  We study the difference between internal categories and internal groupoids in
terms of generalised Mal'tsev properties---the weak Mal'tsev property on the
one hand, and $n$-permutability on the other. In the first part of the article
we give conditions on internal categorical structures which detect whether the
surrounding category is naturally Mal'tsev, Mal'tsev or weakly Mal'tsev. We
show that these do not depend on the existence of binary products. In the
second part we focus on varieties of algebras.
