HoTT Summer School - August 7 to 10

Topics and Instructors

  • Synthetic homotopy theory: Egbert Rijke (University of Illinois, USA)
  • Semantics of type theory: Jonas Frey (Carnegie Mellon University, USA)
  • Cubical methods: Anders Mörtberg (Carnegie Mellon University, USA and Stockholm University, Sweden)
  • Higher topos theory: Mathieu Anel (Carnegie Mellon University, USA)
  • Formalization in Agda: Guillaume Brunerie (Stockholm University, Sweden)
  • Formalization in Coq: Kristina Sojakova (Cornell University, USA)



The summer school will take place in the Giant Eagle Auditorium, room number A51 in CMU’s Baker Hall building (see conference map).

We will start off Wednesday (August 7) morning with coffee at 8:45 AM in front of the lecture hall, the first course will begin at 9:15 AM. The following days courses will start at 9 AM.


Summer school program


Familiarity with basic ideas and concepts of homotopy type theory will be helpful for all courses. We recommend that beginners have a look at the first sections of Egbert Rijke’s course notes and/or the HoTT book.

For the formalization classes you will need to have the appropriate versions of the Agda and Coq proof assistants installed (or use an online version in case of Coq), see the course descriptions below for more details.

Course descriptions and materials

Synthetic homotopy theory – Egbert Rijke

In this introduction to synthetic homotopy theory we start by introducing the rules of dependent type theory. Using the homotopy interpretation of type theory we develop many notions that might also be familiar from homotopy theory, such as equivalences, contractible types, and the univalence axiom. Thus we will be doing homotopy theory in type theory. This is also called synthetic homotopy theory. In the second lecture we introduce the first higher inductive type: the circle. Furthermore, we show that there is a type theoretic analogue of the fundamental cover of the circle, characterizing its identity type. In the third lecture we introduce homotopy groups of types, and show how to construct the Hopf fibration. Since the circle is a 1-type, the Hopf fibration can be used to show that the homotopy groups of the 2-sphere and the 3-sphere agree from the 3rd homotopy group onwards.

Course notes.

Semantics of type theory – Jonas Frey

This course will give an overview of the categorical and homotopical semantics of dependent type theory. I will start out by presenting interpretations in locally cartesian closed categories and display map (fibration) categories first on a non-strict, informal level, and then introduce the notion of category with families after discussing the coherence problems of the non-strict approach. After sketching the soundness proof of the interpretation of type theory in categories with families, I will speak about strictification of non-strict models.

Prerequisites: Knowledge of basic category theory – in particular the concepts of adjoint functor and cartesian closed category – will be helpful. See for example Awodey’s textbook Category theory (2006).

The rules of type theory.


Cubical methods – Anders Mörtberg

Cubical methods have played an important role in the development of HoTT during the last few years. The original motivation for considering these methods was to give constructive justifications to HoTT, in particular to the univalence axiom. These developments have also led to other major advances in the field, including proof assistants with native support for computational univalence, semantics and a syntactic schema for HITs, canonicity results, and a variety of independence results. This course will give an introduction to these methods, both from a syntactic perspective in the form of cubical type theory and from a semantic perspective in the form of cubical set models.

Course notes.

Agda code.

Higher topos theory – Mathieu Anel

I will give a presentation of the notion of ∞-topos by means of the higher algebraic theory of logoi.

  • Course 1: Homotopy theory and localization of categories (homotopy, fractions, model structure, Dwyer-Kan localization)
  • Course 2: The theory of logoi (descent, universe, definitions, monadicity, free logos, ∞-topos, comparison with 1-topos)
  • Course 3: Features of logoi (quotient, classifying logoi, truncations, ∞-connected objects, modalities)

Reference: Anel, M and Joyal, A. Topo-logie


Formalization in Agda – Guillaume Brunerie

This course is an introduction to the Agda proof assistant. I will show you the [main features of Agda, how to develop proofs interactively with it, and how to use it for formalization of mathematics and of homotopy type theory. There will also be various practical exercises.

In order to try Agda for yourself while following the course (highly recommended), you will need a laptop with Agda version (or newer) and Emacs (a text editor used by Agda for editing proofs interactively). See the installation instructions here or here.


Agda code for class 1.

Formalization in Coq – Kristina Sojakova

This course will focus on using the HoTT version of the Coq proof assistant to formalize arguments from homotopy type theory. We will go over a few simpler examples to illustrate the basic techniques and use of the library. The students will then have the option to pursue more challenging exercises on their own and/or contribute to the formalization of the HoTT book.

We encourage everyone intending to do serious formalization to consider getting a Linux distribution (e.g., Ubuntu). The installation instructions for the HoTT version of Coq on Linux can be found here, and in the excerpt here.

See also the additional Coq instruction notes for Linux users.

On Windows: there is now a beta version of Coq 8.10 that includes the HoTT library and the HoTT version of CoqIDE, courtesy of Michael Soegtrop. Here is the 64-bit version, and here is the 32 bit version.

For people who just want to get through the course, there is an option to use the HoTT version of Coq from within a browser, that should work on the latest version of Chrome and/or Firefox.

Coq file for 2nd session.

Paper: Path spaces of higher inductive types in homotopy type theory.

Open problems

Here is the list of open problems from the discussion on the last day.