- quotient inductive-inductive types (QIIT) are Simultaneous definition of types and indexed types with built-in quotients.
- Expressive quotients are to be reduced; this is related to W Type noted.
- Higher Inductive Type (Homotopy Type Theory): QIITs are “set-level” versions of Higher Inductive Type, but higher inductive types themselves are not well understood within HoTT.
- Theory of signatures, containers, see damato2025-formalizing-containers
- “Reduction” topics: Is it possible to encode QIITs using only inductive types and quotients?
- There appears to be an argument against this, implying that quotients are not a conservative extension in this context.
- Symmetries for Coinductive Type (e.g., infinite unlabelled rooted trees) may require some version of the axiom of choice.
- Motivation: Fibrancy in HoTT (having a J Eliminator); for W-types, proving fibrancy of QIIT requires showing equality is fibrant (potentially coinductively), but this is not yet detailed in any existing paper.