广义空间上的高阶范畴层
Sheaves of Higher Categories on Generalised Spaces
AI总结：
本文研究同伦层沿Yoneda嵌入的右Kan扩张，证明其保持同伦层并诱导等价，应用于微分几何向量丛下降、丛格拉斯曼下降及光滑场论值提取。
摘要：
我们研究了从范畴上的同伦层到其自由余完备化（即其预层范畴）的同伦右Kan扩张。原始范畴上的任何预拓扑都会在自由余完备化上诱导出一个典范的广义覆盖预拓扑。我们证明，对于这些预拓扑，沿Yoneda嵌入的同伦右Kan扩张保持取值于（足够好的）单纯模型范畴中的同伦层。此外，我们证明这诱导了原始范畴上的空间层与其自由余完备化上的保余极限空间层之间的等价。我们在几何和拓扑中给出三个应用：首先，我们证明微分几何向量丛沿微分几何空间的次约化下降。其次，我们推导出各种带联络的丛格拉斯曼满足$(\infty,2)$-范畴下降。最后，我们研究流形上的光滑博尔德型场论中的光滑微分同胚作用。我们展示这些光滑作用如何使我们能够从场论在博尔德范畴生成对象上的值一致地提取其在任何对象上的值。
英文摘要：
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.
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