Weak subobjects and the epi-monic completion of a category

The notion of weak subobject, or variation, was introduced by Grandis (Cahiers Topologie Géom. Différentielle Catégoriques 38 (1997) 301–326) as an extension of the notion of subobject, adapted to homotopy categories or triangulated categories, and well linked with their weak limits. We study here some formal properties of this notion. Variations in the category X can be identified with (distinguished) subobjects in the Freyd completion FrX, the free category with epi-monic factorisation system over X, which extends the Freyd embedding of the stable homotopy category of spaces in an abelian category (Freyd, in: Proceedings of Conference on Categ. Algebra, La Jolla, 1965, Springer, Berlin, 1966, pp. 121–176). If X has products and weak equalisers, as HoTop and various other homotopy categories, FrX is complete; similarly, if X has zero-object, weak kernels and weak cokernels, as the homotopy category of pointed spaces, then FrX is a homological category (Grandis, Cahiers Topologie Géom. Différentielle Catégoriques 33 (1992) 135–175); finally, if X is triangulated, FrX is abelian and the embedding X→FrX is the universal homological functor on X, as in Freydʹs original case. These facts have consequences on the ordered sets of variations.