Instructor Insights

Instructor Insights pages are part of the OCW Educator initiative, which seeks to enhance the value of OCW for educators.

Instructor Insights

Below, Dr. David Spivak describes various aspects of how he teaches 18.S097 Applied Category Theory.

The desire to know and understand a subject comes from the depth of ourselves, from something we don't have control over, from a sense of beauty and importance that the subject brings out in us.

— David Spivak

OCW: Why did you decide to include real-world applications in the course? What impact did this decision have on the students’ learning experience?

David Spivak: People learn best when they have a clear and personal sense of wanting to know. Unlike the desire for a good grade, the desire to know and understand a subject comes from the depth of ourselves, from something we don't have control over, from a sense of beauty and importance that the subject brings out in us. Category theory is beautiful naturally, so we felt like the most important thing for inspiring people was to emphasize its importance. 

Category theory has a history of being applied in mathematics itself. It helps structure many subdisciplines of math, and allows these subdisciplines to communicate with each other in a shared language. We wanted to show that this structuring and interoperability extends far beyond mathematics. When people see how it can be used in databases, programming languages, resource theories, quantum information, collaborative design, systems engineering, materials science, etc., they start to see the breadth of possibilities and consider how they can be part of the research effort. They start to think “categorically.”

Beyond that, applying category theory to real-life scenarios provides the students with their own ground to stand on. We all know what baking a pie is like—at least the rough idea—so if “morphism composition in symmetric monoidal categories” has something to do with pie baking, we can more easily follow the story and grasp how the mathematical particulars fit in.

OCW: What was your approach to problem-solving with students?

David Spivak: Every day after class we'd have an hour-long “fun time.” Students would come up and ask questions, talk about insights they had, etc., both with us (the instructors) and with other students. Usually around 20 students would stay after for a little while, and by the end of the fun time there would still be around 5–10 people.

The homework problems we gave were of three sorts. The first consisted of questions that tested students’ grasp of basic concepts, like “How many functors are there from the two-element linear order to itself?” (There are three!) Questions of this kind don't take creativity, they just require that you understand the definitions and work them through. Answering these simple questions gives students a grounded feeling and helps ensure that everyone is on track.

The second sort of homework problem involved interesting applications of ideas. For example, we asked a question about Gricean pragmatics, in which we challenged the students to show how pragmatic speaking and listening—where one gets the most information out of each utterance—can be described using adjunctions. 

The third type of problem was more open-ended, asking students to tell us about where a mathematical concept might arise in their own line of work, their own major or subject of interest. This was often interesting to us, though perhaps not quite as much as we would have hoped. 

OCW: What advice do you have for other educators facilitating a similar course or learning experience, especially one that is condensed in format, as 18.S097 Applied Category Theory was?

David Spivak: We covered the whole book in 14 hours; it would have been better with 21 or 28. Having the extra time in the same room after class was probably a big help, and it precluded the need for office hours. Of course, that was time we could have spent doing other things like research, but sometimes good students become research assistants, and it can be more than worthwhile.

 

Curriculum Information

Prerequisites

No prerequisites; instructor permission required

Requirements Satisfied

Unrestricted elective credits

Offered

Most years during IAP

The Classroom

  • A large classroom with tiered rows of tablet desks, and large windows along one side wall.

    Lecture

    Classes were held in a lecture hall equipped with about 80 tablet desks, sliding blackboards, and an A/V system.

 

Assessment

Grade Breakdown

The students' grades were based on the following activities:

The color used on the preceding chart which represents the percentage of the total grade contributed by problem sets. 75% Problem sets
The color used on the preceding chart which represents the percentage of the total grade contributed by class participation. 25% Class participation

Student Information

35 students took this course when it was taught in IAP 2019.

Breakdown by Year

Approximately 1/6 first-years, 1/3 sophomores, 1/6 juniors, 1/6 seniors, and 1/6 graduate students

Breakdown by Major

Approximately 50% math majors, 20% EECS majors, 30% other majors

 

How Student Time Was Spent

During an average week, students were expected to spend 17.5 hours on the course, roughly divided as follows:

Lecture

5 hours per week
  • Met 5 times per week for 1 hour per session; 14 sessions total; mandatory attendance.
  • Sessions included real-world applications.
 

Extra Discussion

2.5 hours per week
  • Met after lecture sessions for non-mandatory problem solving.
 

Out of Class

10 hours per week
  • Students were assigned one problem set each week to be completed outside of class.
 

Semester Breakdown

WEEK M T W Th F
1 Lecture session scheduled. Lecture session scheduled. Lecture session scheduled. Lecture session scheduled. Lecture session scheduled.
2 No classes throughout MIT. Lecture session scheduled and a problem set due. Lecture session scheduled. Lecture session scheduled. Lecture session scheduled.
3 Lecture session scheduled and a problem set due. Lecture session scheduled. Lecture session scheduled. Lecture session scheduled. Lecture session scheduled and a problem set due.
Displays the color and pattern used on the preceding table to indicate dates when classes are not held at MIT. No classes throughout MIT
Displays the symbol used on the preceding table to indicate dates when problem sets are due. Problem set due
Displays the color used on the preceding table to indicate dates when lecture sessions are held. Lecture session