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Homotopy Meaningful Hybrid Model Structures

Ames, Aaron D. (2007) Homotopy Meaningful Hybrid Model Structures. In: Topology and Robotics. Contemporary Mathematics Series. No.438. American Mathematical Society , Providence, RI, pp. 121-144. ISBN 978-0-8218-4246-1 . https://resolver.caltech.edu/CaltechAUTHORS:20100819-104123296

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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."


Item Type:Book Section
Additional Information:© 2007 American Mathematical Society. The author is indebted to Mariusz Wodzicki for the many insightful discussions on model categories and their role in hybrid systems, and to Shankar Sastry for his support. This research was supported in part by NSF award #CCR-0225610.
Funders:
Funding AgencyGrant Number
NSFCCR-0225610
Series Name:Contemporary Mathematics Series
Issue or Number:438
Classification Code:2000 Mathematics Subject Classification. Primary 18G55, 18A25; Secondary 18D99, 18A30
Record Number:CaltechAUTHORS:20100819-104123296
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20100819-104123296
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19509
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:19 Aug 2010 23:55
Last Modified:03 Oct 2019 01:58

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