On 02/12/2011, at 12:17 PM, David Leduc wrote:
Thank you very much but I am afraid I am already stuck at the first sentence. Why such a diagram in Cat would be a theory?
A (limit) sketch in the sense of Ehresmann is a family of cones on a category C. A cone is a natural transformation pointing right in a triangle whose top horizontal side is X ------> 1 and whose bottom vertex is C. A cone is the special case of what I called a theory with A = X and B = D = 1 (the terminal category). Now let A be the coproduct (disjoint union) of all the categories X in the sketch and let B = D be the discrete category obtained by adding up all the 1s, one for each index of the family. Take u : A --> B to be the induced map on the coproducts. Take t to be the identity. The cones give a single natural transformation tau using the 2-universal property of coproduct. So a sketch on C is the same as one of my theories with B discrete and t an identity. The extra flexibility of having B not necessarily discrete and tagging on the t was aiming at theories in the sense of John Isbell [General functorial semantics. I. Amer. J. Math. 94 (1972), 535–596; MR0396718 (53 #580)]. Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]