Dear Peter We thought of "other" of course. But that word has no agreed on mathematical definition. Students who think it means "distinct" will be confused when told that it is a special application of the UMP that yields the graph of a map as a section of a projection. (Perhaps best if "self" is a special case of "other" ?) Sammy always scolded Jon, Fred, Myles, and me that such "helpful" explanations make difficult the digestion and mathematical use of simple clear definitions. (I don't think this excludes explanation in a separate paragraph or footnote). Bill
From: peasthope@shaw.ca Date: Fri, 29 Apr 2011 11:56:48 -0800 To: categories@mta.ca CC: peasthope@shaw.ca Subject: categories: Re: Explanations
Charles & everyone,
Earlier peasthope wrote, "...changing a few words of a sentence can make a concept obvious rather than nebulous". Revise that to "obvious rather than difficult".
From: Charles Wells <charles@abstractmath.org> Date: Fri, 22 Apr 2011 09:37:44 -0500
Can you give specific examples? I suspect that in most cases the change introduces a useful metaphor that was hidden before.
Here is a small example from the _Conceptual Mathematics_ of Lawvere and Schanuel. No offense to the authors or the book. It's an indispensible and invaluable resource.
L&S page 292, "Definition ... equalizer ... and for each x:T-->X ... there is exactly one e:T-->E ... ." "For all T" is implicit.
http://en.wikipedia.org/wiki/Equalizer_(Mathematics) , "In category theory ... defined by a universal property, ... object E and morphism eq ... such that, given any other object O and morphism m ... ."
For me, the reference to "any other object O" helps. The definition in the Wikipedia seems to reveal the "universality" of the equalizer better. The diagram also helps.
A trivial issue for most readers but a small detail can make a difference for a student.
Regards, ... Peter E.
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