Subject: categories: Injectives via KZ-monads

From: "Martin Escardo" <[email-masked]>
Date: Tue, 4 Aug 1998 18:01:18 +0100

The following short paper is available from my home page:

	http://www.dcs.ed.ac.uk/home/mhe/pub/papers/top97.ps.gz
or
        http://www.dcs.ed.ac.uk/home/mhe/papers.html

(It is an updated version of a paper which was previously circulated
in other lists.)

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		Injective spaces via the filter monad
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An injective space is a topological space with a strong extension
property for continuous maps with values on it. A certain filter space
construction embeds every T_0 topological space into an injective
space.  The construction gives rise to a monad. We show that the monad
is of the Kock-Zoberlein type and apply this to obtain a simple proof
of the fact that the algebras are the continuous lattices (Alan Day,
1975, Oswald Wyler, 1976).  In previous work we established an
injectivity theorem for monads of this type, which characterizes the
injective objects over a certain class of embeddings as the
algebras. For the filter monad, the class turns out to consist
precisely of the subspace embeddings.  We thus obtain as a corollary
that the injective spaces over subspace embeddings are the continuous
lattices endowed with the Scott topology (Dana Scott, 1972).  Similar
results are obtained for continuous Scott domains, which are
characterized as the injective spaces over dense subspace embeddings,
via the proper filter monad.
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Two notes (and some questions concerning credit)
=========

(i) Bob Flagg and I have also considered the following variations on
the filter monad (a report is being written)

  (a) Category: T_0 exponentiable spaces 
          (= core-compact = open sets form a continuous lattice)

      Restriction on filters: Scott open.

      => Associated maps: "semi-proper" embeddings
          (= right adjoint of the frame maps preserve directed joins)

      => Algebras (and hence injectives over semi-proper):
          continuous meet-semilattices with Scott topology. 

      (Corollary: continuous meet-semilattices and Scott continuous
       functions form a CCC. Was this known before?)

      This characterization of the algebras was previously known
      (Andrea Shalk--anyone else?), but the "KZ-method" outlined 
      in the above abstract gives a much shorter proof.

  (b) Category: T_0 spaces 

      Restriction on filters: prime.

      => Associated maps: flat embeddings
          (= right adjoint of the frame maps preserve finite joins)

      => Algebras (and hence injectives over flat):
          compact, stably locally compact spaces. 

      (A localic version is given via the ideal monad. What Johnstone
       refers to as Joyal's Lemma appears as a special case of this.)

      (I don't know what was previously known about this.)

      (A result by Isbell (in his paper "Flat = prosupersplit")
      implies that the flat embeddings form the largest class of 
      embeddings over which the CSLCSs are injective, because
      finite spaces are (trivially) CSLCSs.)

  (c) Category: T_0 spaces 

      Restriction on filters: completely prime.

      => Associated maps: "completely flat" embeddings
          (= right adjoint of the frame maps preserve all joins)

      => Algebras (and hence injectives over completely flat):
          sober spaces. 

  (d) Category: T_0 locally connected spaces  

      Restriction on filters: filters of connected open sets.

      => Associated maps: "locally dense" embeddings
          (= frame maps preserve connectedness (and hence right 
           adjoints preserve disjoint unions))

      => Algebras (and hence injectives over locally dense):
          L-domains. 

      (This was obtained by Bob, based on some previous work by
       Paul Taylor (and Andrea Shalk) on the algebras. Again, the 
       KZ-method gives a simpler proof of the characterization.)

(ii) The filter monad is formally analogous to the so-called
continuation monad, as it is observed (with the formal details of the
analogy) in the paper being advertised.

I would like to also mention that the general injectivity result for
KZ-monads referred to in the above abstract was established in the
paper

        http://www.dcs.ed.ac.uk/home/mhe/pub/papers/injective.ps.gz

which is (mainly) about continuity of the extension process (answering
a question by Scott in his 1972 paper on continuous lattices).

Comments are wellcome.

Martin
=================================================================
Martin H. Escardo, Department of Computer Science, LFCS
King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland
office:	2606 (JMCB) fax: +44 131 667 7209 phone: +44 131 650 5135
mailto:[email-masked]          http://www.dcs.ed.ac.uk/home/mhe
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