nLab
sheaf of n-types

The term sheaf of $n$-types is another word for an ∞-stack that has trivial homotopy groups above degree $n$.

Under the homotopy hypothesis a homotopy n-type is just an ∞-groupoid that has trivial homotopy groups above degree $n\in \mathbb{N}$. Hence a “sheaf of $n$-types” is just a special case of a “sheaf of $\mathrm{\infty }$-groupoid”s which is usually called an $n$-stack .