Dear categorists, in "Combining algebraic effects 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 ------------------------------------------------------------- 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/ ]
in "Combining algebraic eï¬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? Martin asked me about this about a year ago, and at the time I came up with something I reckoned was a counterexample, though I recall that the details were pretty foul. Do you want me to try to reconstruct it?
Nathan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Here's a slightly artificial counterexample: let C be the category of finite sets whose cardinality is a power of 2, and all functions between them. The covariant power-set functor restricts to a functor C --> C, and has a monad structure whose algebras are semilattices. If the tensor product of this monad with itself existed, its algebras would be bisemilattices, i.e. sets with two semilattice structures which "commute with each other" in the obvious sense. Free bisemilattices exist, but they don't necessarily have cardinality a power of 2: by my calculation, the free bisemilattice on two generators has seven elements. So the free-bisemilattice functor doesn't exist as an endofunctor of C. Peter Johnstone ----------------------- On Wed, 28 Jul 2010, Sergey Goncharov wrote:
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/ ]
There are examples in Ernie Manes's 1967 thesis. Perhaps the simplest (although it piggybacks on the non-existence of free complete boolean algebras that had been know for only a few years at the time) is that the tensor product of the complete sup semilattice triple with itself doesn't exist. The triple takes a set X to 2^X and can be interpreted also as the complete inf semilattice triple. On the other hand, I think Manes showed that the tensor product of the beta triple with itself exists, but is one of the two inconsistent triples, the one that fixes the empty set and takes all non-empty sets to one point. (The other inconsistent triple takes all sets to one point.) On Wed, 28 Jul 2010, Sergey Goncharov wrote:
Dear categorists,
in "Combining algebraic eects 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,
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Isn't P * P isomorphic to P, by the Eckmann-Hilton argument? On 29 July 2010 20:29, Michael Barr <barr@math.mcgill.ca> wrote:
There are examples in Ernie Manes's 1967 thesis. Perhaps the simplest (although it piggybacks on the non-existence of free complete boolean algebras that had been know for only a few years at the time) is that the tensor product of the complete sup semilattice triple with itself doesn't exist. The triple takes a set X to 2^X and can be interpreted also as the complete inf semilattice triple. On the other hand, I think Manes showed that the tensor product of the beta triple with itself exists, but is one of the two inconsistent triples, the one that fixes the empty set and takes all non-empty sets to one point. (The other inconsistent triple takes all sets to one point.)
On Wed, 28 Jul 2010, Sergey Goncharov wrote:
Dear categorists,
in "Combining algebraic eects 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,
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (5)
-
Michael Barr -
N.Bowler＠dpmms.cam.ac.uk -
Prof. Peter Johnstone -
Richard Garner -
Sergey Goncharov