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