American University
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Sheaf Morphisms Induced by Dynamical Systems

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posted on 2023-08-03, 13:48 authored by Jackson Williams

Interrelated sets of data can be modeled in the form of a mathematical object called a sheaf. Essentially, a sheaf over a collection of sets represents a hypothesis about exactly how the data in those sets affect one another. This structure has many useful properties for data analysis and is the optimal way to think about the relationships between all sorts of data sets. Now similarly, a dynamical system acting on a set is a way of modeling relationships within a set of data and how they progress over time. Dynamical systems model many physical events or other patterns that can be found in the real world and can be used to make predictions about the future state of a set of data or even of many related sets of data. Bringing these two modeling concepts together we will demonstrate how, firstly, a dynamical system over a set of data can be represented in the form of a sheaf, and that doing so will provide all the analytic tools of sheaf theory to the study of that dynamical system. Secondly we will show that when such a sheaf is constructed, the progression of the time parameter of the dynamical system induces functions on the stalks of that sheaf and this set of new functions is in fact a sheaf morphism. Lastly we will show a few examples of this construction being done and how in some cases it can be done in more than one way.



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