Welcome to the course Duality Theory: Connecting Logic, Algebra, and Topology! It is given during the winter semester 2025/6 at LMU Munich as part of the Master in Logic and Philosophy of Science. (Past editions: winter 2023/4, winter 2024/5.)
Motivation
This course is an introduction to duality theory, which is an exciting area of logic and neighboring subjects like math and computer science. The fundamental theorem is Stone’s duality theorem stating that certain algebras (Boolean algebras) are in a precise sense equivalent to certain topological spaces (totally disconnected compact Hausdorff spaces). The underlying idea is that the two seemingly different perspectives—the algebraic one and the spatial one—are really two sides of the same coin:
- formulas/propositions vs. models/possible worlds,
- open sets of a space vs. points of the space,
- properties of a computational process vs. denotation of the computational process.
In terms of content, the focus of the course will be to introduce the mathematical theory, after a philosophical motivation. In terms of skills, the aim is to learn how to apply the tools of duality theory. We will illustrate this with applications—especially to philosophical phenomena—that make use of dualities by combining the often opposing advantages of the two perspectives.
General information
We have one session per week during the semester:
- The lecture given by me, Levin Hornischer, on Thursdays from 10:00 to 12:00 in room 028 (Ludwigstr. 31). Note the sharp (aka “s.t.”) times!
Lecture Notes
The lecture notes are written as the course progresses. You find the latest edition here: duality.pdf.
Formalities
All the organizational details for the course are described in this file: formalities.pdf.
Schedule
The schedule below describes in which week we will cover which material in the lecture, and it recommends exercises. The future part of the schdule is preliminary and will be updated according to what we actually have done in the lecture.
| Week | Date | Chapter | Lecture | Exercise |
|---|---|---|---|---|
| 1 | 16.10.2025 | 1 | Intro to the course and 1.1.1. | Intro to partial orders (ex 2.a) |
| 2 | 23.10.2025 | 1 | Finishing sec. 1.1 | More on partial orders (ex 2.b) |
| 3 | 30.10.2025 | 1-2 | Finishing ch. 1 and until beginning of 2.1.2 | Ex 2.c & 2.d |
| 4 | 06.11.2025 | - | cancelled | Bonus session on order-adjunctions: ex. 2.e & 2.f |
| 5 | 13.11.2025 | 2 | Finishing ch. 2 | Ex. 2.g-i |
| 6 | 20.11.2025 | 3 | Sec. 3.1.1 | Ex. 2.a-b |
| 7 | 27.11.2025 | 3 | Until sec. 3.2.1 | Ex. 3.c-e |
| 8 | 04.12.2025 | 3 | Almost finishing sec. 3.2.3 | Ex. 3.f-g |
| 9 | 11.12.2025 | 3 | Finishing ch. 3 | Ex. 3.h-j (no tutorial in this week) |
| 10 | 18.12.2025 | 4 | Sec. 4.1 and until 4.2.2 | Ex. 4.a-c |
| 11 | 08.01.2026 | 4 | Finishing sec. 4.2 | 4.d-e (we did 4.d in the lecture) |
| 12 | 15.01.2026 | 4 | Sec. 4.3.1-2 | 4.f and,if interested, attempting the proofs in sec. 4.3.3-5 yourself |
| 13 | 22.01.2026 | 5 | Sec. 5.1 and sketch of 5.2 | Ex. 5.b |
| 14 | 29.01.2026 | 6 | Generalizations | Ex. 6.a-b |
| 15 | 05.02.2026 | - | Exam | - |
Assessment
There will be a written exam in the last session, which determines your final grade. I plan to provide a practice exam further into the course.
(In the previous year, the assessment was via an essay. Should you want to take a look at the potential essay topics back then, you can go here.)