Categorical anatomy of closed graph and open mapping theorems

Recent development of the theory of general topological vector spaces (without local convexity, hence without duality theory), as in [1], has exposed more of the essential (distinctly topological) nature of a good deal of the locally convex theory, resulting in a remarkably high survival rate of theorems, with (it is claimed) greater simplicity and elegance of proofs. This is especially true of the closed graph theorem. Associated concepts, such as barrelledness and the various kinds of completeness, can be described and related in a manner which demands extension to uniform spaces and beyond. In the setting of categories and functors, we believe these ideas acquire an elegance and unity which considerably widens their scope while shedding light on their homeground in functional analysis. The aim of this paper, which takes a much broader perspective than Pelletier [10], is to distill some of this categorical essence, and to point out new problems, both topological and categorical, which present themselves along the way.