Re: Isbell & MacLane on the insufficiency on skeletal categories

["Fred E.J. Linton" <fejlinton@usa.net>]

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/ ]