Weak subobjects and the epi-monic completion of a category

Abstract The notion of weak subobject , or variation , was introduced by Grandis (Cahiers Topologie Geom. Differentielle Categoriques 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 Fr X , 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 Ho Top and various other homotopy categories, Fr X is complete; similarly, if X has zero-object, weak kernels and weak cokernels, as the homotopy category of pointed spaces, then Fr X is a homological category (Grandis, Cahiers Topologie Geom. Differentielle Categoriques 33 (1992) 135–175); finally, if X is triangulated, Fr X is abelian and the embedding X → Fr X is the universal homological functor on X , as in Freyd's original case. These facts have consequences on the ordered sets of variations.

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