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 Shop pages http://carnot.yi.org/ accessible as long as the old drives survive. Personal pages http://members.shaw.ca/peasthope/ . [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 Shop pages http://carnot.yi.org/ accessible as long as the old drives survive. Personal pages http://members.shaw.ca/peasthope/ .
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
Hi, Peter, Actually, the word "other" below introduces a red herring: there is in fact every reason *not* to wish to restrict attention only to objects O *other* than E or T or X -- indeed, I can imagine that there might be settings in which there are *no* objects "other than" E or T or X, in which case the Wikipedia verbiage quoted paints you into a corner you really *don't* want to be in :-) . Cheers, -- Fred ------ Original Message ------ Received: Sat, 30 Apr 2011 03:30:49 PM EDT From: peasthope@shaw.ca To: categories@mta.ca Cc: peasthope@shaw.ca Subject: categories: Re: Explanations 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 Shop pages http://carnot.yi.org/ accessible as long as the old drives survive. Personal pages http://members.shaw.ca/peasthope/ .
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I think this is a perfect example of when something is or is not an explanation. It's both in this case: to the cogniscenti, it is perfectly clear; the novice is going to head off on the wrong track. On 4/30/11 15:58, Charles Wells wrote:
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. <snip>
-- Dr. D. E. Stevenson Associate Professor Director, Insitute for Modeling and Simulation Applications School of Computing, Clemson University [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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.
-- Telephone 1 360 450 2132. bcc: peasthope at shaw.ca Shop pages http://carnot.yi.org/ accessible as long as the old drives survive. Personal pages http://members.shaw.ca/peasthope/ .
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (5)
-
Charles Wells -
Clemson Steve -
F. William Lawvere -
Fred E.J. Linton -
peasthope＠shaw.ca