Is the Heyting algebra of global elements of the classifier in an elementary topos always complete? (Of course, the classifier is complete internally; here, I mean externally complete.) I suspect not, but I can't presently think of a natural counterexample. Thanks, Lutz Schröder -- ------------------------------------------------------------------ PD Dr. Lutz Schröder office @ Universität Bremen: Senior Researcher Cartesium 2.051 Safe and Secure Cognitive Systems Enrique-Schmidt-Str. 5 DFKI-Lab Bremen FB3 Mathematik - Informatik Robert-Hooke-Str. 5 Universität Bremen D-28359 Bremen P.O. Box 330 440 D-28334 Bremen phone: (+49) 421-218-64216 Fax: (+49) 421-218-9864216 mail: Lutz.Schroeder@dfki,de www.dfki.de/sks/staff/lschrode ------------------------------------------------------------------ ------------------------------------------------------------- Deutsches Forschungszentrum für Künstliche Intelligenz GmbH Firmensitz: Trippstadter Strasse 122, D-67663 Kaiserslautern Geschäftsführung: Prof. Dr. Dr. h.c. mult. Wolfgang Wahlster (Vorsitzender) Dr. Walter Olthoff Vorsitzender des Aufsichtsrats: Prof. Dr. h.c. Hans A. Aukes Amtsgericht Kaiserslautern, HRB 2313 -------------------------------------------------------------
On Tue, 20 Nov 2007, Lutz Schroeder wrote:
Is the Heyting algebra of global elements of the classifier in an elementary topos always complete?
The answer is no: any Boolean algebra, complete or not, can occur as Sub(1) in a topos (see Exercise 9.11 in my old Topos Theory book), and any quotient of a complete Heyting algebra (by a finitary Heyting congruence -- such quotients needn't be complete) can occur, by use of the filterpower construction (cf. my paper with Murray Adelman "Serre classes for toposes", Bull.Austral.Math.Soc. 25 (1982), 103-115). There are examples due to Peter Freyd of Heyting algebras which can't occur as Sub(1) in a topos generated by subobjects of 1. For a long time it was an open problem whether any Heyting algebra can occur as Sub(1) in a topos (without the restriction on generators): Dito Pataraia has recently announced a positive solution to this problem. I have heard a seminar talk about his solution, and seen half of a preprint, but haven't yet managed to understand the other half of his construction. Peter Johnstone
Of course not ! Elementary toposes are models of a first order theory (this is also true for elementary toposes with an NNO) . Thus there are countable models. In such a model the answer is no. Le 20 nov. 07 à 13:10, Lutz Schroeder a écrit :
Is the Heyting algebra of global elements of the classifier in an elementary topos always complete? (Of course, the classifier is complete internally; here, I mean externally complete.) I suspect not, but I can't presently think of a natural counterexample.
Thanks,
Lutz Schröder
-- ------------------------------------------------------------------ PD Dr. Lutz Schröder office @ Universität Bremen: Senior Researcher Cartesium 2.051 Safe and Secure Cognitive Systems Enrique-Schmidt-Str. 5 DFKI-Lab Bremen FB3 Mathematik - Informatik Robert-Hooke-Str. 5 Universität Bremen D-28359 Bremen P.O. Box 330 440 D-28334 Bremen phone: (+49) 421-218-64216 Fax: (+49) 421-218-9864216 mail: Lutz.Schroeder@dfki,de www.dfki.de/sks/staff/lschrode ------------------------------------------------------------------
------------------------------------------------------------- Deutsches Forschungszentrum für Künstliche Intelligenz GmbH Firmensitz: Trippstadter Strasse 122, D-67663 Kaiserslautern
Geschäftsführung: Prof. Dr. Dr. h.c. mult. Wolfgang Wahlster (Vorsitzender) Dr. Walter Olthoff
Vorsitzender des Aufsichtsrats: Prof. Dr. h.c. Hans A. Aukes
Amtsgericht Kaiserslautern, HRB 2313 -------------------------------------------------------------
participants (3)
-
JeanBenabou -
Lutz Schroeder -
Prof. Peter Johnstone