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

Abstract: 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.