just an addendum on bimonoids

12 Oct 2001

      Dear friends,

a short follow-up to the discussion thread initiated by Jules Bean.

The history of categories where each object is also a bimonoid (that
is, as Tom Leinster pointed out, those self dual categories such that
for each object "a" there is an arrow "a -> a \otimes a" and an arrow
"a -> e", wher "e" is the unit of the tensor product \otimes) is quite 
a long one.  They appeared with various names (and eventually with 
just either the monoid or the co-monoid structure) such as categories 
of relations, Cartesian bicategories, etc, etc... We tried to give some 
references to all that in a recent, applicative paper of ours, "Some 
algebraic laws for Spans (and their connections with Multirelations)", 
call it [SAL] for short, and more are present in the papers of the our 
group mentioned there (see also the forthcoming paper "normal form for 
algebras of connections", to appear in TCS, and the references in
there).

@inproceedings{BG:SAL,
    author = {Bruni, R. and Gadducci, F.},
    title ={Some algebraic laws for spans 
            (and their connections with multirelations)},
    booktitle = {Proceedings of RelMiS 2001, 
                 Relational Methods in Software},
    editor = {Kahl, W. and  Parnas, D.L. and Schmidt, G.},
    series = {ENTCS},
    volume = {44},
    number = {3},
    year = {2001},
}

The paper [SAL] is just a straighforward exercise in Set theory. 
Nevertheless, it gives concrete, and very simple examples, of
categories where objects are bimonoids, but the axioms these
two structures satisfy are different. On the one hand, you find what
could be called the generalization of Frobenious algebras, where the
axiom F indeed holds
...
*  *      *   *
 \/       |\  |
  |   =   | \ |
  |       |  \|
 /\       |   |
*  *      *   *
and what could be called the generalization of Hopf algebras, where 
instead a different axiom holds, as represented by
...
*  *      *    *
 \/       |\  /|
  |   =   | \/ |
  |       | /\ |  
 /\       |/  \|
*  *      *    *
Hope it may be of help.

Fabio and Roberto

PS Note that any category where each object is a bimonoid, satisfying
the F axiom, is also compact closed: a new thread in itself... :-)

PPS About analogies for the differences between the axiomatisations,
Jules proposed
...
I have an intuitive justification for wanting these to be different,
if people aren't offended by slightly silly analogies. Think of the
networks (which is what I call them) as river networks. They have to
flow downhill (down the page). They can join as tributaries do, or
split into distributaries. Then in the 'X' all the water has possibly
mixed; we can't assume it will divide the same way.  In the 'N' on the
other hand, all of the water which came in on the right, has
definitely gone out on the right.
Or just think about relations and equivalence relations: in the latter,
you always need to close by transitivity. Less of an analogy, in fact,
than of a theorem... Anyhow, for the network analogy, see in particular 
the book "Network Algebra" by Stefanescu, Springer, 2000.

just an addendum on bimonoids

Roberto Bruni