categories: Re: Chu spaces vs Toposes (and maybe Stone Spaces)

[Vaughan Pratt <pratt@cs.stanford.edu>]

>From: Galchin Vasili <vngalchin@yahoo.com>
>I have been reading some of Prof. Pratt's papers on Chu spaces. I am
>just trying to understand Topos. Both seem to generalize the notion
>of topological spaces. Is there any relationship beteween Chu spaces
>and Toposes?

About all that a topos has in common with Chu(V,k) is that both
are symmetric monoidal closed categories with all finite limits.
Among the conditions that hold for every topos but only for degenerate
Chu categories is cartesian closedness.  Conversely, all Chu categories
but only degenerate toposes are self-dual.

Not a strong basis for a relationship.  Marriage would seem out of the
question, maybe half a game of correspondence chess.

There are a couple of things worth mentioning about Chu spaces
vs. topological spaces that may help in understanding why that connection
does not extend nicely to toposes.

First, Chu spaces are to topological spaces as (universal) algebras are
to groups.  The former (in each fiefdom) relaxes the specific signature
and axioms of the latter while retaining its underlying machinery: open
sets and continuous functions in the first fiefdom, n-ary structures
and homomorphisms in the second.

As noted by Lafont and Streicher [LICS'91], Chu(Set,2) embeds Top, much as
the category of ternary structures and their homomorphisms embeds Grp (by
"embeds" and "subcategory" I mean "fully embeds" and "full subcategory"
throughout).  A topological space may be understood as a partial algebra
equipped with a "signature" consisting of an operation (or proper class
of operations if one takes arity into account) for taking the limit of
a sequence.  This operation is the only one to survive the requirement
that the family of open sets of a topological space be closed under
arbitrary union and finite intersection.

Strengthening "finite" to "arbitrary" restricts the domain of the limit
operation to finite sequences, the "signature" of a preordered set.
Further requiring closure under complement removes (discretizes) the
partial order.  Conversely, when the closure conditions are removed, all
heaven breaks loose with even finitary partial operations being admitted.

Note the contravariance between strength of condition on the open sets
and size of language: more restrictions means less signature.

An altogether different connection with topology involves Stone duality.
The easy part of this duality is to match up a Boolean algebra B to a
quasitopological space (one whose open sets need only be closed under
finite union and finite intersection, with continuity defined standardly)
whose points are the ultrafilters of B (Boolean homomorphisms from B
to the 2-element Boolean algebra) and whose open sets are the elements
of B (taking membership of such a point in such an open set to be
the converse of membership of elements of B in ultrafilters of B).
The Lafont-Streicher embedding of Top in Chu(Set,2) extends without
change of wording to embed Qtop (quasitopolical spaces, no connection
with Qtips).  Furthermore Bool embeds in Chu(Set,2) in an obvious way,
such that the corresponding representation of the above dualization of
B is then nothing more than the transposition of this representation of
B yielding the Lafont-Streicher representation of the dual of B.

The hard bit in Stone's theorem is to show that the process of generating
a topological space from this dual (i.e. the result of closing the open
sets under arbitrary union) embeds these dualizations in Top (fully as
always), and moreover as totally disconnected compact Hausdorff spaces.

The question then arises, what is Stone's theorem morally?  On an Arcturan
planet teeming with mathematicians, the fundamental theorem of arithmetic
is surely as it is on Earth (not necessarily under the same name), but
what about Stone duality for Boolean algebras?  What useful purpose is
served by the hard bit of Stone's theorem?  Why take the (hard) second
step when one could as usefully stop after performing the (very easy)
transposition?

It is sometimes said that Stone's theorem makes a precise connection
between geometry and algebra.  It seems to me that the connection
between geometry and totally disconnected topological spaces is a much
bigger stretch than between topological spaces and quasitopological
spaces.  If connecting algebra to geometry were the primary motivation
behind Stone duality, one would only go to the considerable bother of
representing a perfectly good and very easily obtained quasitopological
space as a topological space if one's religion--or mathematics--forbade
entertaining the former as a geometrical notion.  While acknowledging
the weight of a century of tradition underlying this proscription here
on Earth, interplanetary mathematical anthropology needs to be able to
interpret more neutrally the morality of mathematical decisions made on
other planets.

My own view is that the connection between algebra and geometry is not
the most important lesson of Stone duality.  The important point of the
theorem Stone proved was that any Boolean algebra B could not only be
represented (via Birkhoff's more general representation of distributive
lattices) as a field of sets (one closed under finite union, finite
intersection, and complement), but that there was a canonical, even
universal, such representation.  Later (I don't know when) it became
clear that Stone's representation applied not just to the objects but
the morphisms, showing that this representation was in fact a duality.
And a strikingly beautiful duality it is, independently of its connection
with geometry.

>From this perspective the better version of Stone's theorem would,
I think, be the one more easily grasped.  What I like so much about
Chu spaces here is that it reduces Stone duality for Boolean algebras
to matrix transposition.  This makes it something one can present in an
introductory text on algebras, lattices, and varieties, or proudly take
home to show to one's loved ones, which is more than can be said of a
lot of mathematics.

A pedagogically ideal illustration of Stone duality is that between
finite distributive lattices with top and bottom and finite posets,
observed explicitly by Birkhoff and implicitly by Stone, both in 1937.
The 5 (16,63...) distributive lattices of height 3 (4,5,...) correspond to
the 5 (16,63,...) posets with 3 (4,5,...) elements.  This duality further
pairs up distributive lattice homomorphisms (respecting 0 and 1) with
monotone functions between posets, a correspondence with a terrific visual
appeal when illustrated with embeddings (which dualize as surjections)
between Hasse diagrams of both the lattices and the posets.

This example can be further extended to a great introduction to Chu
spaces.  Here are the five Chu spaces over 2 corresponding to the five
dual pairs of 3-element posets and height-3 distributive lattices.

     stuvwxyz   stvwxz   stvxz   stwxz   stxz
a    10101010   100100   10000   10100   1000   
b    11110000   111000   11100   11000   1100
c    11001100   110110   11010   11110   1110

The rows of the first represent the elements of the discrete tripleton
{a,b,c} while its columns represent the elements of the 3-cube (8-element
Boolean algebra) {s,t,u,v,w,x,y,z} with s at the top and z at the bottom.

Deleting the columns violating a <= c yields the second, in which it can
be seen at a glance that row a <= row c; its columns form the 6-element
distributive lattice dual to the poset ({a,b,c},a<=c).

The remaining three matrices are obtained by further deletions to satisfy
a <= b, b <= c, or both, in that order.  In every case their columns
form a distributive lattice, the last being the 4-chain (qua lattice)
dual to the 3-chain (qua poset).

Deleting being the opposite of adjoining, these rightward-moving
transformations can be understand forwardly as monotone functions acting
as the identity on the underlying set, or, dually, backwardly as lattice
embeddings.

Easy but fun, whence pedagogically ideal.

Talking of long-standing tradition, Mike Barr and I decided on the phone
one day in 1992 to call the objects of the Chu construction Chu spaces.
(In our respective published accounts of this conversation I've attributed
the actual suggestion to Mike and vice versa.)

I now regret that the term "couple" did not occur to either of us.  An apt
description of a Chu space is as a related pair of complementary objects.
"Couple" seems a fine word for such a notion.  I've started using it in
my papers as a synonym for Chu space.  Now if only people would realize
that couples are just as important as triplesxxxxxxxmonads.

Vaughan Pratt