A tensor-hom adjunction in a topos related to vector topologies and bornologies
Let \(\mathcal{M}\) be the topos of \(M\)-sets, where \(M\) is the monoid of continuous maps \(f:N\rightarrow N\) such that \(f(\infty)=\infty \), \(N=\mathbb{N}\) \(\cup \left\{ \infty \right\} \) being the one point compactification of the discrete space of natural numbers. This topos is used as a universe of presheaves, and the sheaf subtoposes \(\mathcal{C}\) (called continuous topos, and adapted from Johnstone' s topological topos) and \(\mathcal{B}\) (called bounded topos, and adapted from Lawvere' s bornological topos) are introduced and studied. A tensor-hom adjunction between certain categories of modules (over the ring of continuous reals, respectivelly bounded reals in \(\mathcal{M}\)) in \(\mathcal{C}\), respectively in \(\mathcal{B}\) is constructed. Finally, using the above construction, an adjunction between adequate categories of topological and bornological real vector spaces is obtained.
scientific article; zbMATH DE number 1538802