EECS 598 005: Category Theory for Computer Scientists

Category theory is a mathematical theory of structures and transformations. While originally developed for applications to pure mathematics, category theory has become a useful tool for studying logic and programming languages.

The breadth of applications and high level of abstraction can make category theory difficult to approach for those without an extensive math background. This course aims to introduce and motivate category theory by focusing on applications in computer science, in particular to logic and programming languages.

Students are expected to have familiarity with basic notions of programming language theory such as abstract syntax, type systems and some form of formal semantics (operational, axiomatic or denotational). Students with a strong math background but minimal programming language background are also welcome to enroll but the focus of the class will be on these computer science applications.

Learning Objectives

After completing this course, students should be able to

Degree Requirements

To see what degree requirements this course satisfies, see the spreadsheet on this page.

Evaluation

There are four components of your grade:

  1. Problem Sets (70%)
  2. In-class presentation of solutions (10%): we will set aside class time to review the solutions to problem sets
  3. Scribing (10%): students will contribute notes written in LaTeX of the in-class lectures to share with the entire class.
  4. Participation (10%): attendance and participation in lecture.

Problem Sets

Problem Sets will be posted below in the schedule with their LaTeX source in the gitlab repo. You can complete them in groups of 2 or 3. You will submit them on Canvas as a pdf. You may include hand-drawn diagrams, but text and equations should be typeset in LaTeX.

Problem sets will be always be due at 11:59pm on the listed due date. Late work will not be accepted.

Students will present solutions to problem sets at the end of certain lecture periods. Signup for this on the gitlab repo.

Scribing

This semester will use a new curriculum without following a single textbook. To help everyone in the class, each lecture will have an assigned scribe who will take notes on the lecture and post a LaTeX copy in the gitlab repo.

Attendance

Participating in this course means attending lecture in person. Lectures will be recorded but this is not a substitute for attendance. Each class will have a sign-in sheet.

Course Schedule

The following schedule of topics and homeworks is tentative and will be updated throughout the semester.

We will have two weeks of no lecture during the semester: the week of MLK day and spring break. There will not be a homework due after spring break but there will be after MLK week (since only one day is an academic holiday).

Readings are chosen to complement the lecture. All readings will be from freely available online sources.

Meeting Date Topic Readings HW Scribe
Wed, Jan 04 Course overview, Propositional Logic Frank Pfenning notes Max S. New
Mon, Jan 09 Models of Propositional Logic Crole Ch 1.1-1.4 PS1 Released Alan Yang
Wed, Jan 11 Soundness, Completeness, Initiality of Heyting Algebra Semantics Zhemin Qu
Mon, Jan 16 NO CLASS - MLK Day
Wed, Jan 18 NO CLASS - POPL
Mon, Jan 23 Simple Type Theory: Syntax and Axiomatic Semantics Crole Ch 4.1-4.3 PS1 Due, PS2 Released Yanjun Chen
Wed, Jan 25 Signatures for STT, Set-theoretic Semantics Crole Ch 3
Mon, Jan 30 Categories, Functors
Wed, Feb 01 Universal Mapping Properties and Yoneda’s Lemma PS2 Due, PS3 Released
Mon, Feb 06 Initiality of STT in Cartesian Closed Categories
Wed, Feb 08 Disjunction Property for STT
Mon, Feb 13 Inductive Datatypes, Axiomatics PS3 Due, PS4 Released
Wed, Feb 15 Inductive Datatypes, Models
Mon, Feb 20 Evaluation Order and Computational Effects
Wed, Feb 22 Concrete Models of Effects, Adjoint Functors PS4 Due
Mon, Feb 27 NO CLASS - SPRING BREAK
Wed, Mar 01 NO CLASS - SPRING BREAK
Mon, Mar 06 Monads and Algebras PS5 Released
Wed, Mar 08 Call-by-push-value
Mon, Mar 13 Recursive programs and datatypes, fixed points
Wed, Mar 15 Basic Domain Theory PS5 Due, PS6 Released
Mon, Mar 20 Solving Domain Equations
Wed, Mar 22 Guarded Domain Theory
Mon, Mar 27 Intuitionistic Linear Logic, Monoidal Categories PS6 Due, PS7 Released
Wed, Mar 29 Models of ILL
Mon, Apr 03 Bunched Implications/Separation Logic
Wed, Apr 05 Dependent Type Theory Ps7 Due, PS8 Released
Mon, Apr 10 Fibrations, Categories with Families
Wed, Apr 12 Presheaf Models PS8 Due
Mon, Apr 17 Subobject Classifiers, Toposes