论文标题
在广义空间上的较高类别的系带
Sheaves of Higher Categories on Generalised Spaces
论文作者
论文摘要
我们研究了同轴轴承的同型右键扩展，即在其自由共完成的类别上，即其在其前置类别的类别中。原始类别上的任何自制性都会引起自由结合的广义覆盖物的规范自我制体。我们表明，关于这些自我示范性，沿Yoneda嵌入YONEDA的右KAN扩展，固定在（足够好的）简单模型类别中。此外，我们表明，这在原始类别的空间系带和其自由共完成的空间延伸带托管之间具有等效性。我们在几何和拓扑中介绍了三个应用：首先，我们证明，差异矢量束沿差异空间的俯冲下降。其次，我们推断出具有连接的各种捆绑胶ger的口味满足$（\ infty，2）$ - 分类下降。最后，我们研究了流形的平滑伴侣型场理论中平滑的差异性作用。我们展示了这些平滑的动作如何使我们能够从其在生成bordism类别的对象的值中相干地提取字段理论的值。
We study the homotopy right Kan extension of homotopy sheaves on a category to its free cocompletion, i.e. to its category of presheaves. Any pretopology on the original category induces a canonical pretopology of generalised coverings on the free cocompletion. We show that with respect to these pretopologies the homotopy right Kan extension along the Yoneda embedding preserves homotopy sheaves valued in (sufficiently nice) simplicial model categories. Moreover, we show that this induces an equivalence between sheaves of spaces on the original category and colimit-preserving sheaves of spaces on its free cocompletion. We present three applications in geometry and topology: first, we prove that diffeological vector bundles descend along subductions of diffeological spaces. Second, we deduce that various flavours of bundle gerbes with connection satisfy $(\infty,2)$-categorical descent. Finally, we investigate smooth diffeomorphism actions in smooth bordism-type field theories on a manifold. We show how these smooth actions allow us to extract the values of a field theory on any object coherently from its values on generating objects of the bordism category.
