Weak subobjects and the epi-monic completion of a category

The notion of weak subobject, or variation, was introduced by Grandis (Cahi ers Topologie Geom. Differentielle Categoriques 38 (1997) 301-326) as an ex tension of the notion of subobject, adapted to homotopy categories or trian gulated categories, and well linked with their weak limits. We study here s ome formal properties of this notion. Variations in the category X can be i dentified with (distinguished) subobjects in the Freyd completion Fr X, the free category with epi-monic factorisation system over X, which extends th e Freyd embedding of the stable homotopy category of spaces in an abelian c ategory (Freyd, in: Proceedings of Conference on Categ. Algebra, La Jolla, 1965, Springer, Berlin, 1966, pp. 121-176). If X has products and weak equa lisers, as Ho Top and various other homotopy categories, Fr X is complete; similarly, if X has zero-object, weak kernels and weak cokernels, as the ho motopy category of pointed spaces, then Fr X is a homological category (Gra ndis, Cahiers Topologie Geom. Differentielle Categoriques 33 (1992) 135-175 ); finally, if X is triangulated, Fr X is abelian and the embedding X --> F r X is the universal homological functor on X, as in Freyd's original case. These facts have consequences on the ordered sets of variations. (C) 2000 Elsevier Science B.V. All rights reserved. MSG: 18A20; 18A35; 18E; 55P; 18E 30.