On Thu, 06 Jun 2013 07:45:03 PM EDT, by Vaughan Pratt <pratt@cs.stanford.edu>:
On 5/25/2013 8:47 AM, Colin McLarty wrote:
Of course is also follows that NxN=N. But it does not follow, and in fact it is refutable, that the projection functions are the identity function 1_N. Isbell's argument is on p. 164 of my copy of CfWM (1998).
Why do you need Isbell's long argument, or even any monoidal structure on Set, to obtain a contradiction here? Just use that NxN is a product and observe that the pair (3,4) in NxN (as a map from 1 to NxN) would have to be both 3 and 4 (as maps from 1 to N) when the projections are the identity.
Even more convincing: The equalizer of those two projections from N x N must be the diagonal in N x N. But if those projections are equal, their equalizer is all of N x N. Thus every map to N x N factors through the diagonal there, i.e., no matter what the object A, for every pair of maps f, g: A --> N, we must have f = g. It will follow that N is terminal. [Or was that your argument, Vaughan, that I somehow did not recognize?] Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]