Much as a topological structure on a set is a notion of which subsets are ‘open’, so a bornological structure, or bornology, on a set is a notion of which subsets are ‘bounded’.
So far, we only discuss bornological topological vector spaces. See bornological set for the general notion of bornological space.
However, we can tell that bornological spaces and certain morphisms between them form a category $Born$.
Every inductive limit of Banach spaces is bornological. (Alpay-Salomon 13, prop. 2.3)
Conversely, every bornological space is an inductive limit of normed spaces, and of Banach spaces if it is quasi-complete (Schaefer-Wolff 99).
Wikipedia about bornological spaces
Jiri Adamek, Horst Herrlich, and George Strecker, Abstract and concrete categories: the joy of cats. free online
Daniel Alpay, Guy Salomon, On algebras which are inductive limits of Banach spaces (arXiv:1302.3372)
H. H. Schaefer with M. P. Wolff, Topological vector spaces, Springer 1999