标题： $(\infty,1)$-站点及相关形式结构 显示英文标题

标题： Notions of $(\infty,1)$-sites and related formal structures

摘要： 我们研究了在给定的$\infty$-范畴$\mathcal{C}$上的高阶站点的各种特征，这些特征在概念上与其经典的普通范畴对应物一致，并从中提取了一些关于$\infty$-到理论的新结果。 首先，从形式化的$(\infty,2)$-范畴理论出发，我们定义了$\infty$-到论上的高阶Lawvere-Tierney算子，这内化了Anel、Biedermann、Finster和Joyal以及Rijke、Shulman和Spitters的左精确模态的参数化版本。 Second, in the spirit of Lawvere's hyperdoctrines, we describe the $\infty$-toposes embedded in the $\infty$-category $\hat{\mathcal{C}}$ of presheaves over $\mathcal{C}$ as the sheaves of ideals of what we call the logical structure sheaf on $\mathcal{C}$. This naturally induces a notion of ''geometric kernels'' on $\mathcal{C}$ which play the part of higher Grothendieck topologies from the given perspective. 最后，我们研究了笛卡尔$\infty$-站点的$(\infty,1)$-范畴。 我们适当将Lurie书中的典范Grothendieck拓扑概念推广到所有几何核，并在最理想的情况下展示了相应的“比较引理”。 然而，我们在Grothendieck拓扑的背景下展示了该引理对应的拓扑版本是失败的。 显示英文摘要

摘要： We study various characterizations of higher sites over a given $\infty$-category $\mathcal{C}$ which are conceptually in line with their classical ordinary categorical counterparts, and extract some new results about $\infty$-topos theory from them. First, in terms of formal $(\infty,2)$-category theory, we define a notion of higher Lawvere-Tierney operators on $\infty$-toposes which internalizes a parametrized version of the left exact modalities of Anel, Biedermann, Finster and Joyal and the left exact modalities of Rijke, Shulman and Spitters. Second, in the spirit of Lawvere's hyperdoctrines, we describe the $\infty$-toposes embedded in the $\infty$-category $\hat{\mathcal{C}}$ of presheaves over $\mathcal{C}$ as the sheaves of ideals of what we call the logical structure sheaf on $\mathcal{C}$. This naturally induces a notion of ''geometric kernels'' on $\mathcal{C}$ which play the part of higher Grothendieck topologies from the given perspective. Lastly, we study the $\infty$-category of cartesian $(\infty,1)$-sites. We generalize the notion of canonical Grothendieck topologies from Lurie's book appropriately to all geometric kernels and show an according ''Comparison Lemma'' in the best case scenario. However, we show that a corresponding topological version of the lemma in the context of Grothendieck topologies fails.