Jacqmin, Pierre-AlainRodelo, Diana2019-11-202019-11-2020171201-561Xhttp://hdl.handle.net/10400.1/13267The purpose of this paper is two-fold. A first and more concrete aim is to characterise n-permutable categories through certain stability properties of regular epimorphisms. These characterisations allow us to recover the ternary terms and the (n + 1)-ary terms describing n-permutable varieties of universal algebras. A second and more abstract aim is to explain two proof techniques, by using the above characterisation as an opportunity to provide explicit new examples of their use: an embedding theorem for n-permutable categories which allows us to follow the varietal proof to show that an n-permutable category has certain properties; the theory of unconditional exactness properties which allows us to remove the assumption of the existence of colimits, in particular when we use the approximate co-operations approach to show that a regular category is n-permutable.engMalcev categoriesOperationsjournal article