There are a number of definitions of sketch around, some of which require it to be a category with finite products. In one of Ehresmann's (and Bastiani's, I believe) there is mentioned the possibility of its being what they called a quasicategory (or some such substructure term) in which composition is a partly defined multi-ary operation (in other words, fgh could be defined without fg or gh being defined). Charles and I realized that this was equivalent to what we called a graph with diagrams, which seemed a more useable notion. So what we called a sketch was a graph with diagrams as well as certain cones and cocones that were singled out to be taken to limits and colimits, resp. Peter Johnstone criticized us for doing the equivalent of replacing groups by generators and relations, which is correct, but it was a conscious decision and there were reasons for it. I had never heard the term "idea" in this connection or we might have used it. But anyway, "sketch" is used in different ways and I guess Charles and I contributed to this, but didn't create it. On Mon, 3 Dec 2001 baez@math.ucr.edu wrote:
Toby Bartels writes:
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?
I'm confused: in my understanding, a sketch basically amounts to a way of giving generators and relations for a category with products, Different sketches give the same category with products, not vice versa. Your example gives two sketches, but one category with products. In this sense, a sketch is more like an "idea" than you seem to be giving it credit for.
By the way, in response to Lawvere's comments:
My use of the term "Platonic idea of X" for the free category/category with products/monoidal category/2-category/whatever on an X was not meant as an endorsement of "Platonism" in the philosophy of mathematics - especially since "Platonism" means many things to many people. It was also not meant to suggest that Plato had this idea. It was basically meant to get people thinking about abstract generals versus concrete particulars.
Best, John Baez