We propose foundations for a synthetic theory of $(\infty, 1) $-categories within homotopy
type theory. We axiomatize a directed interval type, then define higher simplices from it and …

We report on the development of the HoTT library, a formalization of homotopy type theory in
the Coq proof assistant. It formalizes most of basic homotopy type theory, including …

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 …

This is a short overview of an experimental library of Mathematics formalized in the Coq
proof assistant using the univalent interpretation of the underlying type theory of Coq. I …

We introduce and develop the notion of displayed categories. A displayed category over a
category C is equivalent to 'a category D and functor F: D→ C', but instead of having a single …

The generality and pervasiveness of category theory in modern mathematics makes it a
frequent and useful target of formalization. It is however quite challenging to formalize, for a …

In homotopy type theory (HoTT), all constructions are necessarily stable under homotopy
equivalence. This has shortcomings: for example, it is believed that it is impossible to define …

We offer an introduction for mathematicians to the univalent foundations of Vladimir
Voevodsky, aiming to explain how he chose to encode mathematics in type theory and how …

Homotopy type theory is a new branch of mathematics, based on a recently discovered
connection between homotopy theory and type theory, which brings new ideas into the very …

We develop bicategory theory in univalent foundations. Guided by the notion of univalence
for (1-) categories studied by Ahrens, Kapulkin, and Shulman, we define and study univalent …