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.
Keywords: Mal'tsev category, Goursat category, n-permutable category, embedding theorem, unconditional exactness property
2010 MSC: 08A30, 08B05, 08C05, 18B15, 18B99, 18C99,18A32
Theory and Applications of Categories, Vol. 32, 2017, No. 45, pp 1563-1587.