Course Meeting Times

Lectures: 5 sessions / week, 1 hour / session

Extra discussion: 5 sessions / week, 1/2 hour / session

This course is offered during the Independent Activities Period (IAP), which is a special 4-week term at MIT that runs from the first week of January until the beginning of February. This course was offered during a 3-week period.

The instructors lead problem discussion and are available for questions right after each lecture during the extra discussion session.


There is no prerequisite. No prior knowledge of category theory is assumed; we will build up from the basics to the advanced theory over the series of lectures.

Course Description

Category theory is a relatively new branch of mathematics that has transformed much of pure math research. The technical advance is that category theory provides a framework in which to organize formal systems and by which to translate between them, allowing one to transfer knowledge from one field to another. But this same organizational framework also has many compelling examples outside of pure math. In this course, we will give seven sketches on real-world applications of category theory.


This course is based on the following textbook, with two lectures on each chapter:

Fong, B. and D. I. Spivak. An Invitation to Applied Category Theory: Seven Sketches in Compositionality. Cambridge University Press, 2019. ISBN: 9781108482295.

An online version is freely available on Cornell University’s e-Print archive site as well as on Lecture Videos and Readings section.

The instructors welcome feedback about the book. There is also a moderated online forum dedicated to discussing the course textbook.

Assignments and Exam

Students taking the course for credit will be required to complete three problem sets. There will be no exam.


Attending lecture is mandatory, unless you have prior permission from one of the instructors. Class participation is 25% of your grade. Problem sets are worth 75%.


1–2 Chapter 1: Generative Effects: Orders and Adjunctions Problem set 1 out 
3–4 Chapter 2: Resources: Monoidal Preorders and Enrichment  
5–6 Chapter 3: Databases: Categories, Functors, and (Co)Limits Problem set 1 due in session 6
Problem set 2 out
7–8 Chapter 4: Co-design: Profunctors and Monoidal Categories  
9–10 Chapter 5: Signal Flow Graphs: Props, Presentations, and Proofs Problem set 2 due in session 10
Problem set 3 out
11–12 Chapter 6: Circuits: Hypergraph Categories and Operads  
13–14 Chapter 7: Logic of Behavior: Sheaves, Toposes, Languages Problem set 3 due in session 14