Topological categories presented by small sets of axioms

On bD-Sets and Associated Separation Axioms

Structured lattices and topological categories of L-sets

5 Goguen’s category of V-sets whose objects are functions with values in a pre-ordered set and morphisms are suitable maps is shown to be a topological construct if and only if the pre-ordered set is a complete lattice; in particular the category Set(L) of L-sets, 7 also considered by Goguen, is topological over Set. The special case when the considered pre-ordered set is L with the “mapping to...

The Axioms for Triangulated Categories

This is an edited extract from my paper [11]. We define triangulated categories and discuss homotopy pushouts and pullbacks in such categories in §1 and §2. We focus on Verdier’s octahedral axiom, since the axiom that is usually regarded as the most substantive one is redundant: it is implied by Verdier’s axiom and the remaining, less substantial, axioms. Strangely, since triangulated categorie...

Point Axioms in Dedekind Categories

A Dedekind category (locally complete division allegory) D is a category satisfying the following:

ON ALGEBRAIC AND COALGEBRAIC CATEGORIES OF VARIETY-BASED TOPOLOGICAL SYSTEMS

Motivated by the recent study on categorical properties of latticevalued topology, the paper considers a generalization of the notion of topological system introduced by S. Vickers, providing an algebraic and a coalgebraic category of the new structures. As a result, the nature of the category   TopSys   of S. Vickers gets clari ed, and a metatheorem is stated, claiming that (latticevalu...