K theory of a simplicial monoidal category, Cofinality theorem

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...Read more