#
Topological functors as total categories

##
Richard Garner

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.

Published 2014-08-12.

http://www.tac.mta.ca/tac/volumes/29/15/29-15.pdf

TAC Home