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Total categories and solid functors. (English) Zbl 0742.18001

The authors prove an important and far-reaching generalization of a theorem of B. J. Day [Cah. Topologie Géom. Différ. Catégoriques 28, 77-78 (1987; Zbl 0626.18001)] solving at the same time two problems posed by G. M. Kelly [ibid. 27, 109-131 (1986; Zbl 0593.18007)]. A category \({\mathcal A}\) is called \({\mathcal E}\)-cocomplete for a class \({\mathcal E}\subseteq{\mathcal A}\), if a pushout of a morphism of \({\mathcal E}\) along an arbitrary morphism exists and is in \({\mathcal E}\) again, if a cointersection of any family of morphisms in \({\mathcal E}\) (with common domain) exists in \({\mathcal A}\) and if any such cointersection belongs to \({\mathcal E}\). With this notion the following theorem is proved. Theorem: Every \({\mathcal E}\)-cocomplete category \({\mathcal A}\), \({\mathcal E}\subseteq {\mathcal A}\), with an \({\mathcal E}\)-generator is total. This results contains Day’s theorem for \({\mathcal E}=Epi ({\mathcal A})\). Crucial for the very elegant proof is the notion of a solid (formerly called “semi-topological”) functor and the lifting properties of these functors. The remainder of the paper contains very interesting converses of this generalization and a fine analysis of the area of validity of this theorem by non-trivial examples. The paper closes with a section in which the theorem is improved for special classes of morphisms, particularly important for applications, namely split resp. regular epimorphisms.
Reviewer: D.Pumplün (Hagen)

MSC:

18A35 Categories admitting limits (complete categories), functors preserving limits, completions
18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
18A32 Factorization systems, substructures, quotient structures, congruences, amalgams
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
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