Technique de descente et théorèmes d'existence en géométrie algébriques. II. Le théorème d'existence en théorie formelle des modules
The category of uniform spaces as a completion of the category of metric spaces
A criterion for the existence of an initial completion of a concrete category universal w.r.tḟinite products and subobjects is presented. For metric spaces and uniformly continuous maps this completion is the category of uniform spaces.
The structure of initial completions
Théories de catégories
Topological Space Objects in a Topos II: ...-Completeness and ...-Cocompleteness.
Topologically-algebraic structure functors full reflective or coreflective restrictions of semitopological functors
Totality of colimit closures
Adámek, Herrlich, and Reiterman showed that a cocomplete category is cocomplete if there exists a small (full) subcategory such that every -object is a colimit of -objects. The authors of the present paper strengthened the result to totality in the sense of Street and Walters. Here we weaken the hypothesis, assuming only that the colimit closure is attained by transfinite iteration of the colimit closure process up to a fixed ordinal. This requires some investigations on generalized notions...
Totality of product completions
Categories whose Yoneda embedding has a left adjoint are known as total categories and are characterized by a strong cocompleteness property. We introduce the notion of multitotal category by asking the Yoneda embedding to be right multiadjoint and prove that this property is equivalent to totality of the formal product completion of . We also characterize multitotal categories with various types of generators; in particular, the existence of dense generators is inherited by the formal product...