Prof. Peter Johnstone wrote:
But I think there is still some point in drawing the second square in A1.1.4, at least in pedagogical terms: until you've seen (or at least visualized) the second square, it's hard for the mind to accept the argument that says af = 1.
Agreed 1000% (UK: 100%). In fact it could serve as a real-world example of the difference an extra square can make in a diagrammatic argument whose verbal counterpart gains nothing from it. As such it would be interesting grist for the mill currently being ground more finely lately by those interested in diagrammatic reasoning, represented here by Sol Feferman who's been taking quite an active interest in it lately. (Even though I'm more of a visual thinker, quite the opposite of Gordon Plotkin for example who considers himself as verbal as I am visual, for some reason I tend to view commutative diagrams as more verbal than visual, perhaps because my ability to visualize makes it clearer to me that they are depicting equations between words and are therefore really verbal entities, appearances notwithstanding. This would help explain my lack of enthusiasm for the second square. But I would still draw it explicitly if teaching the lemma, just like you.)
And, as someone (I forget who, but it may have been Mike Barr) pointed out long ago, one can (well, almost) define the variety of groups as the variety defined by a single binary operation satisfying a single equation; 1 < 3, but no sane group-theorist would do it that way.
Substitute "Boolean algebra" for "group" and one finds in Stephen Wolfram's ANKS the same claim: one binary operation, satisfying one equation; in this case 1 < 12 or thereabouts when defining a Boolean algebra as a complemented distributive lattice. I'm sure Stephen is under no delusions as to the pedagogical benefits of his definition. Unfortunately neither definition is correct because both varieties so defined have as their initial object the "empty group," resp. "empty Boolean algebra," one, resp. two, elements too few. Or did you not count e when you said "one binary operation?" Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]