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：我们在构造性与谓语性单值基础（亦称同伦类型论）中发展域论。所谓谓语性工作，是指我们不采用Voevodsky的命题分层公理。我们的工作在构造性意义下不依赖排中律或（可数）选择公理。域论研究所谓定向完全偏序集（dcpo）及其上的Scott连续映射，在编程语言语义学、高阶可计算性与拓扑学中有广泛应用。在谓语性基础中处理规模问题的常见方法是使用信息系统、抽象基或形式拓扑而非dcpo，以及使用逼近关系而非Scott连续函数。在我们的类型论方法中，我们接受dcpo可能是大型的，并通过类型宇宙来处理这一情况。先验地，人们可能认为dcpo的复杂构造会导致宇宙层级的不断攀升，从而在谓语性框架下不可实现。我们证明此类构造可在谓语性设定中完成。我们通过编程语言语义学中的应用来阐述该发展：PCF的Scott模型与无类型λ演算的Scott $D_\infty$ 模型的可靠性与计算完备性。我们还提供了连续与代数dcpo的谓语性解释，以及小基及其圆整理想完备化的相关概念。事实上，非平凡dcpo具有大型承载子是不可避免且具有谓语性设定特征的，我们在关于单值基础的构造性与谓语性局限性的补充章节中对此进行了说明。我们对单值基础中域论的叙述几乎全部实现了形式化，仅有少数例外。证明助手Agda推断宇宙层级的能力对我们的研究目的具有不可估量的价值。