Re: "maps to"

[John Duskin <duskin@math.buffalo.edu>]

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>[This is not exclusively category theory, I know, but I think the categories
>list members are the sort of people who might know the answer.]
>
>Does anyone know the origins of "|->" as symbol for "maps to", e.g. defining
>a function by
>    x |-> 3x + 5   ?
>
>Steve Vickers.

	I don't know who invented the "vertical bar-arrow" notation 
but I do have a pretty good idea of how and why it caught on, and it 
is related to the rise of category theory where the use of arrows 
proliferates and a real need arises for some notational distinction 
between using an arrow to denote a mapping X--->F(X) as, for example, 
the value at an object X of a natural transformation from the the 
identity functor to some endofunctor F and the correspondence 
X--->F(X) which defines the functor itself.
	In the early sixties, Grothendieck started using a "very 
wiggly arrow"  X~~~>F(X) to denote the correspondence (curiously, 
always drawn in by hand--not with repeated "~" on the typewriter) and 
this notation caught on, at least with those who studied with him at 
that time, and even survives today (e.g., in Mike Artin's recent and 
beautiful "undergraduate" text, Algebra). Moreover, in the early 
seventies, seeing those very noticeable wiggles in a book or paper 
immediately provoked attention ("What the hell is that?") and made 
the book or paper look "very up to date" (or,more derisively, "very 
trendy"). Undergraduate Calculus texts had already been using arrows 
(" x--->3x+5" instead of "y=3x+5" ) not only to try to  convey the 
idea of a mapping but also to look "very up to date". Being told that 
the "correct" notation for this was now "x~~~>3x+5", publishers were 
faced with the very expensive problem (in those days before Tex) of 
re-typesetting their heretofore trivially revised ("just mix up the 
exercise problem numbers and correct a few misprints") 5th and 6th 
editions with a 7th edition in which every every arrow "--->" would 
have to be replaced by " ~~~>" if the text were to remain "up to 
date".
  	Someone--I wish I knew whom-- had the authority and the 
temerity to suggest that " x|--> 3x+5" would be a less intrusive and 
certainly less expensive solution to both the logical and the 
economic problem. It caught on almost immediately and survives to 
this day. The "very wiggly arrow" would still be a pretty good 
notation to indicate a " lax" or pseudo-functor, but as far as I can 
see, it hasn't been adopted even for that purpose. It does survive 
(if somewhat diminutively) as "\leadsto" in the "arrow and pointers" 
math fonts of Tex.


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<excerpt>[This is not exclusively category theory, I know, but I think
the categories

list members are the sort of people who might know the answer.]


Does anyone know the origins of "|->" as symbol for "maps to", e.g.
defining

a function by

   x |-> 3x + 5   ?


Steve Vickers.

</excerpt>

	I don't know who invented the "vertical bar-arrow" notation but I do
have a pretty good idea of how and why it caught on, and it
<italic>is</italic> related to the rise of category theory where the
use of arrows proliferates and a real need arises for some notational
distinction between using an arrow to denote a mapping X--->F(X) as,
for example, the value at an object X of a natural transformation from
the the identity functor to some endofunctor F and the correspondence
X--->F(X) which defines the functor itself.

	In the early sixties, Grothendieck started using a "very wiggly arrow"
 X~~~>F(X) to denote the correspondence (curiously, always drawn in by
hand--not with repeated "~" on the typewriter) and this notation caught
on, at least with those who studied with him at that time, and even
survives today (e.g., in Mike Artin's recent and beautiful
"undergraduate" text, <underline>Algebra)</underline>. Moreover, in the
early seventies, seeing those very noticeable wiggles in a book or
paper immediately provoked attention ("What the hell is that?") and
made the book or paper look "very up to date" (or,more derisively,
"very trendy"). Undergraduate Calculus texts had already been using
arrows (" x--->3x+5" instead of "y=3x+5" ) not only to try to  convey
the idea of a mapping but also to look "very up to date". Being told
that the "correct" notation for this was now "x~~~>3x+5", publishers
were faced with the very expensive problem (in those days before Tex)
of re-typesetting their heretofore trivially revised ("just mix up the
exercise problem numbers and correct a few misprints") 5th and 6th
editions with a 7th edition in which every every arrow "--->" would
have to be replaced by " ~~~>" if the text were to remain "up to
date".

 	Someone--I wish I knew whom-- had the authority and the temerity to
suggest that " x|--> 3x+5" would be a less intrusive and certainly less
expensive solution to both the logical and the economic problem. It
caught on almost immediately and survives to this day. The "very wiggly
arrow" would still be a pretty good notation to indicate a " lax" or
pseudo-functor, but as far as I can see, it hasn't been adopted even
for that purpose. It does survive (if somewhat diminutively) as
"\leadsto" in the "arrow and pointers" math fonts of Tex.