Bornological LB-spaces and idempotent adjunctions
Jack Kelly, Lenny Neyt, Sven-Ake Wegner
TL;DR
The paper compares two concepts of bornological LB-spaces, $\text{bLB}$ and $\text{LBb}$, and analyzes their relationship through the idempotent adjunction between bornologification $ (\cdot)^{\operatorname{b}} $ and topologification $ (\cdot)^{\operatorname{t}} $. It introduces norming and Banach disks, the ultrabornologification $ (E,\mathcal{B})^{\operatorname{ub}} $, and the notion of pre-Banach disks to give a precise description of when a bornological space arises as a bornologification of a Hausdorff LB-space. The main results establish NBorn $\simeq$ Pre, characterize the range of $ (\cdot)^{\operatorname{b}}$ on $\mathbf{LB}^{\text{Haus}}$ via two conditions, and show $ \bfbLB = \textbf{LBb} \cap \mathbf{NBorn}$; LS$_w$-spaces and nuclear LB-spaces provide concrete examples. The paper also derives abstract consequences of idempotent adjunctions, proving subobject stability for $\bfbLB$ and extending the framework to bornological modules over Banach rings, with potential applications to condensed mathematics and derived analytic geometry.
Abstract
The notion of an LB-space was introduced by Grothendieck in his 1953 thèse, referring to a countable colimit of Banach spaces taken within the category of locally convex topological vector spaces, and refining prior work done by Dieudonné, Schwartz and Köthe. Recently, two different notions of `bornological LB-spaces' emerged: one, given by Stempfhuber, refers to countable colimits of Banach spaces as well, but now taken in the category of bornological vector spaces. The other one, given by Bambozzi, Ben-Bassat and Kremnizer, refers to bornologifications of regular LB-spaces, i.e., of LB-spaces in the Grothendieck sense having the additional property that every bounded subset of the colimit is contained and bounded in one of its Banach steps. In this note, we show that the two notions are distinct, but nevertheless closely related. This involves, in particular, an intimate study of the idempotent adjunction of the bornologification and topologification functors.