On Connectivity Spaces

This paper presents some basic facts about the so-called connectivity spaces. In particular, it studies the generation of connectivity structures, the existence of limits and colimits in the main categories of connectivity spaces, the closed monoidal category structure given by the so-called tensor product on integral connectivity spaces; it defines homotopy for connectivity spaces and mention briefly related difficulties; it defines smash product of pointed integral connectivity spaces and shows that this operation results in a closed monoidal category with such spaces as objects. Then, it studies finite connectivity spaces, associating a directed acyclic graph with each such space and then defining a new numerical invariant for links: the connectivity order. Finally, it mentions the not very wellknown Brunn-Debrunner-Kanenobu theorem which asserts that every finite integral connectivity space can be represented by a link.

💡 Research Summary

The paper introduces and systematically develops the theory of connectivity spaces, a categorical framework that abstracts the notion of “connectedness” beyond classical topology. A connectivity structure on a set X is a family 𝓚 ⊆ 𝒫(X) containing ∅ and X and closed under arbitrary unions; this definition generalizes the usual topological connectedness and allows one to treat a wide variety of combinatorial and geometric objects uniformly.

The first major contribution is the generation process. Given any collection 𝓐 ⊆ 𝒫(X), the authors construct the smallest connectivity structure ⟨𝓐⟩ containing 𝓐. They prove that this generation is monotone, idempotent, and preserves morphisms, showing that every connectivity space can be viewed as a free object generated by a family of subsets. This parallels the free‑algebra construction in universal algebra and provides a concrete tool for building examples.

Next, the paper studies categorical limits and colimits in the category Cnx of all connectivity spaces and in its subcategory Icnx of integral connectivity spaces (those in which every point belongs to a non‑trivial connected subset). By interpreting intersections and unions of families of subsets as limits and colimits, the authors prove that Cnx is both complete and cocomplete. In Icnx they define a tensor product ⊗: for spaces A and B, A⊗B is the set A×B equipped with the smallest connectivity structure making the projections continuous. This tensor product satisfies associativity, has a unit (the one‑point space), and admits an internal hom