The following arose when I was thinking about a talk at the Midwest Topology Seminar:
Background
I want to consider a generalization of a Waldhausen-like structure on a category $C$ with 0-object $\ast$. Namely, I will have a category $\text{co}C$ consisting of "cofibrations," but I will relax the usual axioms to be:
Cof1: The isomorphisms of $C$ are cofibrations;
Cof2: The arrow $\ast \to A$ is a cofibration for any object $A$ of $C$.
Cof3': Pushouts of cofibrations along cofibrations are always defined, and all arrows in the the pushout will be cofibrations.
(Note that Cof3' is different from Waldhausen's usual axiom. In particular, quotient objects are not necessarily defined.)
For my purposes, I am happy to work with the isomorphisms as the weak equivalences.
Using the above structure, one can apply a construction due to Thomason to form a simplicial category $iT.C$. An object in simplicial degree $n$ is a string of cofibrations $$ C_\bullet \quad \equiv \quad C_0 \rightarrowtail C_1 \rightarrowtail \dots \rightarrowtail C_n $$
A morphism is filtration preserving map $C_\bullet \to D_\bullet $, such that each $C_k \to D_k$ is a cofibration and each square
$$ C_{k-1} \to C_k $$ $$ \downarrow \qquad \quad \downarrow $$ $$ D_{k-1} \to D_k $$ is a pushout (Note: I do not impose the condition that the vertical maps be weak equivalences in this set up.)
The simplicial structure is defined by dropping the $k$-th term of a filtration or inserting the identity in the appropriate place.
Waldhausen showed that Thomason's construction is homotopy equivalent to the $wS.$-construction whenever the above structure arises from a bona fide Waldhausen category structure on $C$. The idea, roughly is given by mapping the filtration above to the filtration $C_1/C_0 \rightarrowtail \cdots \rightarrowtail C_n/C_0$, provided we incorporate quotient data into the definition. (See Waldhausen LNM1126, p.334.)
My Question
My question is basically whether a version of the additivity theorem holds in this context. To formulate this, consider the category whose objects are pushout squares of objects of $C$: $$ A \to B $$ $$ \downarrow \quad \quad \downarrow $$ $$ C \to D $$ in which all displayed maps are cofibrations. Morphisms are defined in the obvious way (natural transformations of such diagrams) It seems to me that this diagram category is also equipped a cofibration structure as above.
Then we may consider the functor which sends this diagram to $$ D \vee A $$ as well as the functor which sends the diagram to $$ B \vee C $$ These are both exact functors, and one can ask:
Question: Are these functors homotopic after passing to $iT_\bullet$ constructions?
