[I have the slight impression that this message got overlooked due to the simultaneous discussion on `free categories with X', to which one might at first sight think it belonged. Hence the reposting --- thanks for your patience...] Dear categorists, it is well known that the Yoneda embedding Y: A -> [A^op,Set] preserves existing function spaces (and also, for instance, the subobject classifier if there is one). Does anybody know if the presheaf category [A^op,Set] contains the `relatively free' cartesian closed category (topos, ...) over A as a (full?) subcategory, generated by Y[A] in the obvious sense? Here, relatively free means universal w.r.t. functors that, like Y, preserve all existing structure. Thanks a lot, Lutz -- ----------------------------------------------------------------------------- Lutz Schroeder Phone +49-421-218-4683 Dept. of Computer Science Fax +49-421-218-3054 University of Bremen lschrode@informatik.uni-bremen.de P.O.Box 330440, D-28334 Bremen http://www.informatik.uni-bremen.de/~lschrode ----------------------------------------------------------------------------- 10-Jul-2002 09:54:03 -0300,1305;000000000000-00000013
Dear categorists, in response to my reposting, I have been told off the list that the answer to my question (can the relatively free ccc be found in the Yoneda extension?) is no, the counterexample being the object classifier [FinSet,Set] -- Regard the dual A of FinSet as a subcategory of [FinSet,Set], and take U to be the singleton set as an object of A; then (U->U)->(U->U) has only two global elements in [FinSet,Set], while in the free ccc (preservation of function spaces does not play a role here, since A has only trivial function spaces) the global elements of that object should be the (Church) natural numbers. Apparently, observations such as this one go 'back' to Lawvere. Best regards, Lutz -- ----------------------------------------------------------------------------- Lutz Schroeder Phone +49-421-218-4683 Dept. of Computer Science Fax +49-421-218-3054 University of Bremen lschrode@informatik.uni-bremen.de P.O.Box 330440, D-28334 Bremen http://www.informatik.uni-bremen.de/~lschrode ----------------------------------------------------------------------------- 15-Jul-2002 20:57:32 -0300,1502;000000000001-00000019
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Lutz Schroeder