This Course at MIT

This Course at MIT pages provide context for how the course materials published on OCW were used at MIT. They are part of the OCW Educator initiative, which seeks to enhance the value of OCW for educators.

Course Overview

This page focuses on the course 18.S996 Category Theory for Scientists as it was taught by Dr. David Spivak in Spring 2013.

This course demonstrated how category theory can be a powerful language for understanding and formalizing common scientific models. The course employed a mathematical tool called an olog (or ontology log), a concept the instructor developed to provide an abstract means for categorizing the general properties of a system. Ologs provide a rigorous mathematical framework for knowledge representation, for the construction of scientific models, and for data storage using linguistic and graphical tools.

During the semester, the power of the tool was tested by its ability to cast taken-for-granted ideas in new light, either by exposing existing weaknesses or flaws in the instructor's and students' understanding, or by highlighting hidden commonalities across scientific fields. In this regard, the course has application for scientists in every discipline.

Course Outcomes

Category theory is especially important in algebra, topology, and algebraic geometry, where it is generally covered in courses in these subject areas, typically at the graduate level. But category theory is rarely taught on its own, at least in the US.

For students, the goal of this course was to get an idea of what category theory is, to improve their ability to see broad patterns in human reasoning, to see how category theory relates to modeling in the sciences, and to gain clarity into their own subjects in the process.

 

Curriculum Information

Prerequisites

None

Requirements Satisfied

H-Level grad credit

Offered

This was a mathematics topic course. Topic courses like this one allow students a class setting in which to study an advanced topic that is not covered in any other course in the mathematics department. Offerings are initiated by members of the mathematics faculty on an ad hoc basis.

The Classroom

  • Photo of a classroom with long tables and chairs all facing multi-level chalkboards at the front of the room.

    18.S996 was taught in a medium-sized classroom with lots of board space. The classroom seats 30 students and has projector capabilities.

 

Student Information

On average, about 18 students take this course each year.

Breakdown by Year

About 1/3 undergraduate students, 2/3 graduate students

Breakdown by Major

About 1/3 math majors, 2/3 students from other science and engineering disciplines.

 
 

How Student Time Was Spent

During an average week, students were expected to spend 12 hours on the course, roughly divided as follows:

In Class

3 hours per week
  • Three class sessions per week; 1 hour per session; 36 sessions total
  • Classes were mainly in lecture format, with some group work. Each class session covered a portion of the textbook, which was written for this course by the instructor. Questions and comments were encouraged.
  • The last 2 weeks were spent on oral presentations given by students to satisfy the final project requirement.
 

Out of Class

9 hours per week
  • Readings and assignments
  • Final project
  • Office hours: Dr. Spivak offered 2 office hours per week, but often spent an additional hour with students before homework was due. Students were highly encouraged to come to office hours.
 

Semester Breakdown

WEEK M T W Th F
1 No classes throughout MIT. No session scheduled. Lecture session. No session scheduled. No session scheduled.
2 Lecture session. No session scheduled. Lecture session; assignment due. No session scheduled. Lecture session.
3 No classes throughout MIT. Lecture session; assignment due. Lecture session. No session scheduled. Lecture session.
4 Lecture session; assignment due. No session scheduled. Lecture session. No session scheduled. Lecture session.
5 Lecture session. No session scheduled. Lecture session; assignment due. No session scheduled. Lecture session.
6 Lecture session. No session scheduled. Lecture session; assignment due. No session scheduled. Lecture session.
7 Lecture session. No session scheduled. Lecture session; assignment due. No session scheduled. Lecture session.
8 No classes throughout MIT. No classes throughout MIT. No classes throughout MIT. No classes throughout MIT. No classes throughout MIT.
9 Lecture session. No session scheduled. Lecture session; assignment due. No session scheduled. Lecture session.
10 Lecture session. No session scheduled. Lecture session. No session scheduled. Lecture session; assignment due.
11 No classes throughout MIT. No session scheduled. Lecture session. No session scheduled. No session scheduled.
12 Lecture session; assignment due. No session scheduled. No session scheduled. No session scheduled. Lecture session; assignment due.
13 Lecture session. No session scheduled. Student presentations in class; assignment due. No session scheduled. Student presentations in class.
14 Student presentations in class. No session scheduled. Student presentations in class. No session scheduled. Student presentations in class.
15 Student presentations in class; assignment due. No session scheduled. Student presentations in class. No session scheduled. No classes throughout MIT.
16 No classes throughout MIT. No classes throughout MIT. No classes throughout MIT. No classes throughout MIT; assignment due. No classes throughout MIT.
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 color used on the preceding table to indicate dates when class sessions are held. Lecture session
Displays the symbol used on the preceding table which indicates dates when assignments are due. Assignment due
Displays the color used on the preceding table to indicate dates when no class session is scheduled. No class session scheduled
Displays the color used on the preceding table to indicate dates when student presentations are held. Student presentations
 

Instructor Insights

Keep everything as grounded as possible; use fully-worked examples.

—Dr. David Spivak

Below, Dr. David Spivak shares some advice for others teaching a similar course

Keep Abstract Material Grounded

Keep everything as grounded as possible; use fully-worked examples. As an analogy, if this were a calculus class, we would not have discussed Riemann sums without considering an actual function (e.g., y=x2), step size (e.g., h=0.5), interval (e.g., [1,4]), and computation of total area. The material in this course is sufficiently abstract and new to the students that they need to see it touch the ground. I also used databases (as discussed extensively in the book) to ground the material.

Provide Good Feedback on Homework

Make sure you have a good grader. The students will learn nothing if they do not do the homework, and feedback is essential.

Using Tools

Some students loved ologs, while others thought they were a bit lame. The point of ologs is to teach people to think about functions in everyday life. Some students found operads to be most interesting and wished we could have covered them earlier. This is certainly possible, at least as motivation.