This is in reply to Toby Bartels, quoted below. I don't believe that those of us who have written about "ideas" in Ehresmann's sense ever conceived that each theory (sketch) was based on one right idea. There is no "correct" idea for a given sketch. I want to add, for those new to the subject, that the word "sketch" has been used with at least three meanings. Ehresmann and his students use it for a structure which is a weakening of the concept of category (the composite may not be defined for all composable pairs) plus specified cones and/or cocones. Many others have used the word sketch to refer to a category with specified cones and/or cocones. Michael Barr and I in our two books used "sketch" to mean a graph with specified cones and/or cocones plus some commutativity conditions on paths; that is in the same spirit as Ehresmann's "idea". --Charles Wells
Andree Ehresmann wrote in part:
He thought first of calling a sketch an idea, but then reserved the word "idea" for the smallest part which helps reconstruct the sketch; for instance for a category, the arrows which 'represent' the domain and codomain maps and the composition law.
There could be multiple ideas that generate the same sketch; how do we decide which is the correct idea among equivalent ones? OTOH, if we take equivalence classes of ideas, then we're taking sketches. For example, one could define the idea of multiplication in a monoid as a binary operation and a nullary operation or alternatively as an operation on finite tuples. The former is more common, but I prefer the latter; who has the right idea?
-- Toby toby@math.ucr.edu
Charles Wells, Emeritus Professor of Mathematics, Case Western Reserve University Affiliate Scholar, Oberlin College Send all mail to: 105 South Cedar St., Oberlin, Ohio 44074, USA. email: charles@freude.com. home phone: 440 774 1926. professional website: http://www.cwru.edu/artsci/math/wells/home.html personal website: http://www.oberlin.net/~cwells/index.html genealogical website: http://familytreemaker.genealogy.com/users/w/e/l/Charles-Wells/ NE Ohio Sacred Harp website: http://www.oberlin.net/~cwells/sh.htm