Essential constancy of directed-colimit-preserving functors on Banach and metric spaces

Determine whether every directed-colimit-preserving functor U from the category Metr of complete metric spaces of bounded diameter ≤ 1 with isometric embeddings to Set, or from the category Banr of Banach spaces with linear isometric embeddings to Set, is naturally isomorphic to a constant functor when restricted to the full subcategories consisting of non-locally compact spaces.

Background

The paper proves a strong rigidity result for Hilbert spaces: any directed-colimit-preserving functor from the category Hilbr (Hilbert spaces with linear isometric embeddings) to Set is essentially constant on infinite-dimensional spaces. This extends previous work showing non-faithfulness of such functors and emphasizes an intrinsic continuity of these structures.

Using this result, the authors show that there is no faithful directed-colimit-preserving functor from the categories Metr (complete metric spaces of diameter ≤ 1 with isometric embeddings) or Banr (Banach spaces with linear isometric embeddings) to Set, strengthening earlier results for related supercategories. However, they explicitly state that it remains unknown whether the stronger conclusion of essential constancy, analogous to the Hilbr case, holds for Metr and Banr—at least when restricted to non-locally compact spaces.