We develop domain theory in constructive and predicative univalent foundations (also known as homotopy type theory). That we work predicatively means that we do not assume Voevodsky's propositional resizing axioms. Our work is constructive in the sense that we do not rely on excluded middle or the axiom of (countable) choice. Domain theory studies so-called directed complete posets (dcpos) and Scott continuous maps between them and has applications in programming language semantics, higher-type computability and topology. A common approach to deal with size issues in a predicative foundation is to work with information systems, abstract bases or formal topologies rather than dcpos, and approximable relations rather than Scott continuous functions. In our type-theoretic approach, we instead accept that dcpos may be large and work with type universes to account for this. A priori one might expect that complex constructions of dcpos result in a need for ever-increasing universes and are predicatively impossible. We show that such constructions can be carried out in a predicative setting. We illustrate the development with applications in the semantics of programming languages: the soundness and computational adequacy of the Scott model of PCF and Scott's $D_\infty$ model of the untyped $\lambda$-calculus. We also give a predicative account of continuous and algebraic dcpos, and of the related notions of a small basis and its rounded ideal completion. The fact that nontrivial dcpos have large carriers is in fact unavoidable and characteristic of our predicative setting, as we explain in a complementary chapter on the constructive and predicative limitations of univalent foundations. Our account of domain theory in univalent foundations is fully formalised with only a few minor exceptions. The ability of the proof assistant Agda to infer universe levels has been invaluable for our purposes.

翻译：我们在构造性和谓词性单值基础（也称为同伦类型论）中发展了域理论。谓词性意味着我们未采用Voevodsky的命题缩并公理。构造性则指我们既不依赖排中律，也不依赖（可数）选择公理。域理论研究所谓的定向完全偏序集（dcpo）及其上的Scott连续映射，在编程语言语义学、高阶可计算性和拓扑学中具有应用。在谓词性基础中处理大小问题的常用方法是使用信息系统、抽象基或形式拓扑而非dcpo，并使用可逼近关系而非Scott连续函数。在我们的类型论方法中，我们接受dcpo可能为大对象，并通过类型宇宙来处理这一问题。直观上，dcpo的复杂构造可能需要不断升级宇宙层级，在谓词性设置中似乎不可行。我们证明了此类构造可在谓词性框架中实现。我们通过编程语言语义学中的应用加以说明：PCF的Scott模型的可靠性与计算充分性，以及无类型λ演算的Scott $D_\infty$ 模型。我们还给出了连续与代数dcpo、以及小基与其取整理想完备化相关概念的谓词性描述。正如我们在关于单值基础构造性与谓词性局限性的补充章节中所解释的，非平凡dcpo具有大承载集的事实是不可避免的，也是我们谓词性框架的特征。我们对单值基础中域理论的阐述已完全形式化，仅有少数例外。证明助手Agda对宇宙层级的自动推断能力对我们的工作至关重要。