On 07/29/2010 03:24 PM, Prof. Peter Johnstone wrote:
Sorry, the previous posting was nonsense -- a bisemilattice is the same thing as a semilattice, by the Eckmann-Hilton argument. However, if you leave out the zero, and consider the "set of nonempty subsets" monad, this time on the category of sets of cardinality 2^n - 1 for some n, you do get a counterexample. This looks fine! But I guess, Eckmann-Hilton argument does not apply to your previous example because it presupposes that the monoidal structures share the unit, which was not the case there, was it?
Thanks a lot, -- Sergey Goncharov, Junior Researcher DFKI Bremen Phone: +49-421-218-64276 Safe and Secure Cognitive Systems Fax: +49-421-218-9864276 Cartesium, Enrique-Schmidt-Str. 5 Email: Sergey.Goncharov@dfki.de D-28359 Bremen Site: www.dfki.de/sks/staff/sergey ------------------------------------------------------------- Deutsches Forschungszentrum fuer Kuenstliche Intelligenz GmbH Firmensitz: Trippstadter Strasse 122, D-67663 Kaiserslautern Geschaeftsfuehrung: 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 ------------------------------------------------------------- [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Sergey, Here is a simple example, based on the ordered class ORD of all ordinals numbers. The subclass LIMIT of all limit ordinals is reflexive in ORD. Let us denote by L the resulting reflection operator. Similarly, the subclass SUCC of all successor ordinals is reflexive in ORD. Let us denote by S the resulting reflection operator. The operators L and S cannot be bounded simultaneously by a reflection operator (= monad) since the classes LIMIT and SUCC have an empty intersection. Best, Andre -------- Message d'origine-------- De: Sergey Goncharov [mailto:sergey@informatik.uni-bremen.de] Date: mer. 28/07/2010 10:02 À: categories@mta.ca Objet : categories: Tensor of monads Dear categorists, in "Combining algebraic e?ects with continuations", by Hyland et al. the authors say carefully: "In general, the tensor product of two arbitrary monads seems not to exist.." without providing a counterexample though, presumably because they did not have any. Was there any progress reported on this issue since then? Or maybe someone can even make up a counterexample right on the nail? Thanks, -- Sergey Goncharov, Junior Researcher DFKI Bremen Phone: +49-421-218-64276 Safe and Secure Cognitive Systems Fax: +49-421-218-9864276 Cartesium, Enrique-Schmidt-Str. 5 Email: Sergey.Goncharov@dfki.de D-28359 Bremen Site: www.dfki.de/sks/staff/sergey [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Sergey Goncharov wrote in part:
But I guess, Eckmann-Hilton argument does not apply to your previous example because it presupposes that the monoidal structures share the unit, which was not the case there, was it?
Actually, the requirements for Eckmann-Hilton are surprsingly weak! As long as you have two binary operations on a given set, each with its own unit, where one is a homomorphism WRT the other, then everything else (associativity, same unit, etc) follows. See http://ncatlab.org/nlab/show/Eckmann-Hilton+argument for details. --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (3)
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Joyal, André -
Sergey Goncharov -
Toby Bartels