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的单值基础（也称为同伦类型论）中发展域论。谓词性工作意味着我们不采用沃耶沃茨基的命题重述公理。我们的工作是构造性的，即不依赖排中律或（可数）选择公理。域论研究所谓的定向完全偏序集（dcpo）及其上的斯科特连续映射，并在程序语言语义学、高阶可计算性和拓扑学中有应用。在谓词性基础中处理大小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斯科特连续函数。在我们的类型论方法中，我们则接受dcpo可能是大的，并通过类型宇宙来处理这一问题。先验来看，dcpo的复杂构造可能导致宇宙层级不断上升，且可能在谓词性设定下无法实现。但我们证明这类构造可以在谓词性环境中完成。我们通过编程语言语义学的应用来阐述该发展：PCF的斯科特模型的可靠性和计算充分性，以及无类型λ演算的斯科特$D_\infty$模型。我们还给出了连续性和代数性dcpo的谓词性描述，以及小基及其圆整理想完备化等相关概念。事实上，非平凡dcpo具有大承载者这一特性在谓词性设定中是不可避免且具有特征性的，我们在关于单值基础的构造性和谓词性局限的补充章节中对此进行了阐述。我们在单值基础中对域论的阐述几乎完全形式化，仅有少数例外。证明助手Agda自动推断宇宙层级的能力对我们的工作起到了不可或缺的作用。