Two-level type theory and applications

We define and develop two-level type theory (2LTT), a version of Martin-L\"of which combines two different theories. refer to them as the inner outer theory. In our case interest, is homotopy (HoTT) may include univalent universes higher inductive types. The traditional form validating uniqueness identity proofs (UIP). One point view on it internalised meta-theory There are motivations for 2LTT. Firstly, there certain results about HoTT meta-theoretic nature, such statement that semisimplicial types up level $n$ can be constructed in any externally fixed natural number $n$. Such cannot expressed itself, but they formalised proved 2LTT, where will variable This inspired by observations conservativity presheaf models. Secondly, 2LTT framework suitable formulating additional axioms one might want add HoTT. idea heavily Voevodsky's Homotopy Type System (HTS), constitutes specific instance HTS has an axiom ensuring numbers behaves like external numbers, allows construction universe this stated simply asking isomorphic. After defining we set collection tools with goal making convenient language future developments. As first application, Reedy fibrant diagrams style Shulman. Continuing line thought, suggest definition (infinity,1)-category give some examples.