2017/11/23 by Christian Frank, Frank, Christian, Andrew Salch +1
Mathematics · #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #K-Theory and Homology (math.KT) #math.AT #math.CT #math.KT
paper · pdf · doi:10.48550/arxiv.1711.08579
arxiv created 2017/11/23 · arxiv updated 2017/11/27
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 A of pointed small categories also vanishes, regardless of finiteness conditions assumed on the objects of 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.