N-dimensional generalization of map and reduce?


Is there any conceptual generalization of higher-order functions like map and reduce but for N-dimensional objects (e.g. arrays or tensors)?

For mapping, I guess it would be a point-wise generalization. In the case of reduction, I have more doubts because the iteration is not trivial. Perhaps it would take an additional argument specifying the dimension (or dimensions?) along which the reduction is applied?

Are such higher-order functions defined in algebra? E.g. do they have a specific name and are studied as such?