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Contemporary Mathematics Homotopy Meaningful Hybrid Model Structures

Summary: Contemporary Mathematics
Homotopy Meaningful Hybrid Model Structures
Aaron D. Ames
Abstract. Hybrid systems are systems that display both discrete and contin-
uous 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."
1. Introduction
Hybrid systems are systems that display both continuous and discrete behavior
and so have important applications to robotic systems, e.g., mechanical systems un-
dergoing impacts such as bipedal robotic walkers are naturally modeled by systems


Source: Ames, Aaron - Department of Mechanical Engineering, Texas A&M University


Collections: Engineering