What is the HoTT Game?
The Homotopy Type Theory (HoTT) Game is a project written by mathematicians for mathematicians interested in HoTT and no experience in proof verification, with the aim of introducing cubical agda as a tool for trying out mathematics in HoTT. You can find it here. Much of the content of the game is based on the HoTT book and lectures by Robert Harper on YouTube.
How does the game work?
The game is a guided sequence of exercises that the user can complete in cubical agda. There are currently two main parts to it:
- Fundamental Group of the Circle: This part has one goal in mind: to show that π₁(S¹) = ℤ from scratch. We use this as an excuse to introduce higher inductive types, univalence, transport, loop spaces, and homotopy levels.
- Trinitarianism: This introduces the basics of dependent type theory and later the extension to HoTT. We try to outline each new concept with logical, type theoretic, and categorical perspectives. Whilst this can be the starting point for learning HoTT, we recommend users to start with Fundamental Group of the Circle.
The format: The website contains all the information the user needs to play the game, but to follow along with the exercises, one must do so in agda installed on their computer. Unfortunately installing agda is often the most challenging part, and we are yet to find a good solutions to this. If you have any difficulties installing agda, please try to reach us with ‘issues’ on the github repository. If you have any suggestions or solutions (such as making a browser format of the game) please let us know as well.
Who are we and why did we make it?
The current development team consists of Joseph Hua (me), Ken Lee, and Bendit Chan. We are masters students in pure mathematics at Imperial College London. We were inspired by the Natural Numbers Game made by our professor Kevin Buzzard introducing undergraduates to lean, and thought making a similar game would be a nice way to review the HoTT we learnt. Moreover, we thought this would be a nice resource for people to try out HoTT in a modern, hands-on way.
If you try out the game we would also love to hear your feedback.