AI 中文总结
该研究探讨局部凸李群胚与李代数胚的严格逆系统，建立函子同构，给出可积巴拿赫-李代数胚逆系统的可积化条件，描述源单连通性障碍，并用射流群胚等实例阐释理论。
AI 中文摘要
我们研究局部凸李群胚与李代数胚的严格逆系统，证明李代数胚与李群胚的严格逆极限继承自然的李结构，并建立函子的自然同构：$\boldsymbol{\text{lim}}\boldsymbol{\text{Lie}}\boldsymbol{\text{Lie}}\boldsymbol{\text{lim}}$。随后考虑可积巴拿赫-李代数胚的逆系统及其逐源单连通的可积化，给出这些可积化构成李群胚严格逆系统的条件，从而其逆极限可积分逆极限代数胚。在源纤维逆序列的额外假设下，我们通过源纤维第二同伦群的导出逆极限$\boldsymbol{\text{lim}}^1\boldsymbol{\text{pi}}_2$描述所得可积化的源单连通性障碍。我们以多个例子阐释该理论：射流群胚、电流群胚与规范群胚的逆极限，包括对应的逆极限阿蒂亚代数胚，以及源自四元数霍普夫丛的源单连通逆极限可积化。
英文摘要
We study strict inverse systems of locally convex Lie groupoids and Lie algebroids. We show that strict inverse limits of Lie algebroids and Lie groupoids inherit natural Lie structures and establish a natural isomorphism of functors $\varprojlim\circ\operatorname{Lie}\cong\operatorname{Lie}\circ\varprojlim$. We then consider inverse systems of integrable Banach-Lie algebroids and their source-simply connected levelwise integrations. We give conditions under which these integrations form a strict inverse system of Lie groupoids and hence their inverse limit integrates the inverse-limit algebroid. Under additional hypotheses on the source-fiber inverse sequence, we describe the obstruction to source-simply connectedness of the resulting integration in terms of the derived inverse limit $\varprojlim^{1}π_2$ of the second homotopy groups of the source fibers. We illustrate the theory with several examples: inverse limits of jet groupoids, current groupoids, and gauge groupoids, including the corresponding inverse-limit Atiyah algebroids and a source-simply connected inverse-limit integration arising from the quaternionic Hopf bundle.
