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

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.