Following a comment of user131781, posted to an answer of this question on MO, I am looking for references to the construction of (co)-free functors from categories into the category of Banach spaces and short maps (which I denote by Ban-1).
... the (originally algebraic) concept of a free object was transferred in a natural way to a plethora of topological versions—-free topological groups, free locally convex spaces ... over a topological space (metric space, uniform space, differentiable manifold, complex manifold) around the middle of the last century. The underlying principle was always the same. You embed your space into a suitable TVS by providing the free vector space with a corresponding structure, then taking the completion which then has a characteristic universal property...
There are countless examples of this ( topological spaces into spaces of measures, metric spaces into free Banach spaces, uniform spaces into uniform measures manifolds into distributions, complex manifolds into analytic functionals,...) and the principle is always the same—-you basically construct an adjoint to a forgetful functor. Of course, you don‘t have to do it or think of it in this way—-most people probably wouldn‘t. However, a small minority is of the opinion that so doing sheds some light on the subject.
Ideally, the source category should itself admit a forgetful functor into Top, the category of topological spaces and continuous functions. However, I'm interested in all such results.
Note:
I would like to point out, that I have never come across a co-free functor with sink as Ban-1. So such a result is also of particular interest to me.
Thanks in advance.
