Parametrized and Functional Differential Geometry

Abstract

The aim of this chapter is to present flexible tools to do differential geometry on spaces of fields, i.e., on spaces of “functions” between generalized “spaces”. We choose here a viewpoint that is directly adapted to homotopical generalizations. Our general approach may be described in the particular case of differential geometry. The category Open \({}_{\mathcal{C}^{\infty}}\) of open subsets \(V\subset \mathbb {R}^{n}\) for varying n, with smooth maps between them, has the following drawbacks: it doesn’t contain enough limits (e.g., solutions spaces for equations or fiber products), doesn’t contain enough colimits (e.g., quotients), and doesn’t contain “spaces of fields”, i.e., spaces of functions. The solution to this problem, introduced by Grothendieck in algebraic geometry and Lawvere and Ehresmann in categorical logic, is to use the Yoneda embeddings of Open \({}_{\mathcal{C}^{\infty}}\) into the functor categories Hom(Open \({}^{op}_{\mathcal{C}^{\infty}}\),Sets) and Hom(Open \(_{\mathcal{C}^{\infty}}\),Sets), that have all the desired limits, colimits and mapping spaces, and to restrict the class of functors (by some limit-commutation conditions) to have some permanence properties for geometric constructions (e.g., transversal fiber products, pastings) with open subsets. In this chapter, we describe the two approaches to geometry and differential calculus that will be used in this book, called parametrized and functional geometry, and apply them to differential, super and graded geometry.