Categorical Homotopy I. Quivers

We quiver-interpret the classical simplicial theory - including the cosimplex category $Δ$, 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 $Δ$, and associate to each a Q-homotopy theory. We provide many examples, including such theories of type $D$.

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