Correction/clarification to previous note: in what follows, the sloppy phrase
... the resulting u' is the same as the result of conjugating all the values of u by the transposition ...
should be replaced by the more accurate expanded version
... the h' (as below) resulting from u' is the same as what Sammy gets by conjugating all the values of the h that arises from u by the transposition ...
Sorry for the sloppy writing. Cheers, -- Fred ------ Original Message ------ Received: Sun, 24 Jun 2012 08:09:45 PM EDT From: "Fred E.J. Linton" <fejlinton@usa.net> To: André" <joyal.andre@uqam.ca>Cc: "categories" <categories@mta.ca> Subject: categories: Re: Two_questions
Salut, André,
I'm ashamed how long it took me to come to this realization, but you were absolutely correct in your surmise that, when one lets ...
... E' be the set obtained from E by adding copy p' of p. There are two embeddings u,u':E-->E', the first u is the inclusion of E in E', and the second u' is defined by putting u'(p)=p' and u'(x)=x for x different than p.
... the resulting u' is the same as the result of conjugating all the values of u by the transposition t that exchanges p with p'. Thus, the ...
... pair of homomorphisms h,h':E!-->E'! the equaliser of which is the stabiliser S(p) of p in E!.
... that arises is exactly the same as the pair Sammy's argument would adduce, and the only respect in which ...
This last argument seems to differ from the argument you have presented.
is that for Sammy it was enough to observe that h and h' differ SOMEWHERE when E is more than just {p}, while what you observe is rather more, namely,
that h and h' actually differ EVERYWHERE other than on S(p), i.e., that the
ONLY place where h and h' do NOT differ is S(p) :-) .
Am I making an error?
Only in thinking that seeming difference makes any real difference :-) .
Cheers, -- Fred
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