CSCI 8980 Higher-Dimensional Type Theory

This is a graduate seminar course on the recent development of higher-dimensional type theory. Type theory serves as an alternative foundation to set theory, with attention to construction. The study of higher-dimensional types was motivated by homotopy theory, a branch of mathematics on topological spaces up to continuous deformation, but has become a thing on its own. In this semester, we will start with Martin-Löf (dependent) type theory, introduce extensions such as univalence and higher inductive types, and eventually cubical type theory, a radical rethink about these extensions. We will also use Agda to assist the learning.

Announcement: important changes in response to COVID-19.


Tentative Schedule

Please review important changes in response to COVID-19.

Topic References Homework/Survey Notes/Code/Slides
Jan 21 Introduction
HoTT 1
Preclass Survey Lecture Notes
23 [Agda] Double-Negation Translation Install Agda Agda Code
28 Principle of Harmony I HoTT 1
PMLTT 2-7,9-12
Preclass Survey due Lecture Notes
30 Principle of Harmony II
[Agda] Agda Pre-basics
Homework 1
Re: Homework 1
Notes by Mahrud
Agda Code
Feb 4 Types with Dependency HoTT 1
PMLTT 4-13
Lecture Notes
6 [Agda] Basics Agda Agda Code
11 Identification I HoTT 1.12,2 Lecture Notes
13 [Agda] Groupoid Laws and Homotopy and Truncation Levels Agda
HoTT 2
Homework 1 due
Homework 2
Agda Code
18 Identification II
HoTT 2.4,2.9,4 Notes by Mahrud
20 [Agda] Paths in Various Types Agda Agda Code
25 Universe & Univalence Lecture Notes
27 [Agda] Paths in Various Types Self Practice Agda
HoTT 7.1 (for Homework 3)
Homework 2 due
Homework 3
Mar 3 Inductive Types HoTT 5 Notes by Calvin & Devon
5 [Agda] Equational Reasoning HoTT 5 3-2-1 Agda Code
Extra Practice for J
10 No Class (Spring break)
12 No Class (Spring break) URGENT: Technology Check-In
17 No Class (Spring break)
Zoom Test Session
19 Inductive Types HoTT 5,6 Homework 3 due
Homework 4
Temporary Slides and Temporary Code
Notes by Calvin & Devon
24 Introduction to Cubical Type Theory (Part I) Slides
Notes by Daniel
26 [Agda] Cubes and Paths in Sigma Types Agda: Library Management
GitHub: agda/cubical
3-2-1 Agda Code
29 Mini-Lecture: Truncation Levels Slides
Apr 1 Composition Structure of Types Slides
2 [Agda] Connections and Partial Elements Homework 4 due
Homework 5
Agda Code
7 Composition Structure of Various Types Slides
Notes by Nathan & Nori
9 [Agda] Groupoid Laws 3-2-1 Agda Code
14 Universe & Univalence, reprised Slides
Notes by Dawn & Jack
16 [Agda] Transport and Heterogeneous Composition Homework 5 due
Homework 6
Agda Code
22 Normalization Slides
Notes by Bowen & Zhuyang
23 [Agda] Univalence 3-2-1 Agda Code
30 [Agda] Homework 6 due
Homework 7
Agda Code
May 2 More on Higher Inductive Types Slides
4 Categorical Aspects of Type Theory Slides
12 No Class (Final exam week) Homework 7 due (12 days)


At the end of the semester, you will have learned cubical type theory (updated).

Install Agda

You need the development version of Agda.


(Updated on 3/13 in response to coronavirus.)


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