Stability properties characterising n- permutable categories

The 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.FNRS [1.A741.14]Centro de Matematica da Universidade de Coimbra [UID/MAT/00324/2013]Portuguese Government through FCT/MCTESEuropean Regional Development Fun