Abstract tangent functors

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As physicists explore aspects of space and time, it is important for them to have mathematical structures to provide a framework for their calculations. Until recently, the standard mathematical structure for many physical objects was the structure known as a smooth manifold, and the abstract study of smooth manifolds became known as differential geometry. But new objects in physics have forced mathematicians and physicists to realize that smooth manifolds were not enough. Recent researchers have moved beyond smooth manifolds, developing a wide array of more complicated mathematical structures such as diffeological spaces, Fermat spaces, convenient vector spaces, and models of synthetic differential geometry. One pressing question, then, is to understand how to evaluate and compare these different extensions of differential geometry. When should researchers in mathematics or physics use diffeological spaces, say, as opposed to convenient vector spaces? To help evaluate these proposals, this research program will develop a theory of what an extension of differential geometry should look like, initially based on studying the abstract properties of the tangent bundle functor. By developing an abstract framework for different extensions of differential geometry, researchers can then compare and contrast the different structures, enabling them to choose one appropriate for their needs or be able to move to new structures as required.