2013/10/03 by Richard Garner, Garner, Richard
Mathematics · #Advanced Topology and Set Theory #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #General Topology (math.GN) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1310.0903
openalex publication_date 2013/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.