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Smooth coalgebra: testing vector analysis

Published online by Cambridge University Press:  14 December 2015

DUSKO PAVLOVIC  and
BERTFRIED FAUSER
DUSKO PAVLOVIC
Affiliation:
Information and Computer Sciences Department, University of Hawaii, HI 96822 Honolulu, USA Email: dusko@hawaii.edu
BERTFRIED FAUSER
Affiliation:
Mathematisch- Naturwissenschaftliche Sektion, University of Konstanz, 78464 Konstanz, Germany Email: bertfried.fauser@gmail.com
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Abstract

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Processes are often viewed as coalgebras, with the structure maps specifying the state transitions. In the simplest case, the state spaces are discrete, and the structure map simply takes each state to the next states. But the coalgebraic view is also quite effective for studying processes over structured state spaces, e.g. measurable, or continuous. In the present paper, we consider coalgebras over manifolds. This means that the captured processes evolve over state spaces that are not just continuous, but also locally homeomorphic to normed vector spaces, and thus carry a differential structure. Both dynamical systems and differential forms arise as coalgebras over such state spaces, for two different endofunctors over manifolds. A duality induced by these two endofunctors provides a formal underpinning for the informal geometric intuitions linking differential forms and dynamical systems in the various practical applications, e.g. in physics. This joint functorial reconstruction of tangent bundles and cotangent bundles uncovers the universal properties and a high-level view of these fundamental structures, which are implemented rather intricately in their standard form. The succinct coalgebraic presentation provides unexpected insights even about the situations as familiar as Newton's laws.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 27 , Special Issue 7: Special Issue: Coalgebraic Logic , October 2017 , pp. 1195 - 1235
Copyright
Copyright © Cambridge University Press 2015 

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