It is a classic result that the geometry of the total space of a principal bundle with reference to the action of the bundle's structure group is codified in the bundle's operation, a collection of derivations comprising the de Rham differential and the contraction and Lie derivatives of all vertical vector fields and obeying the six Cartan relations. In particular, connections and gauge transformations can be defined through the way they are acted upon by the operation's derivations. In this paper, the first of a series of two extending the ordinary theory, we construct an operational total space theory of strict principal 2--bundles with regard to the action of the structure strict 2--group. Expressing this latter via a crossed module $(\mathsans{E},\mathsans{G})$, the operation is based on the derived Lie group $\mathfrak{e}[1]\rtimes\mathsans{G}$. In the second paper, an original formulation of the theory of $2$--connections and $1$-- and $2$--gauge transformations based on the operational framework worked out here will be provided.

中文翻译：

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主2束的操作全空间理论I：操作几何框架

一个经典的结果是，主丛的总空间的几何形状与丛结构群的作用有关，被编纂在丛的运算中，一组导数包括所有的 de Rham 微分和收缩和李导数。垂直向量场并遵守六个嘉当关系。特别是，连接和规范转换可以通过操作派生对它们的作用方式来定义。在本文中，作为扩展普通理论的两个系列中的第一个，我们构建了严格主 2-丛的操作全空间理论关于结构严格 2-群的作用。通过交叉模块 $(\mathsans{E},\mathsans{G})$ 表示后者，该运算基于导出的李群 $\mathfrak{e}[1]\rtimes\mathsans{G}$。在第二篇论文中，将提供基于此处制定的操作框架的 $2$--connections 和 $1$-- 和 $2$--gauge 转换理论的原始公式。