  We propose a definition of higher sheaf with respect to an arbitrary set of
maps $\Sigma$ in an $\infty$-topos $\mathcal{E}$. We then show that the
associated reflection $\mathcal{E} \to {\rm Sh}(\mathcal{E},\Sigma)$ is
left-exact so that the subcategory of sheaves with respect to $\Sigma$ is
itself an $\infty$-topos. Furthermore, we show that the reflection $\mathcal{E}
\to {\rm Sh}(\mathcal{E},\Sigma)$ may be characterized as the left-exact
localization generated by $\Sigma$. In the course of the proof, we study the
interaction of various types of factorization systems, and make essential use
of the notion of a \emph{modality}, that is, a factorization system whose left
class is stable by base change.
