  We quiver-interpret the classical simplicial theory - including the cosimplex
category $\Delta$, Dold-Kan correspondence, and Hochschild homology - as 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 $\Delta$, and associate to each a Q-homotopy theory.
We provide many examples, including such theories of type $D$.
