categories: Re: Why binary products are ordered

["zdiskin" <zdiskin@email.msn.com>]

OOPS! In my previous posting i've made some mixed-up and talked mainly about
notation for diagrams while the actual question is whether product  is
commutative (symmetric) operation in Sets. Below is a sketch of possible
solution.

First of all, product is a diagram operation and explicit presentation of
their syntax would be useful; so, let me present a bit of technicalities.

Let G be some base category (think of the category of graphs). The arity of
a diagram operation F (think of  products, push-outs...) is a G-morphism s:
S^in --> S with S the shape of the operation and S^in the input shape.

For example, for binary products, S is the graph    [x] <--p-- [u] --q-->
[y]
where [ ]'s denote nodes and arrows'  names are  written right on the
arrows,
S^in is the two-node discrete graph  [x]  [y],  and morphism s is its
embedding  into S.

A G-object C (think of a category)  is an F-algebra if for any diagram d:
S^in --> C there is a unique diagram  F^C(d): S-->C such that s;F^C(d) = d
(input data are preserved); below i'll write just F for F^C.

Well, categorically we normally have many isomorphic F(d)'s,  let's skip
this  for a while. And as for products of sets (ie, F is Prod, C is Sets),
by using names p and q that are already given in the shape, we really have a
canonical choice. Namely, for any d: [x]  [y] --> Sets with x.d=A, y.d=B, we
define
u.Prod(d) = { t: {p,q} -->  A \cup B |  p.t \in A, q.t \in B } with evident
p.Prod(d) and q.Prod(d).

F is called commutative on algebra C if  any isomorphism i: S^in --> S^in
has an isomorphic extension i+: S --> S with s;i+ = i;s  s.t. for any d:
S^in --> C,   F(i;d)  =  i+ ; F(d)

Now it is seen that Prod is commutative in Sets because any isomorphism i:
S^in --> S^in  has an evident required extension i+, i+(u) = u and i+ acts
on the arrows p,q, . exactly like i acts on their targets x,y,.. The
illusion of non-commutativity of products in Sets arises because of
disregarding the diagrammatic nature of the operation: isomorphism i on the
nodes in the shape acts on d but the corresponding i+ acting on the arrows
is not taken into consideration.

 A peculiarity of what was described above is that  names of items in the
shape are used in the definition of the operation. So, for example, the
following two shapes
[x] <-- weight -- [u] -- height --> [y]   and  [x] <-- red -- [u] -
blue -->[y]
though isomorphic yet different and determine the two different product
operations producing isomorphic yet different results from the same input.
Thus, what we usually call the product operation is  something like
operation schema parameterized by pairs of names. Then expressions like
Prod[weight:A, height:B] denote a concrete instance of the shape schema and,
simultaneously, a diagram d in Sets with
weight.codom.d = A and height.codom.d = B.

Now back to AxB vs. BxA. The writing AxB is normally unambiguous, it means
[1st:A] x [2nd:B] and  encodes
(i) the standard shape instance [x] <-- 1st -- [u] -- 2nd --> [y] and
(ii) the diagram d: [x]  [y] --> Sets,  x.d = A, y.d=B.

In contrast, the writing  BxA is really ambiguous: it may mean either
(a) [1st:B] x [2nd:A] that encodes the same standard shape instance
     [x] <-- 1st -- [u] -- 2nd --> [y] but now the diagram is x.d = B,
y.d=A;
or it may mean
(b) [2nd:A] x [1st:B] that encodes the non-standard shape instance
[x] <-- 2nd-- [u] -- 1st  --> [y] and again the diagram x.d = B, y.d=A.

It seems that  thinking naïvely-set-theoretically people tend to see in
writing BxA the case (a) and then, of course, BxA is not equal to AxB. And
thinking categorically, people tend to see in writing BxA the case (b) and
then BxA is the same as AxB. So, the problem 'ordered vs. unordered products
' is a  problem of poor notation for product operation like BxA  while the
operation itself is a *commutative*  diagram operation as defined above.
Actually it was already said in early postings but now it became more
transparent.

Finally, some speculation. It seems that parameterization of (shapes of )
operations by names, and use of these names in operation's definitions,  is
applicable to other standard categorical operations and can be considered in
a general setting. Probably,  it could be made precise by some kind of
internalization ...?

Zinovy Diskin