Re: a coalgebras over fields question.

[Todd Trimble <trimble1@optonline.net>]

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

----- Original Message -----
From: "Bisson, Terrence P" <bisson@canisius.edu>
To: <categories@mta.ca>
Sent: Wednesday, August 24, 2011 12:58 AM
Subject: categories: a coalgebras over fields question.


> Hi,  I have heard that naive questions are allowed at the cat list, so
> here  goes:
>
> The diagonal map in spaces often gives a co-commutative diagonal map in
> homology,
> so I want to understand the special properties of the category of
> co-commutative coalgebras.
>
> It seems to be a "well-known fact" that
>  the category of co-commutative coalgebras over a field is a cartesian
> closed category,
> but I can't seem to find much discussion of the internal hom.
> Can you suggest any reference?
>
> Thanks, Terry Bisson
>



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