Functorial Semantics 24

Info. Lectures of 90 minutes each. First semester of 2024. Meeting once a week. The lectures will be delivered at Humanisten from 13.00 to 15.00.

Lectures are decorated by additional material. The will open one of the two graded homework.

Date	Room	Date	Room	Date	Room	Date	Room
25/03	J577	08/04	no!	22/4	J577	13/05	J577
03/04	J562	16/04	J577	29/4	J577	21/05	J577

Syllabus. (1) Foundational aspects of the Yoneda Embedding. (2) Lawvere theories and Universal Algebra. (3) Sketches and Accessible Categories.

Description. Functorial semantics emerged after the impulse of Bill Lawvere as a framework to organize universal algebra. Later it became one of the leitmotives of categorical logic, touching on many of its most foundational and fundamental aspects. The course is a tour of this tradition, starting precisely from Lawvere's original contribution and landing on the most sophisticated aspects of the french school of Sketches.

Exam. The exam comprises of the homework () (50%) and an oral examination (50%). Each of these parts of the exam is graded from 1 to 10. The student's final mark will be U is the weighted mean is strictly below 5, G if the weighted mean is between 5 and 7, VG if the weighted mean is strictly above 7.

Audience and Prerequisites. The course is designed for master students. A good knowledge of the language of category theory will be a prerequisite for the audience (the equivalent of the whole (!) book by Riehl in bibliography), yet we will re-discover some of its content from a more foundational perspective.

	Tentative structure	Material	Reference
1.	A prehistory of fun(ctorial semantics): Boole
2.	Lawvere theories.
3.	Varieties.
4.	Finitary monads.
5.	The Structure-Semantics duality.
6.	Sketches.
7.	Accessibility and Presentability.
8.	Gabriel Ulmer duality.

Bibliography.

Category Theory in Context by Emily Riehl.
The history of categorical logic 1963–1977 by Marquis and Reyes.
Handbook of categorical algebra, Vol. I, II & III by Francis Borceux.
The category theoretic understanding of Universal Algebra: Lawvere Theories and Monads by Hyland and Power.
Locally Presentable and Accessible Categories by Rosicky and Adamek.
Further readings.

Comments. The course is dense, and has no intention of completely cover the bibliography above. Instead, one of its main intents is to introduce the students to this corpus of knowledge, making sure that they reach a sufficient level of independence and maturity.