From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/665 Path: news.gmane.org!not-for-mail From: Peter Freyd Newsgroups: gmane.science.mathematics.categories Subject: Reddy's question Date: Tue, 3 Mar 1998 07:32:55 -0500 (EST) Message-ID: <199803031232.HAA05517@saul.cis.upenn.edu> NNTP-Posting-Host: main.gmane.org X-Trace: ger.gmane.org 1241017108 26791 80.91.229.2 (29 Apr 2009 14:58:28 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:58:28 +0000 (UTC) To: categories@mta.ca Original-X-From: cat-dist Tue Mar 3 13:09:26 1998 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id JAA10032 for categories-list; Tue, 3 Mar 1998 09:44:26 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 56 Xref: news.gmane.org gmane.science.mathematics.categories:665 Archived-At: Uday Reddy poses the following (with a few changes in notation]: >Consider a monoid in a CCC. The operations of interest >are natural transformations E:[-,M] -> M that satisfy the >following equations (in the internal language of the CCC): > > E_A(\x.e) = e > E_A(\x. a * gx) = a * E_A(g) > E_A(\x. gx * a) = E_A(g) * a > E_A(\x. E_B(\y.hxy)) = E_B(\y. E_A(\x.hxy)) I wonder if naturality is really desired: it would seem to force M to be trivial. By the familiar Yoneda-lemma argument, E must be constant as far as the "points" of [A,M] are concerned. (Actually one doesn't need the argument, just the lemma itself; consider the transformation that E induces between set-valued functors (-,M) -> (1,M); Yoneda says it must be constant.) The condition E_A(\x.e) = e forces just which constant it is. That is, for any f:A -> M it will be the case that E_A will send f to e. But then either condition E_A(\x. a * gx) = a * E_A(g) or E_A(\x. gx * a) = E_A(g) * a will force M to be trivial. (It's clear in the \-calculus notation. But that argument would be implicitly using the fact that E_A is constant and we officially know only that it's constant on points. So take, say, the second condition. It says in diagrammatic language: 1 x K P [A,*] [A,M] x M -----> [A,M] x [A,M] ---> [A,MxM] -----> [A,M] | | | E_A x 1_M | E_A | | * M x M ----------------------------------------> M where K is the standard "constant-map" operator that's adjoint to the projection AxM -> M and P is the standard operator that defines MxM (given products in Set). Specialize to A = M and precompose with : M -> [M,M] x M where f doesn't matter. If the commutative rectangle above is chased clockwise one obtains the map constantly valued e. Chased counterclockwise one obtains the identity map. And, of course, it didn't matter what f is.)