[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.