Date: Mon, 29 Aug 1994 15:34:59 +0500 (GMT+4:00)
Subject: Top\op is a quasi-variety 

Date: Sat, 27 Aug 94 14:47:11 EDT
From: Michael Barr <barr@triples.math.mcgill.ca>

    by Michael Barr and M. Cristina Pedicchio
 
    We show that there is a certain variety (= category tripleable over
sets) and a simple Horn sentence in it of the form phi(u) = phi(v) ==>
psi(u) = psi(v) whose category of models is equivalent to the opposite
of topological spaces.  The theory consists of that of frames together
with a unary operation we denote ' (it is a kind of pseudocomplement)
satisfying a small set of equations plus an equation scheme that forces
all intervals of the form [u /\ u',u \/ u'] to be complete atomic
boolean algebras with the Sup and ' as operations.  The underlying set
functor on Top\op takes a space to the set of all pairs (U,A) where U is
open and A is an arbitrary subset of U. The frame operations are the
usual, while (U,A)' = (U,U - A).  The Horn clause is u \/ u' \/ 1' = v
\/ v' \/ 1' ==> u \/ u' = v \/ v'.

[note from moderator: Michael says the paper will be available by ftp from 
triples.math.mcgill.ca    soon.]




Date: Fri, 2 Sep 1994 14:07:56 +0500 (GMT+4:00)
Subject: Re: Top\op is a quasi-variety

Date: Thu, 1 Sep 1994 17:24:13 +0200 (MET DST)
From: Paul Johnson <johnson@karlin.mff.cuni.cz>

Dear Categories:

 
 
>     by Michael Barr and M. Cristina Pedicchio
>  
>     We show that there is a certain variety (= category tripleable over
> sets) and a simple Horn sentence in it of the form phi(u) = phi(v) ==>
> psi(u) = psi(v) whose category of models is equivalent to the opposite
> of topological spaces.  


There is a related result which (since it makes no mention of Horn
clauses, this result of Barr and Pedicchio apparently extends,
yet, nonetheless) may be of interest:


For each topological space B, the contravariant hom-functor Top(-,B) :
Top^op -----> Sets is of descent type, if and only if, the space B
contains a two-point indiscrete subspace, that is, a pair of points which
in the topology on B are contained in the same open sets, and a copy of
the Sierpinski space. The first condition is sufficient to ensure that,
for each topological space X, the canonical counit map X -----> B^Top(X,B)
is an injection, and the second ensures that it is the embedding of a
subspace. That these conditions are each necessary is easily checked. 
Each such space gives rise to an algebraic theory Th_B, as well as a
comparison Top^op -----> B-Alg, exhibiting Top^op as a full reflective
subcategory of, and hence tripleable over, the category of algebras B-Alg. 



> The theory consists of that of frames together
> with a unary operation we denote ' (it is a kind of pseudocomplement)
> satisfying a small set of equations plus an equation scheme that forces
> all intervals of the form [u /\ u',u \/ u'] to be complete atomic
> boolean algebras with the Sup and ' as operations.  The underlying set
> functor on Top\op takes a space to the set of all pairs (U,A) where U is
> open and A is an arbitrary subset of U. The frame operations are the
> usual, while (U,A)' = (U,U - A).


This particular algebraic theory to which you refer is perhaps
exactly that arising from the space B having three points, one of
which is closed, with open complement (containing the other two points)
as the only non-trivial open.


>The Horn clause is u \/ u' \/ 1' = v
> \/ v' \/ 1' ==> u \/ u' = v \/ v'.
> 
> [note from moderator: Michael says the paper will be available by ftp from 
> triples.math.mcgill.ca    soon.]

Best regards to all (especially FEJL), Paul.