Re: Non-cartesian categorical algebra

Re: Non-cartesian categorical algebra

[Ross Street]

There is an embedding theorem on which we have put Cayley's name:
if M is a monoid in a closed category then the structural coretraction
M --> [M,M] into the endohom is a nice monoid map.

A bicategorical version of this gives a nice module (distributor) A
--|--> A^{op} #A
for any (pro)monoidal V-category A. This leads to a monoidal
embedding of any
such A into the category of A-bimodules. (E.g. see Section 4 of
Pastro-St:
http://www.tac.mta.ca/tac/volumes/21/4/21-04.pdf
however Brian Day also knew about these things.)

So the abstract case is not so much more abstract. I think Peter
Johnstone says
somewhere that one view of the Abelian Category Embedding Theorem is not
so much that it means we should use module-proofs to work in abelian
categories
but rather, when working in categories of modules, we might as well
work in an
abelian category. I think the same applies here for monoidal categories.
The coring people I have spoken to seem quite comfortable with
this development. Luckily we all have our own sources of motivation.

Ross

On 15/09/2008, at 10:57 PM, Joost Vercruysse wrote:

> 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.

Re: Non-cartesian categorical algebra

[Joost Vercruysse]

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.

Non-cartesian categorical algebra

[George Janelidze]

Dear Colleagues,

I would like to make a remark concerning my CT2008 talk.

First let me recall: A lot of mathematics (e.g. of Galois theory) can be
done in the context of adjoint functors between abstract categories with
finite limits - and since one gets all finite limits our of finite products
and equalizers, one can try a further generalization with monoidal structure
plus equalizers. The point was that this seemingly primitive old idea
actually works very seriously and should be taken as the idea of developing
non-cartesian categorical algebra. And "non-cartesian" is the right idea of
"non-commutative" and "quantum", although what Ross Street means by
"quantum" is more involved and also important. In particular non-cartesian
internal categories are to be taken seriously.

At the end of my talk Jeff Egger told us that he knows someone studied such
generalized internal categories, and later sent me an email with the name:
Marcelo Aguiar; and gave the home page address
http://www.math.tamu.edu/~maguiar/ , and... I realized that it is the third
time I am informed about this work! Recently (winter 2007) I spend two very
nice months in Warsaw invited by Piotr Hajac, and discussing mathematics
with him, Tomasz Brzezinski, Tomasz Maszczyk, and a few others - and, among
other interesting things, Tomasz Brzezinski showed me Marcelo Aguiar's
website, including PhD, where those generalized internal categories were
studied. I also recall now an email message from Steven Chase (from 2002)
where he mentions "...the notion of a category internal to a monoidal
category which was developed by my former doctoral student, Marcelo Aguiar,
in his thesis, "Internal Categories and Quantum Groups" (available on
line...".

In fact the whole story begins, in some sense, with the book [S. U. Chase
and M. E. Sweedler, Hopf algebras and Galois theory, Lecture Notes in
Mathematics 97, Springer 1969], which does not use monoidal categories yet,
but very clearly shows that the commutative case is much easier (for Galois
theory) because it makes tensor product (of algebras) (co)cartesian. There
are many other important further contributions by other authors of different
generations. Knowing them personally, I can name Bodo Pareigis, Stefaan
Caenepeel, Peter Schauenburg, and the aforementioned Polish mathematicians
(although Tomasz Brzezinski is in UK now), but I am not ready to give any
reasonably complete list. 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"...

George Janelidze