Subject: categories: Preprint: Comparing cartesian closed categories of (core) compactly generated spaces

From: Alex Simpson <[email-masked]>
Date: Tue, 13 Jan 2004 16:28:30 +0000 (GMT)

The following preprint may be of interest to some categories readers.
---

        Comparing cartesian closed categories of
           (core) compactly generated spaces

      Martin Escardo, Jimmie Lawson and Alex Simpson

  It is well known that, although the category of topological
  spaces is not cartesian closed, it possesses many cartesian
  closed full subcategories, e.g.: (i) compactly generated
  Hausdorff spaces; (ii) quotients of locally compact Hausdorff
  spaces, which form a larger category; (iii) quotients of locally
  compact spaces without separation axiom, which form an even
  larger one; (iv) quotients of core compact spaces, which is at
  least as large as the previous; (v) sequential spaces, which are
  strictly included in (ii); and (vi) quotients of countably based
  spaces, which are strictly included in the category (v).

  We give a simple and uniform proof of cartesian closedness for
  many categories of topological spaces, including (ii)-(v), and
  implicitly (i), and we also give a self-contained proof that (vi)
  is cartesian closed. Our main aim, however, is to compare the
  categories (i)-(vi), and others like them.

  When restricted to Hausdorff spaces, (ii)-(iv) collapse to (i);
  and most non-Hausdorff spaces of interest, such as those which
  occur in domain theory, are already in (ii). Regarding the
  cartesian closed structure, finite products coincide in (i)-(vi).
  Function spaces are characterized as coreflections of both the
  Isbell and natural topologies. In general, the function spaces
  differ between the categories, but those of (vi) coincide with
  those in any of the larger categories (ii)-(v). Finally, the
  topologies of the spaces in the categories (i)-(iv) are analysed
  in terms of Lawson duality.

  http://www.dcs.ed.ac.uk/home/als/Research/comparing.ps.gz
  http://www.dcs.ed.ac.uk/home/als/Research/comparing.pdf

---
Alex Simpson, LFCS, School of Informatics, Univ. of Edinburgh
Email: [email-masked]           Tel: +44 (0)131 650 5113
Web: http://www.dcs.ed.ac.uk/home/als  Fax: +44 (0)131 667 7209