Transport of finiteness structures and applications

We describe a general construction of finiteness spaces which subsumes the interpretations of all positive connectors of linear logic. We then show how to apply this construction to prove the existence of least fixpoints for particular functors in the category of finiteness spaces: these include the functors involved in a relational interpretation of lazy recursive algebraic datatypes along the lines of the coherence semantics of system T.

💡 Research Summary

The paper “Transport of finiteness structures and applications” develops a unified construction of finiteness spaces that subsumes the relational interpretations of all positive connectives of linear logic. The authors begin by recalling the notion of a finiteness structure: a pair (X, 𝔽) where X is a set and 𝔽 ⊆ ℘(X) is a family of subsets closed under non‑emptiness, downward inclusion, and double‑complement. Such structures capture the resource‑sensitivity inherent in linear logic.

The central technical contribution is the definition of a transport operation. Given two finiteness structures (X, 𝔽) and (Y, 𝔾) and a binary relation R ⊆ X × Y, R is called transportable when both forward and backward images preserve finiteness: for every A ∈ 𝔽, the image R