[论文解读] Differential bundles and fibrations for tangent categories

J.R.B. Cockett, G. S. H. Cruttwell|arXiv (Cornell University)|Jun 27, 2016

Homotopy and Cohomology in Algebraic Topology参考文献 18被引用 11

一句话总结

本文在切范畴中引入了微分丛与纤维化结构，无需假设标量乘法或局部平凡性，即对光滑向量丛进行了推广。在显示切范畴中，微分丛与纤维范畴中的微分对象一一对应，并证明了微分丛的纤维化结构构成一个一致的微分切纤维化结构，通过纤维化结构统一了笛卡尔微分范畴与切范畴。

ABSTRACT

Tangent categories are categories equipped with a tangent functor: an endofunctor with certain natural transformations which make it behave like the tangent bundle functor on the category of smooth manifolds. They provide an abstract setting for differential geometry by axiomatizing key aspects of the subject which allow the basic theory of these geometric settings to be captured. Importantly, they have models not only in classical differential geometry and its extensions, but also in algebraic geometry, combinatorics, computer science, and physics. This paper develops the theory of "differential bundles" for such categories, considers their relation to "differential objects", and develops the theory of fibrations of tangent categories. Differential bundles generalize the notion of smooth vector bundles in classical differential geometry. However, the definition departs from the standard one in several significant ways: in general, there is no scalar multiplication in the fibres of these bundles, and in general these bundles need not be locally trivial. To understand how these differential bundles relate to differential objects, which are the generalization of vector spaces in smooth manifolds, requires some careful handling of the behaviour of pullbacks with respect to the tangent functor. This is captured by "transverse" and "display" systems for tangent categories, which leads one into the fibrational theory of tangent categories. A key example of a tangent fibration is provided by the "display" differential bundles of a tangent category with a display system. Strikingly, in such examples the fibres are Cartesian differential categories demonstrating a -- not unexpected -- tight connection between the theory of these categories and that of tangent categories.

研究动机与目标

• 在不假设标量乘法或局部平凡性的前提下，将微分几何中光滑向量丛的概念推广至抽象切范畴。
• 厘清切范畴中微分丛与微分对象（向量空间的推广）之间的精确关系。
• 通过横截与显示系统发展切范畴的纤维化理论，实现微分构造的结构一致性。
• 证明在显示切范畴中，微分丛的纤维化继承了协调的微分结构，从而将切范畴与笛卡尔微分范畴联系起来。

提出的方法

• 将微分丛定义为配备满足特定公理的竖直提升映射的加法丛，这些公理由经典微分几何中的竖直提升映射推广而来。
• 利用横截与显示系统管理拉回与切函子之间的相互作用，确保与切结构的兼容性。
• 定义切纤维化结构，并证明在显示切范畴上，微分丛的纤维化构成一个微分切纤维化结构。
• 通过代换函子与拉回操作，证明微分丛在基变换下保持不变，推广了经典拉回不变性。
• 证明显示切范畴中该丛纤维化的每个纤维本身都是一个笛卡尔切范畴，从而支持内部微分结构。
• 通过公理[CDS.1]与[CDS.2]证明微分结构的一致性，表明陈类和局部竖直切丛满足所需的相容性条件。

实验结果

研究问题

• RQ1 在不假设标量乘法或局部平凡性的前提下，如何在切范畴中抽象地推广向量丛？
• RQ2 切范畴中微分丛与微分对象之间的确切关系是什么？
• RQ3 拉回与切函子在切范畴中如何相互作用？何种结构控制这种相互作用？
• RQ4 在何种条件下，微分丛的纤维化能继承一致的微分结构？
• RQ5 切范畴中的显示系统如何实现对微分丛及其一致性的纤维化处理？

主要发现

• 在切范畴中，微分丛在不依赖标量乘法或局部平凡性的前提下，通过满足特定公理的竖直提升映射得到推广。
• 在显示切范畴中，对象M上的微分丛与该丛纤维化在M上的纤维范畴中的微分对象一一对应。
• 显示切范畴中该丛纤维化的每个纤维本身都是一个笛卡尔切范畴，支持内部微分结构。
• 在显示切范畴上，微分丛的纤维化构成一个微分切纤维化结构，其微分结构满足公理[CDS.1]与[CDS.2]。
• 微分丛的拉回在基变换下保持不变，且该结果通过纤维范畴之间强切函子的范畴论方法得以概念化证明。
• 通过零截面代换构造的局部竖直切丛满足一致性条件[CDS.2]，确保了微分结构的相容性。