing out $A\_1$ from the sequence to obtain


$A\_1/A\_1 \rightarrow A\_2/A\_1 \rightarrow \cdots $


However, this does not really make sense since choices of quotients are involved; in particular a choice of zero object $A\_1/A\_1$. You may make arbitrary choices, but then you only obtain a simplicial object up to equivalence: For example, $d\_0 s\_0$ is not the identity functor, but merely isomorphic to it. 


This problem is solved by considering a different category: We plug more information into our objects by declaring an object to be a sequence


$\* \rightarrow A\_1 \rightarrow A\_2 \rightarrow \cdots \rightarrow A\_n$


together with choices of quotients $A\_i/A\_j$, $i \geq j$. This yields an equivalent category, but now we have a shot to get an honest simplicial object in categories.


But then a problem in the degeneracy maps arises: In a degenerate sequence of the form


$\* \rightarrow \cdots A\_k = A\_k \rightarrow \cdots$


we now also have to specify the quotient $A\_k/A\_k$, and again making arbitrary choices will not cut it. Letting this quotient be the chosen zero object works when one also restricts to sequences starting with this chosen zero object. I think this is the main technical reason to pick a zero object once and for all.


Such evil concepts are actually quite common in $K$-theory: Often, it is not enough to have diagrams of categories commuting up to natural isomorphism, so one has to wriggle the definitions of the involved categories a bit to obtain different, equivalent categories with a strictly commuting diagram of functors. 


