... i.e. a structure where the tensor product is given by a profunctor instead of a mere functor.

@Martin: I think I can write down some monoidal structure, but I have trouble coming up with a symmetric one. By the way, if $\bullet$ denotes a quiver with one vertex and no edges, do I read it correctly that $\bullet \otimes \bullet = \varnothing$? If so then your monoidal structure is not a Day convolution in the standard sense since then product of representables would be representable. However, it is a theorem of Day that every closed monoidal structure on a category of (co)presheaves is a Day convolution with respect to some promonoidal structure on the indexing category...

I don't know how to answer this from the perspective of the theory of quivers, but categorically we can see quivers as $\mathrm{Set}$-valued functors on the category with two objects and parallel morphisms between them. Then your tensor product seems to be the Day convolution product (see ncatlab.org/nlab/show/Day+convolution) for a particular monoidal structure on this indexing category.

@Dylan: this category has wrong homotopy type in general. It is actually isomorphic to the category of functors from the monoid $\mathbb{N}$ to $C$. If you take $C$ to be a category with two objects and two parallel arrows between them, then $C$ is homotopy equivalent to a circle, so its loop space should be countably infinite discrete (up to homotopy), but your construction gives a finite category.

Could you give an example of a category which satisfies the definition above but not Waldhausen's original definition? And secondly, do you really want to drop stability of acyclic cofibrations under pushouts? (Or did you just drop it because you assumed that weak equivalences are isomorphisms?)

It suffices to unravel the definitions of matching objects, in this case it is rather simple and indeed $A \to B \leftarrow C$ is injective if and only if both $A \to B$ and $C \to B$ are split surjective with injective sources.

Yes, it does. The easiest way to state the definition is that $J$ is inverse if there is functor $J^\mathrm{op} \to \mathbb{N}$ which reflects identities and your $J$ of course admits such a functor. (Sometimes it is useful to generalize by replacing $\mathbb{N}$ by some arbitrary ordinal).

Every Reedy category has a direct part (morphisms that raise the degree) and an inverse part (those that lower the degree). An inverse category is a Reedy category with trivial direct part so that all non-identity morphisms lower the degree. This assumption was used to conclude that "Reedy $\mathcal{L}$-cofibrations" are just levelwise monomorphisms i.e. monomorphisms.

Yes, I admit that my proof is an overkill. Especially that it relies on the extra assumption that there are enough injective objects. But it really is more general, it gives characterization of injectives in $\mathcal{A}^J$ where $\mathcal{A}$ is any abelian category with enough injectives and $J$ is inverse.

The second entry of your formula for r has a free occurance of a. I guess it should be $\inf_{a \in A} p_I r_Y (f(a),t)$.