re: a coalgebra over fields question

re: a coalgebra over fields question

[Anders Kock]

Hi,

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 am in a posoition of a 1981 M.Sc. thesis by Ernst K. Pedersen, giving
the details of the constructions involved.)


Anders


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re: a coalgebra over fields question

[Tom Leinster]

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


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re: a coalgebra over fields question

[Todd Trimble]

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/ ]