Categorical Homotopy I. Quivers

TL;DR

This paper develops a quiver-based interpretation of classical homotopy theories, generalizing simplicial, cyclic, and cubical theories through Q-homotopy frameworks and exploring new types of quivers.

Contribution

It introduces a novel quiver-interpretation of homotopy theories and generalizes these concepts to various quiver types, expanding the categorical framework.

Findings

Unified quiver-based framework for classical homotopy theories

Generalizations to different quiver types including D

Construction of new categories as analogs of Δ

Abstract

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 $D$.