Abstract tangent functors

当物理学家探索空间和时间的各个方面时，拥有数学结构为他们的计算提供框架是很重要的。 直到最近，许多物理对象的标准数学结构是称为光滑流形的结构，而光滑流形的抽象研究被称为微分几何。 但是物理学中的新对象迫使数学家和物理学家认识到光滑流形是不够的。 最近的研究人员已经超越了光滑流形，开发了一系列更复杂的数学结构，如几何空间，费马空间，方便的向量空间和综合微分几何模型。 一个紧迫的问题，然后，是了解如何评估和比较这些不同的扩展微分几何。 什么时候数学或物理学的研究人员应该使用拓扑空间，比如说，与方便的向量空间相反？ * 为了帮助评估这些建议，这个研究计划将发展一个微分几何的扩展应该是什么样子的理论，最初基于研究切丛函子的抽象性质。 通过为微分几何的不同扩展开发一个抽象的框架，研究人员可以比较和对比不同的结构，使他们能够选择一个适合他们需要的结构，或者能够根据需要转移到新的结构。********

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. ********