Abstract tangent functors

当物理学家探索空间和时间的各个方面时，对他们来说，拥有数学结构为他们的计算提供一个框架是很重要的。直到最近，许多物理对象的标准数学结构是被称为光滑流形的结构，而对光滑流形的抽象研究成为众所周知的微分几何。但物理学中的新对象迫使数学家和物理学家认识到，光有光滑的流形是不够的。最近的研究人员已经超越了光滑流形，发展了一系列更复杂的数学结构，如微分空间、费马空间、便利向量空间和合成微分几何模型。因此，一个紧迫的问题是理解如何评估和比较微分几何的这些不同扩展。数学或物理学的研究人员应该在什么时候使用微分空间，而不是方便的矢量空间？*为了帮助评估这些建议，这个研究计划将发展一种理论，即微分几何的扩展应该是什么样子的，最初是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，使他们能够选择适合他们需要的结构或能够根据需要转移到新结构。********

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. ********