Previously we looked at several examples of -ary operations on concrete categories . In every example except two, was a representable functor and had finite coproducts, which made determining the -ary operations straightforward using the Yoneda lemma. The two examples where was not representable were commutative Banach algebras and commutative C*-algebras, and it is possible to construct many others. Without representability we can’t apply the Yoneda lemma, so it’s unclear how to determine the operations in these cases.

However, for both commutative Banach algebras and commutative C*-algebras, and in many other cases, there is a sense in which a sequence of objects approximates what the representing object of “ought” to be, except that it does not quite exist in the category itself. These objects will turn out to define a pro-object in , and when is pro-representable in the sense that it’s described by a pro-object, we’ll attempt to describe -ary operations in terms of the pro-representing object.

The machinery developed here is relevant to understanding Grothendieck’s version of Galois theory, which among other things leads to the notion of étale fundamental group; we will briefly discuss this.