When I first introduced the notions of algebraically complete categories (every endo-functor has an initial algebra) and algebraically compact categories (every endo-functor has an initial algebra naturally isomorphic to a terminal co-algebra) I thought that such things would exist in the classical foundations only in degenerate form. It was a surprise when I noticed that the categories of countable sets and of countably dimensioned vector spaces are algebraically complete. I'm now surprised by:
The category of separable Hilbert spaces and linear operators of
bound at most 1 is algebraically compact.
As in the earlier cases one doesn't really need the controlling cardinal number to be aleph-naught. To avoid using the axiom of choice one can state the more general result by taking an arbitrary Hilbert space A and defining A by:
Then:
A is algebraically compact.
The theorem holds for both the real and complex cases.
In the proof I use the surprising (to me) fact that in the categories in question every half-invertible map has a unique half-inverse. (That is, every map has at most one left-inverse and one right-inverse.) All the other examples from nature that I can think of are categories in which the only half-invertibles are invertible. Are there others?