黎曼几何中的dagger范畴
Dagger Categories in Riemannian Geometry
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中文总结 AI 辅助
本文在黎曼几何中研究dagger范畴，证明dagger编码度量与黎曼结构，分类微分算子范畴上的dagger，并揭示其与度量联络及格林公式的联系。
中文摘要 AI 辅助
范畴上的dagger为每个态射$f \colon X \to Y$分配一个态射$f^\dagger \colon Y \to X$，以逆变且对合的方式，正如转置逆转实矩阵。我们证明，在线性代数和微分几何的范畴上，dagger编码了度量结构和黎曼结构。众所周知，在单个向量空间上，自同态环的反对合是一个形式的伴随，该形式由标量决定至多一个因子。但更一般地，在全体有限维向量空间的范畴上，一个dagger是标量对合连同每个空间上的唯一幺模Hermitian形式的伴随。同样的结论对交换环上的有限生成投射模成立，因此，通过Serre-Swan对应，对向量丛也成立，其中dagger是丛度量的选择。dagger结构在范畴上表达了正交性的概念：正交补和法丛获得泛性质，Hermitian模构成dagger紧致范畴。此外，李群、李群胚及其无穷小对应上的dagger捕捉了酉表示和度量联络：一个联络是度量的，当且仅当其平行移动是一个dagger函子，或者无穷小地，当且仅当其Atiyah序列的分裂与dagger交换。最后，我们分类了李-林哈特代数微分算子局部分次范畴上的dagger。它们对应于度量-散度结构，由实结构、散度以及每个有限生成投射模上的幺模Hermitian形式组成，并且它们作为形式伴随作用于算子；格林公式表达了算子与其伴随之间的差为一个散度。
英文摘要
A dagger on a category assigns to every morphism $f \colon X \to Y$ a morphism $f^\dagger \colon Y \to X$, contravariantly and involutively, in the way the transpose reverses a real matrix. We show that on the categories of linear algebra and differential geometry, daggers encode metric and Riemannian structures. As is known, on a single vector space, an anti-involution of the endomorphism ring is the adjoint of a form that is determined up to a scalar. But more generally, on the category of all finite-dimensional vector spaces, a dagger is the adjoint for an involution of the scalars together with a unique unimodular Hermitian form on every space. The same holds for finitely generated projective modules over a commutative ring, and hence, through the Serre-Swan correspondence, for vector bundles, where a dagger is a choice of bundle metrics. The dagger structure expresses notions of orthogonality categorically: orthogonal complements and normal bundles acquire universal properties, and Hermitian modules form a dagger compact category. Moreover, daggers on Lie groups, Lie groupoids and their infinitesimal counterparts capture unitary representations and metric connections: a connection is metric precisely when its parallel transport is a dagger functor, or, infinitesimally, when its splitting of the Atiyah sequence commutes with the daggers. Finally, we classify the daggers on the locally graded category of differential operators of a Lie-Rinehart algebra. They correspond to metric-divergence structures, consisting of a real structure, a divergence, and a unimodular Hermitian form on every finitely generated projective module, and they act on operators as formal adjoints; Green's formula expresses the difference between an operator and its adjoint as a divergence.
发表机构
- University of Oxford（牛津大学）
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