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).

  • $\begingroup$ 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
    Commented Jan 31, 2013 at 13:34

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.