Dear all, The fact that the 'right' morphisms of topos theory (geometric morphisms) a= re not structure preserving maps gives rise to the debate. With an entirely= localic approach the situation can be recovered; but at the cost of moving= from 'elementary topos' as primitive to 'category of locales'. Here geomet= ric morphisms can be represented as adjunctions that commute with the doubl= e power locale monad and it is the double power locale monad that gives the= structure in categories of locales. So there are avenues worth exploring w= hich may provide a way of marrying up the structural disconnect between obj= ects and morphisms that is implicit in topos theory. = Christopher = = -----Original Message----- From: Marta Bunge [mailto:martabunge@hotmail.com] = Sent: 02 November 2016 11:18 To: categories@mta.ca Cc: Steve Vickers; Patrik Eklund; Joyal, Andr=E9 Subject: categories: Re: Grothendieck toposes Dear all,
It is marvelous that the two notions should be so related. But it is be better to keep them appart before uniting them. Otherwise the miracle disappear in confusion.
The above is a quotation from a recent posting by Andre Joyal. To the risk= of boring everyone I offer the following comment on it here. There is no = need to talk about miracles in mathematics, not even as some sort of analog= y. Why not instead give credit to the very important insight of an elementa= ry topos as embodying both the logic and the geometry? There are two notion= s of morphism between elementary toposes, not a preferred one - the geomet= ric and the logical. One structure - to wit that of an elementary topos, ca= n be seen in two different ways depending on what the mathematical uses one= wants to give it. There is no confusion here - just richness. Let me be m= ore specific. Thinking of an elementary topos S as the chosen "set theory", a Grothendiec= k topos (including any category of the form Sh(X) for X a locale in S, but = more generally as a category of sheaves on a site in S) can be recovered as= a pair (E, e) where E is another elementary topos and e: E -> S a bounded= geometric morphism. Thinking of elementary toposes from the logical point = of view, and so of logical morphisms between them, there are other ideas an= d constructions that profit from this point of view - for instance a formu= lation and proof of realizability by means of Artin-Wraith glueing. Both the geometric and the logical are sides of the same coin. The notion o= f an elementary topos (or "topos" for short) is simple yet powerful and unt= il now it has served most of the mathematical purposes for which it was int= ended and more. Best wishes, Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill University Montreal, QC, Canada H3A 2K6 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/people/bunge ************************************************ ________________________________ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]