发表机构
Macquarie University, School of Mathematical and Physical Sciences（麦考瑞大学数学与物理科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在切触范畴中发展内部李群-李代数对应，构造李函子，证明其切可表示性对应李群，导出微分李代数，恢复微分与代数几何中的经典李对应。
AI 中文摘要
经典李理论建立了李群（光滑流形范畴中的内部群对象）与李代数之间的对应关系，仿射概型中的群对象也存在类似对应。光滑流形和仿射概型均构成切触范畴，切触范畴为微分几何提供了范畴论框架，因此自然可问李对应是否可完全由切触结构构造。本文基于Cockett与Schwarz的工作，在切触范畴中发展了内部李群-李代数对应：将李群对象定义为切触范畴中在单位元处具有切空间的群对象；对任意群对象构造李函子，证明其切可表示性恰在该群对象为李群时成立；可表示性导出内部李代数，其底对象为单位元处的切空间；证明该李代数是微分李代数，其双线性李括号由伴随表示诱导，并将该构造扩展为从李群到微分李代数的函子；最终在微分几何与代数几何中均恢复了通常的李对应。
英文摘要
Classic Lie theory establishes a correspondence between Lie groups, which are internal group objects in the category of smooth manifolds, and Lie algebras. An analogous correspondence also exists for group objects in affine schemes. Both smooth manifolds and affine schemes form tangent categories, which provide a categorical context for differential geometry. Therefore, it is natural to ask whether the Lie correspondence can be constructed entirely from the tangent structure. Building on work of Cockett and Schwarz, we develop an internal Lie group-Lie algebra correspondence in tangent categories. We introduce Lie group objects as group objects in a tangent category which admit a tangent space at the unit. For any group object, we construct a Lie functor and show it is tangentially representable exactly when the group object is a Lie group. Representability then yields an internal Lie algebra whose underlying object is the tangent space at the unit. We prove this Lie algebra is a differential Lie algebra, with a bilinear Lie bracket induced by the adjoint representation, and we extend the construction to a functor from Lie groups to differential Lie algebras. Finally, we recover the usual Lie correspondences in both differential and algebraic geometry.
