  Using dependent type theory to formalise the syntax of dependent type theory
is a very active topic of study and goes under the name of "type theory eating
itself" or "type theory in type theory." Most approaches are at least loosely
based on Dybjer's categories with families (CwF's) and come with a type CON of
contexts, a type family TY indexed over it modelling types, and so on. This
works well in versions of type theory where the principle of unique identity
proofs (UIP) holds. In homotopy type theory (HoTT) however, it is a
long-standing and frequently discussed open problem whether the type theory
"eats itself" and can serve as its own interpreter. The fundamental underlying
difficulty seems to be that categories are not suitable to capture a type
theory in the absence of UIP.
  In this paper, we develop a notion of $\infty$-categories with families
($\infty$-CwF's). The approach to higher categories used relies on the
previously suggested semi-Segal types, with a new construction of identity
substitutions that allow for both univalent and non-univalent variations. The
type-theoretic universe as well as the internalised syntax are models, although
it remains a conjecture that the latter is initial. To circumvent the known
unsolved problem of constructing semisimplicial types, the definition is
presented in two-level type theory (2LTT).
  Apart from introducing $\infty$-CwF's, the paper explains the shortcomings of
1-categories in type theory without UIP as well as the difficulties of and
approaches to internal higher-dimensional categories.

Logic in Computer Science;Category Theory;Logic