That sequence is quite useful, for some reason it usually doesn't appear in algebraic topology books.

I have also wondered why (algebraic) topologists talk so little about ANRs. I am a homotopy theorist and I learned about them from the $C^{\ast}$-algebraists in my department. Maybe it's because any ANR has the homotopy type of a countable CW-complex.

Have a look at Staffeldt's "Fundamental theorems of algebraic K-theory", a google search will find it. The problem for you would be that it uses Waldhausen categories and not symmetric monoidal categories but I hope it's heldpful anyway.

People have certainly been interested in the categorified version of your question (jtopol.oxfordjournals.org/content/4/3/…). Do you have some more context for your question? Is there a particular category of modules or something that you want to understand?

@Kevin: I'm following the conventions I could find in the literature (papers of Segal, Lydakis and Schwede). Anyway, what I want is (covariant) functors from a skeletal subcatory of finite pointed sets. @some guy: I didn't think of that, I'll look into it.

In the Quillen model structure on simplicial sets it does. I don't know what happens with the Joyal model structure. Next time it might be a good idea to specify which model structure you want for your simplicial sets if you don't want the standard one.

If we consider a bisimplicial set as a simplicial object in simplicial sets then a levelwise weak equivalence induces a weak equivalence on diagonals, right? Sorry if I'm missing the point here.

What about taking the nerve in each simplicial degree and then taking the diagonal of the resulting bisimplicial set?

How much of homological algebra can you do in this setting? It seems to me that most homological arguments require the axiom of choice (to lift elements along epis). Do you run into problems when working with these enormous groups?

Nitpick: what is $N$ in $m \geq N$?