Homotopy Meaningful Hybrid Model Structures

Abstract

Hybrid systems are systems that display both discrete and continuous behavior and, therefore, have the ability to model a wide range of robotic systems such as those undergoing impacts. The main observation of this paper is that systems of this form relate in a natural manner to very special diagrams over a category, termed hybrid objects. Using the theory of model categories, which provides a method for "doing homotopy theory" on general categories satisfying certain axioms, we are able to understand the homotopy theoretic properties of such hybrid objects in terms of their "non-hybrid" counterparts. Specifically, given a model category, we obtain a "homotopy meaningful" model structure on the category of hybrid objects over this category with the same discrete structure, i.e., a model structure that relates to the original non-hybrid model structure by means of homotopy colimits, which necessarily exist. This paper, therefore, lays the groundwork for "hybrid homotopy theory."