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Let $X=(d\mapsto X_d)$ be a simplicial symmetric monoidal category. We define the $K$-theory space of $X$ to be $K(X)=|d\mapsto K(X_d)|$, the geometric realisation of the simplicial space $d\mapsto K(X_d)$.

Classically (i.e. for non-simplicial categories) we have the cofinality theorem that states that a full and cofinal functor $Y\to X$ between symmetric monoidal categories induces an isomorphism on K-theory in all higher degrees (>0). Here, $F: Y\to X$ is cofinal if for all $x_1\in X$ there exist $x_2\in X$ and $y\in Y$ such that $x_1+x_2\cong F(y)$.

In my situation I have full functors $A_d \to X_d$ for all $d$, where $A_d$ is contractible. If these functors were cofinal then $K(X_d)$ were discrete for all $d$ and $K(X)$ would simplify significantly.

But these functors are unfortunately only cofinal mod simplicial identities, by what I mean that given $x_1\in X_n$ we can only find $x_2\in X_n$ and $y\in A_0$ such that $x_1+x_2\simeq y$. Here, $\simeq$ means that there exists a $n+1$-simplex in $X$ which has $x_1+x_2$ as a face and $y$ (either considered a 0-simplex or as a degenerated $n$-simplex) as its opposing vertex.

Can I still conclude that $K(X)\cong|d\mapsto K_0(X_d)|$? I somehow jump between arguments concerning the categories $X_d$ seperately and as part of the simplicial set $X$.

Edit: It would be great already if somebody could tell me where I can find a detailed proof of the cofinality theorem. Higher algebraic K-theory II refers to a paper by Gersten which is apparently quite hard to find (this is, you have to pay for it and there seems to be no uni-login).

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  • $\begingroup$ 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. $\endgroup$
    – K.J. Moi
    Jan 31, 2013 at 13:34

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I am not yet able to leave comments, so I'll post this as an answer. I have a copy of the Gersten paper (obtained with some difficulty), if you contact me I can email a scan to you. You may also find this survey useful, if you are willing to read French.

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