Sheaves of Higher Categories on Generalised Spaces

We study the extension of higher presheaves on a category $C$ to its free cocompletion $\hat{C}$. Any pretopology on $C$ induces a canonical pretopology of generalised coverings on $\hat{C}$. We show that with respect to these pretopologies the homotopy right Kan extension along the Yoneda embedding $Y \colon C \to \hat{C}$ maps higher sheaves to higher sheaves. Moreover, we show that that this induces an equivalence between sheaves of spaces on $C$ and a colimit-preserving sheaves of spaces on $\hat{C}$. 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.Comment: 41 pages; v2: Substantially rewritten, presentation improved, Section 5 adde