Introduction to the language of stacks and gerbes

Sheaf Theory for Étale Geometric Stacks

We generalize the notion of a small sheaf of sets over a topological space or manifold to define the notion of a small stack of groupoids over an étale topological or differentiable stack. We then provide a construction analogous to the étalé space construction in this context, establishing an equivalence of 2-categories between small stacks over an étale stack and local homeomorphisms over it. We go on to characterize small sheaves and gerbes. We show that ineffective data of étale stacks is completely described by the theory of small gerbes. Furthermore, it is shown that étale stacks (and in…

拓扑群胚的扩张以及 Serre、Hurewicz 态射

Extension of topological groupoids and Serre, Hurewicz morphisms

该研究提出拓扑群胚扩张概念，将其关联于拓扑栈上的 gerbe，并考察 Serre、Hurewicz 栈上 gerbe 的性质。

Butterflies II: Torsors for 2-group stacks

We study torsors over 2-groups and their morphisms. In particular, we study the first non-abelian cohomology group with values in a 2-group. Butterfly diagrams encode morphisms of 2-groups and we employ them to examine the functorial behavior of non-abelian cohomology under change of coefficients. We re-interpret the first non-abelian cohomology with coefficients in a 2-group in terms of gerbes bound by a crossed module. Our main result is to provide a geometric version of the change of coefficients map by lifting a gerbe along the "fraction" (weak morphism) determined by a butterfly. As a pra…

主无穷-丛——一般理论

Principal infinity-bundles -- General theory

本文在任意无穷-拓扑斯中建立了主无穷-丛的一般理论，证明其与非阿贝尔上闭链自然等价并由非阿贝尔层超上同调分类，同时将相伴无穷-丛、无穷-格贝与扭曲上同调纳入统一框架。

流形上的高阶几何结构与 Deligne 上同调的规范理论

Higher Geometric Structures on Manifolds and the Gauge Theory of Deligne Cohomology

本文构造了流形上高阶几何结构的光滑高阶对称群及其模 $\infty$-栈，研究其同伦类型与等价判据，并应用于高阶 $\mathrm{U}(1)$-联络与 NSNS 超引力解的模 $\infty$-栈。

The van Est Map on Geometric Stacks

We generalize the van Est map and isomorphism theorem in three ways, and we discuss conjectured connections with homotopy theory, including a proposal of a category which unifies differentiable stacks, Lie algebroids and homotopy theory. In Part 2 of this thesis we generalize the van Est map from a comparison map between Lie groupoid cohomology and Lie algebroid cohomology to a (more conceptual) comparison map between the cohomology of a stack $\mathcal{G}$ and the cohomology of a simple foliation $\mathcal{H}\to\mathcal{G}$. In Part 1 we generalize the functions that we can take cohomology of…