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![[parent]](http://images.planetmath.org:8080/images/uparrow.png) Viewing Correction to 'representable functor'
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uses of representability by jay
Correction id: 1811
Filed on: 2003-04-23 00:18:04
Status: Accepted on 2005-09-16 14:10:34
Type: Addendum
Correction text:
(i would first like to underscore the comments of the other pending correction.)
it would nice to add why this notion is (extremely!) important. namely, there is a philosophy of Grothendieck which suggests that we _define_ certain objects by having the characterizing property that they _represent_ certain functors.
for this matter, you might mention under what conditions, for two objects X and Y, we know that X^\bullet is naturally isomorphic to Y^\bullet (respectively for the covariant functors X_\bullet and Y_\bullet). that is, in what categories do the functors that X represents, determine X up to (unique?) isomorphism?
next, you might want to mention the object in X^\bullet(X) = Hom(X,X) identified with id_X. this will be your "universal object"--explain what a universal object is, and why its existence (up to unique isomorphism) is the same thing as the representability of the functor.
now you are in a good setup to explain how certain objects are defined as those which represent certain functors, and how in many cases we can (with our bare hands) quite vividly construct the respresenting ("universal") objects. do a construction or two. good examples are:
1) (co)products, free products
2) fibered (co)products
3) tensor products as universal bilinear maps
4) direct/inverse limit functors
5) the module of Kahler differentials as the universal R-derivation |
Comment from object owner mathcam:
Hi Jay,
For the sake of completing the correction, I've just addressed in this correction the importance of the idea of representability. Many of your other comments (e.g. some explicit constructions) probably merit their own entries.
Thanks,
Cam |
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