Topological Functors as Total Categories

A notion of central importance in categorical topology is that of topological functor. A faithful functor E → B is called topological if it admits cartesian liftings of all (possibly large) families of arrows; the basic example is the forgetful functor Top→ Set. A topological functor E → 1 is the same thing as a (large) complete preorder, and the general topological functor E → B is intuitively thought of as a “complete preorder relative to B”. We make this intuition precise by considering an enrichment base QB such that QB-enriched categories are faithful functors into B, and show that, in this context, a faithful functor is topological if and only if it is total (=totally cocomplete) in the sense of Street–Walters. We also consider the MacNeille completion of a faithful functor to a topological one, first described by Herrlich, and show that it may be obtained as an instance of Isbell’s generalised notion of MacNeille completion for enriched categories.