Categorical Homotopy I. Quivers

Abstract

We quiver-interpret the classical simplicial theory -including the cosimplex category ∆, Dold-Kan correspondence, and Hochschild homologyas a certain Q-homotopy theory of type A. For the cyclic and cubical theories, we proceed analogously. Subsequently, we present far-reaching generalizations, using different types of quivers. Moreover, we explain how to construct certain categories as analogs of ∆, and associate to each a Q-homotopy theory. We provide many examples, including such theories of type D.