切触范畴中的嘉当演算
Cartan calculus in tangent categories
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中文总结 AI 辅助
研究切触范畴中构建嘉当演算所需结构，核心是由交换环对象\(R\)进行标量乘法，使切丛成\(R -\)模。在此基础上向量场李代数满足相关法则，各对象有李 - 莱因哈特形式的嘉当演算，还给出多种示例范畴。
中文摘要 AI 辅助
我们确定了在Rosický和Cockett-Cruttwell意义下的切触范畴中构建所有对象上的嘉当演算所需的结构。缺少的要素是由交换环对象\(R\)进行的标量乘法，它起到光滑实线的作用，为每个对象的切丛配备与切触结构兼容的\(R -\)模结构。我们证明在这些公理下，向量场的李代数通过导数作用于\(R -\)值函数环并满足莱布尼茨法则。即切丛是一个抽象李代数胚，向量场的李代数是函数环上的李 - 莱因哈特代数。因此，每个对象都带有由谢瓦莱 - 艾伦伯格复形及其微分、内导数和李导数给出的李 - 莱因哈特形式的嘉当演算。示例包括光滑流形、\(G -\)流形、李群胚、对数流形、射影流形、弹性微分空间、仿射和一般概型、分次流形以及仿射\(C^\infty -\)概型的切触范畴。
英文摘要
We determine the structure needed in a tangent category in the sense of Rosický and Cockett-Cruttwell to construct the Cartan calculus on all objects. The missing ingredient is a scalar multiplication by a commutative ring object $R$, playing the role of the smooth real line, which equips the tangent bundle of every object with the structure of an $R$-module compatible with the tangent structure. We show that under these axioms the Lie algebra of vector fields acts by derivations on the ring of $R$-valued functions and satisfies the Leibniz rule. In other words, the tangent bundle is an abstract Lie algebroid, so that the Lie algebra of vector fields is a Lie-Rinehart algebra over the ring of functions. Consequently, every object carries a Cartan calculus of Lie-Rinehart forms, given by the Chevalley-Eilenberg complex together with its differential, inner derivative, and Lie derivative. Examples include the tangent categories of smooth manifolds, $G$-manifolds, Lie groupoids, log manifolds, pro-manifolds, elastic diffeological spaces, affine and general schemes, graded manifolds, and affine $C^\infty$-schemes.
