Page 373 - Advanced Linear Algebra

Tensor Products 357

)
2 > is closed under composition of functions, which is an associative
operation.
)
3 For any  > and  < , the composition k is defined and belongs to
<.
S 1
f 1 W 1
A S 2 W 2
f 2 W 3
f 3 S 3

Figure 14.2

We refer to > as the measuring family and its members as measuring
functions.

A pair ²:Á ¢( ¦ :³ , where :  I and  < has the universal property for
the family as measured by , or is a universal pair for < ²Á > ³ , if for every
>
<
<
¢ ( ¦ ? in , there is a unique ¢ : ¦ ? in > for which the diagram in
Figure 14.1 commutes, that is, for which
~ k

or equivalently, any  < can be factored through . The unique measuring
function is called the mediating morphism for .

Note the requirement that the mediating morphism be unique. Universal pairs

are essentially unique, as the following describes.
Theorem 14.1 Let ²:Á ¢( ¦ :³ and ²;Á ¢( ¦ ;³ be universal pairs for
²Á<> ³. Then there is a bijective measuring function  > for which : ~ ; .
In fact, the mediating morphism of with respect to and the mediating

morphism of with respect to are isomorphisms.

Proof. With reference to Figure 14.3, there are mediating morphisms ¢: ¦ ;
and ¢; ¦ : for which
~ k
~ k
Hence,
~ ² k ³ k

~ ² k ³ k
However, referring to the third diagram in Figure 14.3, both k¢ : ¦ : and

the identity map ¢: ¦ : are mediating morphisms for and so the uniqueness