  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.
