Quotient inductive-inductive types (QIITs) are generalized inductive types which allow sorts to be indexed over previously declared sorts, and allow usage of equality constructors. QIITs are especially useful for algebraic descriptions of type theories and constructive definitions of real, ordinal and surreal numbers. We develop new metatheory for large QIITs, large elimination, recursive equations and infinitary constructors. As in prior work, we describe QIITs using a type theory where each context represents a QIIT signature. However, in our case the theory of signatures can also describe its own signature, modulo universe sizes. We bootstrap the model theory of signatures using self-description and a Church-coded notion of signature, without using complicated raw syntax or assuming an existing internal QIIT of signatures. We give semantics to described QIITs by modeling each signature as a finitely complete CwF (category with families) of algebras. Compared to the case of finitary QIITs, we additionally need to show invariance under algebra isomorphisms in the semantics. We do this by modeling signature types as isofibrations. Finally, we show by a term model construction that every QIIT is constructible from the syntax of the theory of signatures.

中文翻译：

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大商和无限商电感-电感类型

商归纳归纳类型 (QIIT) 是广义归纳类型，它允许对先前声明的排序进行索引，并允许使用等式构造函数。QIIT 对于类型理论的代数描述和实数、序数和超现实数的构造定义特别有用。我们为大型 QIIT、大型消元、递归方程和无限构造函数开发了新的元理论。与之前的工作一样，我们使用类型理论描述 QIIT，其中每个上下文代表一个 QIIT 签名。然而，在我们的例子中，签名理论也可以描述它自己的签名，模宇宙大小。我们使用自我描述和 Church 编码的签名概念引导签名的模型理论，而不使用复杂的原始语法或假设现有的内部签名 QIIT。我们通过将每个签名建模为代数的有限完整 CwF（具有族的类别）来为描述的 QIIT 提供语义。与有限 QIIT 的情况相比，我们还需要在语义上的代数同构下显示不变性。我们通过将特征类型建模为等纤维来实现这一点。最后，我们通过术语模型构造表明，每个 QIIT 都可以根据签名理论的语法构造。