Upper and lower limits of scales and the interchange of quantifiers
We define lower and upper limits for a family of structured subobjects of a fixed ambient object, as the direct limit of the meets over tails and the inverse limit of the joins over tails, and construct the canonical comparison morphism between them. The family need not be a diagram: its members are related only through the ambient object. We distinguish two senses in which the comparison may be invertible, before and after forgetting the structure, and prove that they coincide exactly when the forgetful functor reflects isomorphisms; they therefore agree for sets, for vector spaces and for Banach spaces, and separate for locally convex spaces. For countable scales of Banach spaces the separation cannot occur when the upper limit is ultrabornological, so under those hypotheses the obstruction is localised in the topology of that limit. Intermediate conditions correspond to factorisations of the forgetful functor, of which bornological convergence is one. In the cases that matter the comparison is an interchange of quantifiers: membership in the lower limit means that one value of the second index serves every value of the first, and membership in the upper limit that every value of the first is served by some value of the second. We exhibit this for spaces of test functions, for weighted Sobolev spaces, for Laurent series and for the adeles, and discuss three further conditions that suggest themselves.