Re: adjoints to lax-idempotent algebra structures

[Martin Escardo <m.escardo@cs.bham.ac.uk>]

Mike,

On 18/11/13 20:40, Michael Shulman wrote:
> Thanks again to everyone who replied with examples and comments.  I've
> created an nLab page which hopefully includes everything I learned:
> http://ncatlab.org/nlab/show/continuous+algebra

Some more examples and comments follow.

In about 1996, I considered the special case in which the
2-categorical structure is a poset-enrichment, because I had in mind
applications to topological spaces and locales wrt the specialization
order. In that case, your Theorem 1 in the nLab has an additional
equivalent condition, when T is a Kock-Zoberlein monad:

    (4) The underlying object of the algebra A is injective wrt to
    T-embeddings, where a T-embedding is a map f:X->Y such that Tf has a
    reflective left adjoint.

(I deemly recall that Jonathon Funk may have mentioned to me in 2004
in Cambridge during a PSSL in a pub conversation that (4) holds for
the general 2-categorical case, but he would have to confirm or reject
that, as I can't find references after a quick search.)

This was in the paper

     (i) Properly injective spaces and function spaces, Topology and its
         Applications, v89, n1-2, pp75-120, 1998, as Theorem 4.2.2.

In this paper and a series of other papers following that, a number of
examples are given in the categories of topological spaces and of
locales, namely

    (ii) Injective Spaces via the Filter Monad, Topology Proceedings,
         1997.

    (iii) Semantic domains, injective spaces and monads, ENTCS, 1999,
          with R. Flagg.

    (iv) Injective locales over perfect embeddings and algebras of the
         upper powerlocale monad. Applied General Topology, volume 4,
         number 1, pp. 193-200, 2003.

Using the theorem with conditions (1)-(4), one can unify Alan Day's
characterization of the algebras of the filter monad as the continuous
lattices with Dana Scott's characterization of the injective T0
topological spaces as the continuous lattices under the Scott
topology.  This is in paper (ii), which also considers the densely
injective topological characterized as the continuous Scott domains
under the Scott topology (again Scott's result), using the proper
filter monad.

Then there are many other examples of classes of filters one may
consider, to get new and previously known injectivity results using
the theorem with the added condition (4).

These examples are in the paper (iii) with Flagg: The stably compact
spaces are the injective spaces with respect to flat embeddings (as
defined in Johnstone's book Stone spaces). Here one uses the monad of
prime filters, and Harold Simmons characterizations of their algebras
as the stably compact spaces (under another name) in his paper "A
couple of triples". The injectives over completely flat embeddings are
the sober spaces (use the monad of completely prime filters). The
injectives over perfect embeddings in the category of core-compact
spaces are the continuous meet-semilattices under the Scott topology
(use the monad of Scott open filters). The injective spaces over
locally dense embeddings in the category of locally connected T0
spaces are the L-domains under the Scott topology (use connected open
filters).  The injectives over open embeddings are the spaces with a
least element in the specialization order. The injectives over closed
embeddings are the spaces with a largest, isolated point in the
specialization order.  The injectives over semi-open embeddings are
algebras of the lower hyperspace.

In the paper (iv), I considered locales. The injectives over perfect
embeddings (=embeddings such that the right adjoint of the frame map
preserves directed joins) are the algebras of the upper powerlocale
monad (perfect + Frobenius = proper). They can also be characterized
by a sort of "continuity" condition. Define U<V for opens of a locale
to mean that every Scott closed collection of opens that covers V has
U as a member (Scott closed = lower set closed under directed
joins). Then a locale is injective/an algebra iff every open V is the
join of the opens U<V and the relation < is multiplicative. In the
section "Remarks" I give many more examples like this. For example, we
get stably compact locales via the ideal-completion monad.

Of course, this is related to the paper by Jiri Adamek, Lurdes Sousa
and Jiri Velebil advertised on the 11th this month.

And Dirk Hofmann did much work related to that.

Best,
Martin



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