Sheaves of $(\infty, \infty)$-categories

Abstract: We provide a functorial presentation of the $(\infty, 1)$-category of sheaves of $(n, r)$-categories for all $-2 \leq n\leq\infty$ and $0 \leq r\leq n+2$ based on complete Segal space objects. In this definition, the equivalences of sheaves of $(\infty, \infty)$-categories are defined inductively, so we also provide a localisation at the coinductive equivalences to define the $(\infty, 1)$-category of sheaves of $\omega$-categories as well. We prove that the $(\infty, 1)$-category of sheaves of $(\infty, \infty)$-categories and the $(\infty, 1)$-category of sheaves of $\omega$-categories both define distributors over the underlying topos of sheaves of spaces. Moreover, we show that these distributors define terminal and initial fixed points with respect to the construction of complete Segal space objects in a distributor. We conclude with a sheafification result: the category of sheaves of $(n, r)$-categories over a site $\mathscr{C}$ can be presented as a strongly reflective localisation of $\mathbf{Fun}(\mathscr{C}^{\mathrm{op}}, \mathbf{Cat}_{(n, r)})$, where the localisation functor preserves fibre products over $\mathbf{Sh}(\mathscr{C})$, with the analogous result also holding for sheaves of $\omega$-categories.