Hi Tom (and others), I was perhaps the primary author of that nLab page. I think the more standard term is actually "measuring coalgebra". Anyway, this is also discussed in Barr's paper that I alluded to in an earlier message: Coalgebras over a commutative ring, J. Alg. 32, 600-610 (1974) The idea is simple enough: the measuring coalgebra of two commutative algebras A, B is a representing coalgebra m(A, B) for which there is an isomorphism Coalg(C, m(A, B)) = Alg(A, hom(C, B)) natural in C, where the hom on the right is a vector-space hom that acquires a natural algebra structure if C is a coalgebra and B is an algebra. But the idea of measuring coalgebra goes far beyond a base of enrichment for commutative algebras. Basically, if you take two vector-space models A, B of any prop, there is a suitable measuring coalgebra construction m(A, B) [Barr, theorem 6.3]. So, "practically anything" can be enriched in cocommutative coalgebras. :-) Best regards, Todd ----- Original Message ----- From: "Tom Leinster" <Tom.Leinster@glasgow.ac.uk> To: "Anders Kock" <kock@imf.au.dk> Cc: <categories@mta.ca>; "Tom Leinster" <Tom.Leinster@glasgow.ac.uk> Sent: Friday, August 26, 2011 3:15 PM Subject: categories: re: a coalgebra over fields question Dear Anders,

Gavin Wraith lectured in 1976 about an important aspect in this connection. I don't know of any published reference. Namely: The category of commutative algebras is enriched over the cartesian closed category of commutative coalgebras.

Does anybody know of a published reference ?

I believe the key term here is "measure coalgebra" or "measuring coalgebra". There's an nLab page on this: http://ncatlab.org/nlab/show/measure+coalgebra It doesn't give any references, and I don't know any. But presumably the authors of that page are reading this and have a better idea than me. André Joyal also mentioned this enrichment in his talk at CT11. Best wishes, Tom [For admin and other information see: http://www.mta.ca/~cat-dist/ ]