Tuesdays 11.00-12.45
BBG 005
In category theory, various kinds of algebraic theories are used to encode and study algebraic objects. The most famous of these are Lawvere theories, which can be used to encode groups, rings, associative algebras, etc. With such techniques one can do universal algebra: in particular, they are a central tool in category theory used to identify nice properties that categories of such algebras have. These days, more complicated variants are important for their tight connection with functional programming languages.
Mastermath Category theory or equivalent.
The primary text we will be following is Algebraic theories (Adámek, Rosický, and Vitale). This will cover Lawvere theories and their many-sorted generalization. After reading (the core of) the primary text, we will cover more advanced topics. These will depend on the interests of the participants; the following is a preliminary list of possible topics and associated readings.
The seminar is aimed at master students interested in the Logic and Foundations of Computing track. As there are few students, we will opt for a hybrid form: seminar/reading group. Every meeting, one of the participants will be responsible for spotting difficulties in the text, gaps in proofs and so on, and for fixing these. This participant (the “speaker”) may also be called to present stuffing at the blackboard. Moreover, each speaker devises 1 homework exercise, which is solved by the other students, graded by the speaker. Simultaneously with handing out the homework, the team hands a “model solution” to the teacher. Students are encouraged to give feedback to each other.
After completion of the course, the student is able to:
| presentations | homework | |
|---|---|---|
| understanding the material | 20% | 50% |
| effective communication of the material | 20% | 0 |
| formulating and correcting homework | 0 | 10% |
For “understanding the material (presentations)”, “effective communication of the material”, and “formulating and correcting homework” we will use the rubric here. Note that for “formulating and correcting homework”, the default will be 10/10 as long as the homework is graded in a timely manner, and the students work with the instructor to correct any problems that arise. For “understanding the material (homework)”, points for each part of each homework problem will be clearly indicated by the speaker(s).
presentations = presentations and following discussion homework = combined output for homework assignments
| Date | Subject | Speaker |
|---|---|---|
| 11 February | 1. Algebraic theories and algebraic categories | Sara |
| 2. Sifted and filtered colimits | ||
| 18 February | 3. Reflexive coequalizers | Harm & Rob |
| 4. Algebraic categories and free completions | ||
| 4 March | 5. Properties of algebras | Sara & Edward |
| 6. A characterization of algebraic categories | ||
| 18 March | 7. From filtered to sifted | |
| 8. Canonical theories | ||
| 25 March | 9. Algebraic functors | |
| 10. Birkhoff’s Variety Theorem | ||
| 1 April | 11. One-sorted algebraic theories | |
| 12. Algebras for an endofunctor | ||
| 8 April | 13. Equational categories of Σ-algebras | |
| 14. S-sorted algebraic categories |