Directed univalence in simplicial homotopy type theory

TL;DR

This paper develops a directed univalence principle within simplicial homotopy type theory, constructing a universe of discrete types with non-trivial homomorphisms, enabling applications in higher category theory and programming languages.

Contribution

It introduces the first types with non-trivial homomorphisms in simplicial type theory and constructs the universe of discrete types, establishing directed univalence.

Findings

Construction of the universe of discrete types $\\mathcal{S}$ with directed univalence

Definition of categories and recovery of category theory results

Development of functorial types using the universe $\\mathcal{S}$

Abstract

Simplicial type theory extends homotopy type theory with a directed path type which internalizes the notion of a homomorphism within a type. This concept has significant applications both within mathematics -- where it allows for synthetic (higher) category theory -- and programming languages -- where it leads to a directed version of the structure identity principle. In this work, we construct the first types in simplicial type theory with non-trivial homomorphisms. We extend simplicial type theory with modalities and new reasoning principles to obtain triangulated type theory in order to construct the universe of discrete types $\mathcal{S}$. We prove that homomorphisms in this type correspond to ordinary functions of types i.e., that $\mathcal{S}$ is directed univalent. The construction of $\mathcal{S}$ is foundational for both of the aforementioned applications of simplicial type theory. We are able to define several crucial examples of categories and to recover important results from category theory. Using $\mathcal{S}$, we are also able to define various types whose usage is guaranteed to be functorial. These provide the first complete examples of the proposed directed structure identity principle.