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

Classically (i.e. for non-simplicial categories) we have the cofinality theorem that states that a full and cofinal functor Y→X between symmetric monoidal categories induces an isomorphism on K-theory in all higher degrees (>0).

Here, F:Y→X is cofinal if for all x1∈X there exist x2∈X and y∈Y such that x1+x2≅F(y).In my situation I have full functors Ad→Xd for all d, where Ad is contractible. If these functors were cofinal then K(Xd) 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 x1∈Xn we can only find x2∈Xn and y∈A0 such that x1+x2≃y. Here, ≃ means that there exists a n+1-simplex in X which has x1+x2 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)≅|d↦K0(Xd)|? I somehow jump between arguments concerning the categories Xd 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).

**Answer**

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.

**Attribution***Source : Link , Question Author : Simon Markett , Answer Author : Tom Harris*