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）中发展域理论。谓词性工作意味着我们不假设Voevodsky的命题缩并公理。我们的工作具有构造性，即不依赖排中律或（可数）选择公理。域理论研究所谓的定向完备偏序集（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有应用。在谓词性基础上处理规模问题的常见方法是使用信息系统、抽象基底或形式拓扑而非dcpo，以及使用可近似关系而非Scott连续函数。在我们的类型论方法中，我们转而接受dcpo可能是大规模的，并通过类型宇宙来处理这一问题。先验地，人们可能预期dcpo的复杂构造会导致宇宙层级不断上升，从而在谓词性框架中不可行。我们证明此类构造可以在谓词性设定中实现。我们通过编程语言语义学中的应用实例说明这一发展：PCF的Scott模型的无矛盾性与计算完备性，以及无类型λ演算的Scott $D_\infty$模型。此外，我们还给出连续与代数dcpo的谓词性描述，以及小基底的圆整理想完备化相关概念。非平凡dcpo具有大规模承载元这一事实在谓词性设定中不可避免且具有特征性，我们将在关于单值基础构造性与谓词性局限性的补充章节中加以阐释。我们对单值基础中域理论的阐述已完全形式化，仅存少数例外。证明助手Agda自动推断宇宙层级的能力对我们的目标具有不可估量的价值。