CW-complexes in the Category of Small Categories

Abstract

We compute the collection of CW-complexes in the model category of small categories constructed by Joyal and Tierney. More generally, if $X$ is a connected topological space, we show that the homotopy category of CW-complexes in Joyal-Tierney's model category of sheaves of sets on $X$ is equivalent to the homotopy category of groupoids. As an application of the ideas, we show that the algebraic $K$-theory groups of the category of pointed small categories are trivial, and more generally, the algebraic $K$-theory groups of any sufficiently "nice" Waldhausen category $\mathcal{A}$ of pointed small categories also vanishes, regardless of finiteness conditions assumed on the objects of $\mathcal{A}$. The vanishing of this $K$-theory implies that there is no nontrivial Euler characteristic defined on pointed small categories and satisfying certain niceness axioms.