n-群胚的exact sequences的Z形图

On the homotopy theory for Lie $\infty$-groupoids, with an application to integrating $L_\infty$-algebras

Lie $\infty$-groupoids are simplicial Banach manifolds that satisfy an analog of the Kan condition for simplicial sets. An explicit construction of Henriques produces certain Lie $\infty$-groupoids called `Lie $\infty$-groups' by integrating finite type Lie $n$-algebras. In order to study the compatibility between this integration procedure and the homotopy theory of Lie $n$-algebras introduced in the companion paper arXiv:1809.05999, we present a homotopy theory for Lie $\infty$-groupoids. Unlike Kan simplicial sets and the higher geometric groupoids of Behrend and Getzler, Lie $\infty$-group…

Cohomology for small categories: $k$-graphs and groupoids

Given a higher-rank graph $Λ$, we investigate the relationship between the cohomology of $Λ$ and the cohomology of the associated groupoid $G_Λ$. We define an exact functor between the abelian category of right modules over a higher-rank graph $Λ$ and the category of $G_Λ$-sheaves, where $G_Λ$ is the path groupoid of $Λ$. We use this functor to construct compatible homomorphisms from both the cohomology of $Λ$ with coefficients in a right $Λ$-module, and the continuous cocycle cohomology of $G_Λ$ with values in the corresponding $G_Λ$-sheaf, into the sheaf cohomology of $G_Λ$.

李群胚上主$2$-丛的Atiyah序列与规范变换

Atiyah sequence and Gauge transformations of a principal $2$-bundle over a Lie groupoid

本文引入李群胚上主$2$-丛概念，构造其Atiyah序列，定义严格与半严格联络，给出联络构造方法与存在性判据，研究规范变换$2$-群的作用并指出半严格联络范畴的扩展对称性。

微分空间中的 Čech-de Rham 阻碍

The Diffeological Čech-de Rham Obstruction

本文利用高阶拓扑斯理论研究微分空间中 Čech-de Rham 映射成为同构的阻碍，构造了各次数的正合序列，并证明了无理环面的 ∞-叠层上同调与群上同调的一致性以及两类带联络微分主丛群胚的等价性。

Higher commutator conditions for extensions in Mal'tsev categories

We define a Galois structure on the category of pairs of equivalence relations in an exact Mal'tsev category, and characterize central and double central extensions in terms of higher commutator conditions. These results generalize both the ones related to the abelianization functor in exact Mal'tsev categories, and the ones corresponding to the reflection from the category of internal reflexive graphs to the subcategory of internal groupoids. Some examples and applications are given in the categories of groups, precrossed modules, precrossed modules of Lie algebras, and compact groups.