categories: Inevitability of ordering products

[John Duskin <duskin@math.buffalo.edu>]

I guess that the point I was trying to make in my post was missed in 
my telegraphic submission. I'll try to make what I was trying to say 
clearer.

If you take the "representable functor" definition of a product of a 
object A with an object B, you say that an object P together with an 
ordered pair of arrows (p_A:P--->A,p_B:P-->B) is a product of A with 
B if for all objects T , the mapping Hom(T,P)--->Hom(T,A)xHom(T,B), 
defined by f |---> (p_A.f,p_B.f) is a bijection. This makes it clear 
that P together with (p_A,p_B) represents the product of A with B and 
P together with (p_B,p_A) represents the product B with A and that 
these are not representations of the same functor but that there is a 
natural isomorphism of such an object with itself which "interchanges 
the two projections". Note also that this distinction remains even 
for a product of A with itself which leads a pr_1 and pr_2 notation 
for the two "projections".

Now, as with all such bijections, one can eliminate all reference to 
sets of morphisms by a "for all x there exists a unique y such 
that..." statement  which here becomes: for all ordered pairs of 
arrows of the form (x:T--->A,y:T--->B) , there exists a unique arrow 
f:T--->P, such that (p_A.x,p_B.y)=(x,y), and the  distinction made in 
in set theory between AxB and BxA," that they are not equal, but 
there is a 'canonical' bijection between them", is perfectly 
maintained entirely within category theory... or is it? The use of 
the ordered pair term "(x,y)" still appears in the replacement first 
order statement, and if the only way that one can use "ordered pair" 
is through von Neumann's clever but rather grotesque (x,y)={x,{x,y}}, 
one has to pull in "Peano's entirely spurious singletons" , as Bill 
Lawvere referred to them.

Now if the purpose of an "underlying logic" us to "codify by making 
explicit the normal habits of reasoning that all mathematicians will 
accept" and that "an object P and arrows p_A:P--->A and p_B:P--->B" 
is  equivalent to "an object P and arrows p_B:P-->B and p_A:P-->A", 
or more starkly, the logical equivalence of,"p_A and p_B"  and "p_B 
and p_A". Then there would seem to be no way that a purely first 
order category theory could make the distinctions that we teach in 
every elementary math course about the distinction between (x,y) and 
(y,x), unless we appeal to informal aspects of everyday language 
where "and" is sometimes commutative and sometimes not, and thus in 
this case rely on "everybody's having met (x,y) long before they have 
ever heard of a 'category' ".

Apparently this subtlety has surfaced before: In the beginning of 
Bourbaki's Theorie des Ensembles,\footnote{which, remember, was 
written by "working mathematicians" who did not consider themselves 
"professional logicians or professional set-theorists", and may even 
have had some contempt for them (among others) if we are to judge 
from a certain fronts piece photograph inserted by Andre Weil into 
the Fascicule de Resultats.} they introduced as "specific signs", in 
addition to those of equality and membership, another sign of weight 
2, \couple xy, ultimately written as (x,y) together with the Axiom 
(A3) :
(for all x)(for all y)(for all x')(for all y' ) (x,y)=(x',y') implies 
x=x' and y=y'. They then define "z is an ordered pair (couple)" by 
"(there exists x)(there exists y)(z=(x,y))", which then gives (x,y) 
its first and second projections because of the way that "there 
exists " is constructed (using their "\tau operator"). The existence 
of the cartesian product of the set A with the set B  then follows, 
as usual, as the set of ordered pairs,since the presence of (x,y) 
allows them to describe formal "relations" R|x,y| as properties of 
the ordered pair  z=(x,y). Only later do they observe, in an 
exercise, that the little von Neumann nest of singletons has the 
property of the axiom A3, but they  use this only to show that 
\couple xy together with A3 is relatively consistent with their other 
axioms for set theory.

  It is clear, however, that \couple xy and A3 could have been 
introduced immediately after they had done quantification and long 
before any of the axioms for \epsilon were introduced. But, after 
all, they were  trying to use their "Theory of Sets" as a foundation 
for all of mathematics, so most people have considered this whole 
business of adding  \couple xy and A3  at so fundamental a level an 
eccentric and superfluous curiosity, and it has all but been 
completely forgotten.

  My point is not to pull category theorists back into the intricacies 
of  Bourbaki 's treatment of logic, but rather to point out that the 
idea of an ordered pair has at least once before been considered a 
notion that properly belongs somewhere anterior to set theory and can 
be used in category theory without fear of the latter suffering from 
any "set-theoretic contamination". In any case, to my eyes, the use 
of "lists and addresses" with their attendant ordering seems to be 
pretty fundamental in computer science.

Amusingly, Grothendieck "pushed" representability in the forlorn hope 
that it would convince working mathematicians that they did not have 
to give up their Cantorian Paradise of " set-theory"  in order to 
make use of the unique insights provided by "category theory", but 
then had to (re-)invent "universes" when the old paradoxes of 
set-theory and the category of all sets, all groups, etc. carpingly 
resurfaced.

Bill Lawvere, in contrast, noticed that when the working axioms of 
set-theory were rephrased in purely "category-theoretic" terms, that 
they, amazingly, all became "first order" statements , thereby 
raising the question of an entirely new way to look at foundational 
questions in which the pesky membership paradoxes could not arise nor 
even be formally expressible. He, in contrast to Grothendieck, 
"pushed" the much more radical move of, effectively, "banning all use 
of Hom-sets" and thereby  made the divide crystal clear.