Lattice-valued categories of lattice-valued convergence spaces

generalized convergence spaces [2, 3]. We show that extending the structure of continuous convergence (which makes SL-GCS a cartesian closed category) from the set of continuous mappings between spaces to a set F of arbitrary map-pings between spaces, one of the axioms satisfied by the objects in SL-GCS may no longer be valid for F. This poses the question: ”How far is F away from being in SL-GCS? ” Using a frame as lattice, this question can be answered if we attach ”grades of continuity ” to the mappings in F. In this way, we are naturally led to the concept of a lattice-valued category in the sense of Šostak [4, 5, 6]. Such an L-category consists of an ordinary category [1] of ”poten-tial objects ” and ”potential morphisms ” together with two L-classes, assigning a grade of being an object and of being a morphisms of the L-category. We describe initial constructions and function spaces of the resulting L-category of L-convergence spaces. Also we use Šostak’s concept of L-category and study ”how far away a lattice-valued convergence space is from being a lattice-valued topological space”