Those interested in similar results might also look at my recent paper On subcategories of the category of Hopf algebras, Arabian Journal of Science and Enginering, (2011), DOI 10.1007/s13369-011-0090-4 and the references therein. A preprint of which as available at http://www.math.uni-bremen.de/~porst/dvis/PORST_Hopf_fin.pdf Regards, Hans Am 24.08.2011 um 18:46 schrieb Todd Trimble:
Dear Terry,
The category of cocommutative coalgebras over a field k has a lot of nice properties: it is not only cartesian closed, it is also extensive and locally finitely presentable (hence also complete and cocomplete, and even total).
If you want a quick proof of cartesian closure based on the adjoint functor theorem, try this paper by Michael Barr:
ftp://ftp.math.mcgill.ca/pub/barr/coalgebra.pdf
which gives a proof that applies more generally to coalgebras over a general commutative ring.
One absolutely crucial fact in this whole business is sometimes called the fundamental theorem for (cocommutative) coalgebras: each is the filtered colimit of its finite-dimensional subcoalgebras. (Here we are working over a field k. The situation is more subtle over a general commutative ring R.) The finite-dim cocommutative coalgebras over k coincide with the finitely presentable objects in CocommCoalg_k, and the category of finite-dim cocommutative coalgebras is dual to the category of finite-dim commutative algebras/k. One concludes (a la Gabriel-Ulmer duality) that there is an equivalence
CocommCoalg ~ Lex(CommAlg_{fd}, Set),
where the right side is the category of left exact functors on the category of finite-dim commutative algebras/k.
From there, one can derive how exponentials should work: if C and D are cocommutative coalgebras, then their exponential D^C is the coalgebra which represents the left exact functor which takes a finite-dim algebra A to the set of coalgebra homomorphisms
A* \otimes_k C --> D
(NB: if C and C' are cocommutative coalgebras over k, then C' \otimes_k C is their cartesian product in CocommCoalg.)
Best regards,
Todd Trimble
-- Hans-E. Porst porst@math.uni-bremen.de FB 3: Mathematics Phone: +49 421 21863701 University of Bremen Secr.: +49 421 21863700 D-28334 Bremen Fax: +49 421 2184856 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]