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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