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.

翻译：单纯形类型论通过引入有向路径类型扩展了同伦类型论，该类型将同态概念内化于类型之中。这一概念在数学领域——使得综合（高阶）范畴论成为可能——以及编程语言领域——催生了结构恒等原理的有向版本——都具有重要应用。在本研究中，我们首次在单纯形类型论中构造了具有非平凡同态的类型。我们通过模态算子与新的推理原理扩展单纯形类型论，从而建立三角化类型论以构造离散类型宇宙 $\mathcal{S}$。我们证明该类型中的同态对应于类型的普通函数，即 $\mathcal{S}$ 具有有向单值性。$\mathcal{S}$ 的构造对前述单纯形类型论的两类应用均具有基础性意义。我们得以定义若干关键范畴实例，并复现范畴论中的重要结论。借助 $\mathcal{S}$，我们还能定义多种保证具有函子性使用的类型。这些成果为所提出的有向结构恒等原理提供了首批完整范例。