TY - JOUR

T1 - Topological functors as total categories

AU - Garner, Richard

N1 - Copyright the Author(s) 2014. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.

PY - 2014/8/12

Y1 - 2014/8/12

N2 - A notion of central importance in categorical topology is that of topological functor. A faithful functor ε → 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 ε → 1 is the same thing as a (large) complete preorder, and the general topological functor ε → 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.

AB - A notion of central importance in categorical topology is that of topological functor. A faithful functor ε → 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 ε → 1 is the same thing as a (large) complete preorder, and the general topological functor ε → 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.

UR - http://www.scopus.com/inward/record.url?scp=84907356468&partnerID=8YFLogxK

M3 - Article

VL - 29

SP - 406

EP - 421

JO - Theory and Applications of Categories

JF - Theory and Applications of Categories

SN - 1201-561X

ER -