Re: Stone duality for generalized Boolean algebras

["F. William Lawvere" <wlawvere@hotmail.com>]

Dear George You ask

would any categorically
thinking

mathematician say that the
category of pointed sets needs further description as the category of  sets  and partial maps?

 

In 1969-1970, recalling

a) the preorigins of sheaf
theory a hundred years ago in the still-non-trivial problem of extending
of partial maps in analysis and topology, and

b) desirous of an instrument for describing
sheafication in finitely algebraic terms

 

Myles and I proposed Ytilda->Omega as one of
the two axioms for an elementary theory of toposes (the other being the Pi
right adjoint to pullback; applying Grothendieck’s method of relativization
using any given model U of those axioms, the 2-category of U-Toposes was
obtained thus capturing precisely the original SGA4 notion by choice of U).  Of
course these axioms were soon shown to be deducible from special cases,

but the importance of classifying partial maps
X..->Y remains.

 

The fact that this construction reduces to
Y+1->1+1 in sets misled some recursion theorists to try to represent partial
recursive maps that way, but

the categorically thinking mathematician noted
that in their category, the complement of the domain is typically not a
subobject of X. As Phil Mulry showed with his Recursive Topos, subobjects  of
Omega provide a precise specification of degrees of complication for the
inclusion of the domain of definition by pulling back along X->Omega.
(Although it would seen that Hilbert schemes as subobjects of Omega might
provide similar representability, that apparently has not been pursued).

 

In the Boolean case Y+1 can be viewed as an action of the
two-element monoid of

idempotents (the instrument for analysis of  objects in in a protomodular category),
in other words the category of partial maps can be embedded in a topos.Over  a general topos, that can be replaced by actions of
Omega as a multiplicative monoid .

 

  Of course
partial maps are special binary relations, but of a qualitatively special  kind
that requires its own status. In Cat, if replace subobjects by discrete
opfibrations, the analogous “partial maps”  (”machines”) turn out  to be representable but give rise
analogously to special distributors.

 

Peter Freyd’s dictum has a dialectical companion. Category theory
can sometimes discern the germ of nontrivial in the trivial.

 





> From: janelg@telkomsa.net
> To: fejlinton@usa.net; categories@mta.ca
> Subject: categories: Re: Stone duality for generalized Boolean algebras
> Date: Mon, 24 Jan 2011 23:15:17 +0200
> 
> Dear Fred,
> 
> Please forgive me, but let us distinguish between serious questions and
> trivialities:
> 
> Andrej Bauer asked:
> 
> "...How exactly does this extend to generalized Boolean algebras?..."
> 
> And the answer is trivial (without quotation marks): The category GBA of
> what he called generalized Boolean algebras is dually equivalent to the
> category 1\STONE of pointed Stone spaces. This follows from Stone duality
> (since GBA is equivalent to BA/2), but also extends it: just as BA is a
> non-full subcategory of GBA, STONE can be considered as a non-full
> subcategory of 1\STONE via the functor that adds base points. And this way
> the dual equivalence between GBA and 1\STONE indeed extends the Stone
> duality.
> 
> Although your first message about BA/2 was written after mine, I am sure you
> know these things (you probably knew them before I knew the definition of  a
> category...)
> 

...


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