We prove that the category of $\mathbb{ Z}_2 ^n$-manifolds has all finite products. Further, we show that a $\mathbb{ Z}_2 ^n$-manifold (resp., a $\mathbb{ Z}_2 ^n$-morphism) can be reconstructed from its algebra of global $\mathbb{ Z}_2 ^n$-functions (resp., from its algebra morphism between global $\mathbb{ Z}_2 ^n$-function algebras). These results are of importance in the study of $\mathbb{ Z}_2 ^n$ Lie groups. The investigation is all the more challenging, since the completed tensor product of the structure sheafs of two $\mathbb{ Z}_2 ^n$-manifolds is not a sheaf. We rely on a number of results on (pre)sheaves of topological algebras, which we establish in the appendix.
Categories: math-ph math.DG math.FA math.MP