Metatheorems about type theories are often proven by interpreting the syntax into models
constructed using categorical gluing. We propose to use only sconing (gluing along a global …

Programming languages can be defined from the concrete to the abstract by abstract syntax
trees, well-scoped syntax, well-typed (intrinsic) syntax, algebraic syntax (well-typed syntax …

One idiosyncratic framing of type theory is as the study of operations invariant under
substitution. Modal type theory, by contrast, concerns the controlled integration of operations …

We develop algebraic models of simple type theories, laying out a framework that extends
universal algebra to incorporate both algebraic sorting and variable binding. Examples of …

We develop the usage of certain type theories as specification languages for algebraic
theories and inductive types. We observe that the expressive power of dependent type …

It is well-known that extensional lambda calculus is equivalent to extensional combinatory
logic. In this paper we describe a formalisation of this fact in Cubical Agda. The …

In this book, we aim to introduce the reader to a modern research perspective on the design
of “full-spectrum” dependent type theories. After studying this book, readers should be …

We study the coherence and conservativity of extensions of dependent type theories by
additional strict equalities. By considering notions of congruences and quotients of models …

We present a new strictification method for type-theoretic structures that are only weakly
stable under substitution. Given weakly stable structures over some model of type theory, we …

We study the conservativity of extensions by additional strict equalities of dependent type
theories (and more general second-order generalized algebraic theories). The …