Instructor Insights

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


No prerequisites; instructor permission required

Requirements Satisfied

Unrestricted elective credits


Most years during IAP


Grade Breakdown

The students’ grades were based on the following activities:

  • 75% Problem sets
  • 25% Class participation

Student Information


35 students

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:


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

Extra Discussion

  • Met after lecture sessions for non-mandatory problem solving.

Course Info

Learning Resource Types

theaters Lecture Videos
assignment Problem Sets
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