Dear Bill, Beautiful! But is not the conclusion exactly opposite?: Let M(X,Y) be the set of all maps X --> Y and let P(X,Y) be the set of partial maps X --> Y. Then P(X,Y) = M(X,Y+1) (where "=" means "natural bijection" of course), and if instead of sets we consider objects in an elementary topos, then - according to your discovery - Y+1 should be replaced with Ytilda (which is actually the definition of Ytilda). So, the whole point is to describe partial maps in terms of maps and not the other way around! I (still) insist that Andrej Bauer's original question has the trivial answer independently given by Fred and me, and that topos-theoretic Ytilda, in spite of its fundamental role in topos theory (e.g. 1tilda = omega), has nothing to do with it. This does not mean that I do not appreciate your nice and deep comments. With best regards, George From: F. William Lawvere Sent: Wednesday, January 26, 2011 7:19 PM To: janelg@telkomsa.net ; fejlinton@usa.net ; categories Subject: RE: categories: Re: Stone duality for generalized Boolean algebras 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. ... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]