A notion of central importance in categorical topology is that of topological functor. A faithful functor $\cal E \to \cal B$ is called topological if it admits cartesian liftings of all (possibly large) families of arrows; the basic example is the forgetful functor $Top \to Set$. A topological functor $\cal E \to 1$ is the same thing as a (large) complete preorder, and the general topological functor $\cal E \to \cal B$ is intuitively thought of as a "complete preorder relative to $\cal B$". We make this intuition precise by considering an enrichment base $\cal Q_\cal B$ such that $\cal Q_\cal B$-enriched categories are faithful functors into $\cal B$, 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.
Keywords: Topological functors, total categories, enriched categories, quantaloids, MacNeille completion
2010 MSC: 18A22, 18D20, 18B30, 06A75
Theory and Applications of Categories, Vol. 29, 2014, No. 15, pp 406-421.