标题： 李范畴基础与乘法埃雷斯曼联络的杨-米尔斯理论 显示英文标题

标题： Fundamentals of Lie categories and Yang-Mills theory for multiplicative Ehresmann connections

摘要： 论文的第一部分较短，探讨了李群胚中的可逆性结构假设。当放弃这一假设时，我们得到李范畴的概念：一个具有兼容微分结构的小范畴。我们引入了各种李范畴的例子，研究它们与李群胚之间的差异和相似之处，并研究由于箭头不可逆而自然出现的概念。论文第二部分的主要目的是提供杨-米尔斯理论的广泛推广，将其从经典的主丛设置扩展到一般的李群胚和李代数胚。主丛的连接概念现在被更一般的乘法埃雷斯曼连接所取代。在实现这一推广过程中，我们对这类连接的理论以及表示上的不变线性连接进行了多项进展。我们开发了它们存在的障碍类，将（水平）外协变导数推广到表示值的博特-舒尔曼-斯塔谢夫和魏尔复形，并检查它们与范埃斯特映射的关系。我们研究了具有同调平凡曲率的乘法连接类，这些连接对于获得所需的推广至关重要。将变分原理应用于这一框架依赖于我们为乘法连接的仿射变形开发的公式。最终，我们将杨-米尔斯理论扩展到非可积且非传递的设置：经典杨-米尔斯方程升级为一个规范不变的方程组，现在描述规范场在（奇异）轨道叶状结构的纵向和横向方向上的动力学。作为例子，我们得到了一个针对$S^1$-束gerbes 的杨-米尔斯理论。 显示英文摘要

摘要： The first and shorter part of this thesis deals with the structural assumption of invertibility in a Lie groupoid. When this assumption is dropped, we obtain the notion of a Lie category: a small category, endowed with a compatible differentiable structure. We introduce various examples of Lie categories, examine their differences and similarities with Lie groupoids, and research the notions emerging naturally from the lack of invertibility of arrows. The aim of the second and principal part of this thesis is to provide a far-reaching generalization of Yang-Mills theory, extending it from the classical setting of principal bundles to general Lie groupoids and algebroids. The notion of a principal bundle connection is now replaced with that of a more general multiplicative Ehresmann connection. In obtaining this generalization, we make various advances to the theory of such connections, as well as invariant linear connections on representations. We develop the obstruction classes for their existence, generalize the (horizontal) exterior covariant derivative to the representation-valued Bott-Shulman-Stasheff and Weil complexes, and inspect their relationship with the van Est map. We research the class of multiplicative connections with cohomologically trivial curvature, which are central to obtaining the desired generalization. Applying the variational principle to this framework rests upon our developed formulae for affine deformations of multiplicative connections. Ultimately, we develop the extension of Yang-Mills theory to a non-integrable and non-transitive setting: the classical Yang-Mills equation is upgraded to a gauge-invariant pair of equations, which now describe the dynamics of gauge fields in both the longitudinal and transversal directions with respect to the (singular) orbit foliation. As an example, we obtain a Yang-Mills theory for $S^1$-bundle gerbes.