Date: Mon, 16 Jan 1995 22:13:18 -0400 (AST)
Subject: is union associative?

Date: Sat, 14 Jan 1995 15:09:39 +0000 (GMT)
From: Ilya Beylin <ilya@cs.chalmers.se>


I suspect that the following question is very natural and
a known solution has to exist, but I cannot find any reference
in books I have.

Consider the following diagram

        A
        |a
        V    b
        W <----- B

and define union of two morphisms a U b := Pushout(Pullback(X))

In the category Set this operation is obviously associative:
for any arrows a,b,c with common target,

        a U (b U c) = (a U b) U c ,

Can anybody tell,  what conditions should be imposed to a
category to guarantee this associativity?

Thank you in advance,
Ilya Beylin

Date: Fri, 20 Jan 1995 14:16:18 -0400 (AST)
Subject: Re: is union associative?

Date: Thu, 19 Jan 1995 13:47:09 -0500
From: Peter Freyd <pjf@saul.cis.upenn.edu>

Subject: Re: is union associative?

Ilya Beylin asked for which categories is union associative, where
union is defined by:
                        Consider the following diagram

                           A
                           |a
                           V    b
                           W <----- B

and define union of two morphisms a U b := Pushout(Pullback(X))

(I'm a little troubled with a non-idempotent operation being called a
union.)

It's true in any pre-topos. For a proof I would specialize to the case
that  W  is the terminator (move to the slice category if it isn't).
Let me use  @  for the binary operation on isomorphism types
characterized by the pushout diagram

             A x B
            /     \
           A       B               (add downward arrows)
            \     /
             A @ B

(where it's understood that the top half is a product diagram}. Given
a subobject  X' in  X  recall that  X/X'  is characterized by the
pushout

               X'
             /   \
            X     Spt(X')
             \   /
              X/X'

where  Spt(X'), the support of  X', is the image of  X; -> 1.  Then

  A @ B   =    (A + B)/(AxSpt(B) + Spt(A)xB)).

Beyond the pre-topos case, I have no idea of just what's needed. For a
counterexampe take the category of  Z-sets (an object is a set with an
automorphism) in which all orbits have three or fewer elements, take
A  to be a two-element orbit, B  to be a three-element orbit. Then
A @ (B @ B)  is the terminator and  (A @ B) @ B  is isomoprhic to
A + 1.  (B @ B  is the terminator and in any category  1 @ X  =  1.
A @ B  is  A + B.)