Re: Non-cartesian categorical algebra

[Joost Vercruysse <jvercruy@vub.ac.be>]

On 14-sep-08, at 15:39, George Janelidze wrote:

Dear George and all,

> There are also things-to-be-corrected happening:
> for instance by far not enough comparisons have been made with the
> Australian work on abstract monoidal categories, and some authors
> use words
> like "coring"...

I hope the following information can be of help here:
Indeed, Marcello Aguilar gave a definition of `internal categories'.
Although the abstract definition of a `coring' looks formally the same
as the one of an internal category (or, if you wish, an internal
cocategory), corings provide examples of these internal cocategories,
but they (usually) refer to a much more concrete situation: a coring
is a co-monoid in the monoidal category of bimodules over a given
(possibly non-commutative) ring, this dualizes usual ring extensions.
The theory of corings is in fact quite young, and grew from a pure
algebraic theory to something more and more categorical in the last
few years (this might cause some confusion, `internal corings', which
can be defined in certain monoidal categories (the regular ones from
aguilar) or bicategories, are indeed the same objects as internal
cocategories, there is no need for two names for the same thing at
this level of generality). Therefore, I find the above remark ``not
enough comparision have been made ...'' indeed correct: I believe that
people from corings can learn from more from the pure category theory
side, and hopefully the other way around as well.

Best wishes,
Joost.