Stabilization of the category of simplicial objects in CAT

In this article, we define two equivalent new model structures on $\mathbf{s}\mathbf{C}\mathbf{a}\mathbf{t}$ the category of simplicial objects in $\mathbf{C}\mathbf{a}\mathbf{t}$. Then we construct the corresponding stable model category of spectra $Sp(\mathbf{s}\mathbf{C}\mathbf{a}\mathbf{t})$ and make some links with the algebraic $K$-theory via the mapping space.