Remarks on decompositions of categories

Abstract

Consider, to begin with, decompositions of complete categories as products or as coproducts. However, a complete category cannot be a coproduct of two or more nonempty categories. In fact, there is a well-known finest decomposition of any category as a coproduct, whose factors are called the (weak) components and are not connected by any morphisms. (Thus, for instance, objects in different components have no product object.) However, one can define strong components (as in graph theory) so that fo; two objects X, Y in the same strong component, both Hom(X, Y) and Hom(Y, X) are nonempty. Two theorems will be proved. A strong component of a complete category has a unique completion; two direct decompositions of a skeletal complete category have a common refinement. For fine points of terminology we refer to [I ]; but a reader will lose little if he simply considers that all categories to be decomposed belong to a given Grothendieck universe whose cardinal number is called x, small means smaller than xo, and small-complete means having limits and colimits of arbitrary small diagrams. (The other completeness notions mentioned are slightly stronger but equivalent for well-powered co-well-powered categories.) Of course results from [I] are used. Precisely, we call a category e strongly connected if no (hom)Wset e(X, Y) is empty. The maximal strongly connected subcategories are called strong components. The strong components partition the objects, of course. They form a skeletal schlicht (partially ordered) category Sc(e), where by definition there is a unique morphism from (t to 6B in Sc(e) if e(A, B) is nonempty for (arbitrary) representative objects A C IaI, BE |1 6; otherwise there is no morphism from Ct to 63. Sc(e) is a reflection of e in the subcategory of Cat composed of skeletal schlicht categories. Obviously (from [1]), for schlicht categories, completeness is equivalent to small-completeness. Obviously: