  We show that varietal techniques based on the existence of operations of a
certain arity can be extended to n-permutable categories with binary
coproducts. This is achieved via what we call approximate Hagemann-Mitschke
co-operations, a generalisation of the notion of approximate Mal'tsev
co-operation. In particular, we extend characterisation theorems for
n-permutable varieties due to J. Hagemann and A. Mitschke to regular categories
with binary coproducts.
