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Homology for dynamical systems
Homotopy theory studies the continuous transformation between topological objects. The focus is on characterizing the essential properties of topological objects that remain unchanged under distortions. Homology theory is the mathematical means of characterizing the number of holes within an object. One of the foundational principles of topology is that homology is invariant under homotopy. This makes homology one of the most useful algebraic characterizations of homotopy invariance.
HmtpyHmlgyOutline1.pdfA surprisng aspect of the topological notions of homotopy and homology is that they can be extended to any category C [1]. The building blocks are outlined in this flowchatt. Δ above represents the combinatorial category of simplexes. Any functor F from Δ to C corresponds to a choice of simplicial objects in C, along with the notion of face maps and projections. A notion of simplicial objects alone is reponsible for providing a Homology theory for C. The order simplicial object, along with some consistency rules (Axioms 1-4), is responsible for a Homotopy theory for C. The remaining key ingredient is a notion of convexity (Axiom 5). This makes homology homotopy invariant.
An exciting new possibility is the construction of Homotopy and Homology theory for categories C other than topology. C can be chosen to be the category of dynamical systems [6. 7]. This provides a vast collection of opportunities to find concrete realizations of abstract Categorical axioms in the realm of Dynamical systems. The challenge is to adhere to the mathematical rules laid out in [1], while at the same time remaining committed to recreating important dynamic phenomenon such as periodic cycles, tori and chaotic sets.
Future directions.
1) Construct notions of simplicial objects in dynamical systems, forming a cascade of elementary dynamical systems of increasing complexity.
2) Construct a notion of convexity for dynamical systems.
3) Determine the Realization functor for dynamical systems, which converts finite collection of homology numbers into simplicial complexes.
4) Creae algorithms analogous to those in topological data analysis [4,5], which coverts timeseries data into simplicial complexes of dynamical systems.
Prerequisites.
1) Foundational mathematics : familiatity with proofs and mathematical rigor; set-theory
2) Category theory : functors, categories, natural transformations
3) Analysis : continuity and differntiability, limits and convergence
4) Measure theory : probability measures, absolute continuity, Markov transitions, Borel sets, abstract Measure theory
Sources
[1] Homology and homotopy for arbitrary categories - S Das (foundational paper presenting the categorical version of homotopy-homology)
[2] Functors induced by comma categories - S Das (exposition on comma categories)
[3] J. Rotman. An introduction to algebraic topology, volume 119. Springer S (Algebraic topology presented in the language of category theory)
[4] Persistence Diagrams as Diagrams - A Categorification of the Stability Theorem - M Lesnick, U Bauer (topological data analysis)
[5] Simplicial Complex Representation Learning - M Hajij (topological data analysis)
[6] Dynamics, data and reconstruction - S. Das, T. Suda
[7]Dynamical systems as enriched functors - S. Das, T. Suda
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