In the expression "any x:T->X" the T depends on x. If you use the arrow notation you seem bound to name the domain of the morphism. You could say "for any x with codomain X there is an e:dom x -> X ..." but in the rest of the sentence you will have to mention the domain again. My impression is that notation "any x:T->X" where T depends on x without that fact being mentioned is common in category theory writing. There is nothing wrong with this if a reader understands the intent. I would call it "suppression of dependence". In the Handbook I talked about suppression of parameters, but this is not suppression of parameters. It is something I had not noticed before. Are there other situations in math where this happens? On Fri, Apr 29, 2011 at 2:56 PM, <peasthope@shaw.ca> wrote:
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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