Re: Explanations

[Charles Wells <charles@abstractmath.org>]

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.
>
> --
> Telephone 1 360 450 2132.  bcc: peasthope at shaw.ca
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