We define and develop two-level type theory (2LTT), a version of Martin-Löf type theory
which combines two different type theories. We refer to them as the 'inner'and the 'outer'type …

Homotopy type theory allows for a synthetic formulation of homotopy theory, where
arguments can be checked by computer and automatically apply in many semantic settings …

Category theory in homotopy type theory is intricate as categorical laws can only be stated"
up to homotopy", and thus require coherences. The established notion of a univalent …

We extend Homotopy Type Theory with a novel modality that is simultaneously a monad and
a comonad. Because this modality induces a non-trivial endomap on every type, it requires a …

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 …

Voevodsky's univalence axiom is often motivated as a realization of the equivalence
principle; the idea that equivalent mathematical structures satisfy the same properties …

Semi-simplicial and semi-cubical sets are commonly defined as presheaves over
respectively, the semi-simplex or semi-cube category. Homotopy Type Theory then …

On the role of semisimplicial types Page 58 On the Role of Semisimplicial Types Nicolai
Kraus University of Nottingham Abstract Constructing semisimplicial types is a well-known …

The definition of algebraic structures on arbitrary types in homotopy type theory (HoTT) has
proven elusive so far. This is due to equalities between elements of a type behaving like …

The semantics of type theory is usually done on set level. An example is our definition of the
intrinsic syntax of type theory [1] as a quotient inductiveinductive type (QIIT). Here we use a …