Topological functors as total categories

Richard Garner*

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    7 Citations (Scopus)
    171 Downloads (Pure)


    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.

    Original languageEnglish
    Pages (from-to)406-421
    Number of pages16
    JournalTheory and Applications of Categories
    Publication statusPublished - 12 Aug 2014

    Bibliographical note

    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.


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