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