Are there famous results that can be stated as "Every functor from category A to concrete category B is representable"? Or at least "Every functor from this class that would a priori seem to be too large is representable"? I don't have any use for such a result, I just want to have more feeling for representability vs nonrepresentability. A more specialized question: the representing object for a representable functor from k-Alg to Grp gets a natural commutative Hopf algebra structure. This can be generalized in two ways: lose representability, and just have a functor, or lose commutativity, and just have a noncommutative Hopf algebra. Has anyone ever seen an example where it was desirable to lose both, somehow? Allen K. ==============================================================================
Are there famous results that can be stated as "Every functor from category A to concrete category B is representable"? Or at least "Every functor from this class that would a priori seem to be too large is representable"?
One version of the Giraud theorem says: A category E is Grothendieck topos if and only if every contravariant functor from E to Set, which takes colimits to limits, is representable. This is somewhere in expose IV of SGA4 (Springer LNM 269); also in the appendix of Barr's Exact Categories (Springer LNM 236)... The questions of this kind were studied more in the early days of category theory. There are several propositions in the form which interests you in Lambek's book on completions (Springer LNM 24). Regards, Dusko ==============================================================================
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Allen Knutson -
pavlovic＠triples.Math.McGill.CA