Reinhard Börger

Cogenerators for Convex Spaces

We give cogenerators for the categories of convex (= finitely superconvex), finitely positively convex, and absolute convex (= finitely totally convex) spaces introduced by Pumplün and Röhrl.

Compact and hypercomplete categories

The paper deals with compact c&ego&s (cf. Isbell [17]), i.e. categories d which have the property that any functor U : d + 9 preserving all existing colimits of Sp has a right adjoint, and with categories called hypercomplere categories which are defined to have limits of all those (not necessarily small) diagrams D in Sp for which the ‘conglomerate’ of natural transfomations dA + D can be indexed by a set for any object A of .sd. For Sp having small horn classes this is, of course, the widest notion of a diagram for which the existence of a limit in d can be expected. (Lambek’s [19] proper diagrams are, in particular, included). If, as Manes [21] did, we call d a SAFT-category, provided the sufficient conditions of Freyd’s Special Adjoint Functor Theorem are fulfilled, one has the implications

A cogenerator for preseparated superconvex spaces

We construct a cogenerator for the category of preseparated superconvex spaces, and we describe separated convex spaces, i.e. convex spaces for which the morphisms into the unit interval separates points.

Multicoreflective subcategories and coprime objects

Multicoreflective subcategories of a given category can be viewed as “categories of connected objects” for suitable connectivity notions. If one tries to define “connected objects” in an absolute way, one is led to the notion of coprime object. Both notions are treated using the same tool, which is parallel to the description of coreflective subcategories by projectivity. These methods can be applied to characterize multicoreflective subcategories of Comp. They also lead to some special results about reflective and multireflective subcategories of the category of unital rings. In particular, the category of powers of 7L is reflective if and only if there exists no uncountable measurable cardinal.

Joins and Meets of Symmetric Idempotents

We show that the poset of symmetric idempotents of a (non-commutative) unital ring with involution in the obvious order need not admit finite joins and meets.

Measures and idempotents in the non-commutative situation

ABSTRACT We investigate measures on sequential orthomodular posets with values in a vector space or a (not necessarily commutative) algebra with reasonable sequential topologies, using a universal property. Unfortunately, the universal measure and the universal multiplicative measure need not coincide any more as in the commutative situation. This may have applications in quantum physics.

Lexicographic sums and fibre-faithful maps

The notion of lexicographic sum is introduced in general categories. Existence criteria are derived, particularly for locally cartesian closed categories and for categories with suitable coproducts. Lexicographic sums satisfy a generalized associative law. More importantly, every morphism can be factored through the lexicographic sum of its fibres. This factorization and the two types of maps arising from it, fibre-trivial and fibre-faithful, are studied particularly for partially ordered sets and forT1-spaces.