A topological reading of inductive and coinductive definitions in Dependent Type Theory

TL;DR

This paper explores the topological interpretation of inductive and coinductive definitions within dependent type theory, establishing new connections between type-theoretic constructs and topological concepts, formalized in Agda.

Contribution

It introduces a topological perspective on coinductive predicates in dependent type theory, complementing previous work on inductive structures, and formalizes the results in Agda.

Findings

Coinductive predicates correspond to coinductively generated positivity relations.

Inductive structures relate to proof-relevant formal covers and complete suplattices.

All proofs are formalized in the Agda proof assistant.

Abstract

In the context of dependent type theory, we show that coinductive predicates have an equivalent topological counterpart in terms of coinductively generated positivity relations, introduced by G. Sambin to represent closed subsets in point-free topology. Our work is complementary to a previous one with M.E. Maietti, where we showed that, in dependent type theory, the well-known concept of wellfounded trees has a topological equivalent counterpart in terms of proof-relevant inductively generated formal covers used to provide a predicative and constructive representation of complete suplattices. All proofs in Martin-L\"of's type theory are formalised in the Agda proof assistant.