Stabilization of the category of simplicial objects in CAT

TL;DR

This paper introduces two new model structures on simplicial categories, constructs a stable model category of spectra, and explores connections with algebraic K-theory, advancing the understanding of categorical and homotopical structures.

Contribution

It defines equivalent new model structures on simplicial categories and develops a stable model category of spectra, linking to algebraic K-theory.

Findings

Two equivalent model structures on sCat are established.

A stable model category of spectra Sp(sCat) is constructed.

Connections between the stable model category and algebraic K-theory are demonstrated.

Abstract

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