At the moment, this is a “place holder” page. I want to learn about the appearance of algebraic theories in functional analysis and shall record what I learn here. A preliminary outline is to find out about the following statements:
We define a functor sending a Banach space to its unit ball. Since linear short maps take the unit ball of into the unit ball of , this is well-defined.
There is a functor in the opposite direction which assigns to a set the “free” Banach space on that set. That is, it assigns to a set the Banach space of all absolutely summable sequences indexed by elements of . It is a standard result that such a sequence must have countable support, no matter how large is.
is left adjoint to .
We need to define the adjunction natural transformations: and . The first is the map which assigns to the sequence which is at and elsewhere. The second is the summation map which assigns to an absolutely summable sequence indexed by its sum, .
This adjunction defines a monad over . Let us spell out the details. The functor sends a set to the unit ball of . That is, an element of is a weighted (formal) sum of elements of , , such that . The unit for the monad sends an element to the delta sequences in . The product, , takes a “sum of sums” and evaluates them. That is, given a formal sum where each is of the form , where .
AS: I think! I need to check exactly how the product works in this example but I’m just getting the basic sketch down first.
The key question is whether or not is (equivalent to) the category of algebras for this monad. That is, is tripleable? If not (as it will turn out), how close is it?
Beck's tripleability theorem gives three conditions for a functor to be tripleable. We already have one (the adjunction), let us show that the second also holds.
Let be a linear short map which induces an isomorphism on the unit balls of and . It is evident that it is therefore a bijection from the underlying set of to that of . Hence, by the open mapping theorem, it is a linear homeomorphism. It remains to show that (so that its inverse is a short map as well). This is simple to show: if we had some with but (if it fails, it must fail that way as is short) then there would be some such that . As is surjective, there is some such that . But as , so , contradicting the injectivity of . (Incidentally, this argument is valid constructively; it is a property of located real numbers that any number that is neither greater nor smaller than must equal .)
AS: To be continued …
The above is essentially my “notes” on reading the following (and whatever necessary to understand the following):
Section 4.4 of Toposes, Triples, and Theories by Barr and Wells