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6 - Connecting Finite-Dimensional, Infinite-Dimensional and Higher Geometry

Published online by Cambridge University Press:  08 December 2022

Alexander Schmeding
Affiliation:
Nord Universitet, Norway
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Summary

In this chapter, we will highlight the interesting connection between finite and infinite dimensional differential geometry. To this end, we shall consider Lie groupoids, which can be understood as elements from higher geometry. The moniker higher geometry stems from the fact that in the language of category theory, these objects form higher categories. Previously we discussed how finite-dimensional manifolds and geometric structures give rise to infinite-dimensional structures such as Lie groups (e.g. the diffeomorphims and groups of gauge transformations) and Riemannian metrics (such as the L^2-metric from shape analysis). While a manifold determines an (in general infinite-dimensional) group of diffeomorphisms, we turn this observation now on its head and investigate whether the underlying finite-dimensional geometric structure is recognisable from its associated infinite dimensional object.

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Publisher: Cambridge University Press
Print publication year: 2022
Creative Commons
Creative Common License - CCCreative Common License - BYCreative Common License - NCCreative Common License - ND
This content is Open Access and distributed under the terms of the Creative Commons Attribution licence CC-BY-NC-ND 4.0 https://creativecommons.org/cclicenses/

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