Smooth Maps and the Category of Smooth Manifolds (Pt. I)

Abstract Nonsense

Point of Post: In this post we define what it means for a map between two manifolds to be smooth.

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Motivation

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We literally defined smooth manifolds to be the topological spaces where we will have a relatively sound meaning of what a “smooth map” is. Thus, it would seem that the first order of business is to fully define and explore this notion of smooth map. The basic idea though is precisely what we have said before. A map between smooth manifold will be smooth if it is smooth locally around each point and its image–when we think about the space locally as Euclidean space.

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The interesting part is that once we define smooth map we will then be able to define the category of (finite dimensional) smooth manifolds. We will then be able to discuss the functor which takes a smooth…

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