Category Theory, Monads, and Computation
Master level course, 4 ECTS. UR1 / ENS Rennes 2023/24 (sif)
Syllabus
The main objective of this course is to introduce a unifying, yet expressive and powerful, point of view to several constructs and structures, many of which the students have already seen. After presenting the basics of category theory, we will investigate the concept of monads in both its theoretical roots and practical use in programming languages. We hope that this course serves as an invitation for master students to enrich their toolbox with openmindedness, elegance and curiosity.
Keywords Category Theory, Monads, Semantics, Functional Programming, Computation, λcalculus
Prerequisite
Licence in mathematics or theoretical computer science or equivalent degree. Functional programming, semantics, programming languages’ theory, logic. openmindedness and curiosity.
Roadmap (tentative)
All courses will take place in B02BE208. Two classes per week: Tuesday and Wednesday.
Introduction – Rethinking Set Theory
 November 14 ETCS revisited (1/2) (8–9:30am)
 November 21 ETCS revisited (2/2) (3–4:30pm)
Part 1: Basics of Category Theory [Slides]
 November 22 Categories and Functors (3–4:30pm)
 November 28 Natural transformations (8–9:30am)
 November 29 TD1 (3–4:30pm)
 December 5 Adjoints (3–4:30pm)
 December 6 Monads and Kleisli Categories (3–4:30pm)
 December 12 Cartesian Closed Categories (8–9:30am)
 December 13 TD23 (3–4:30pm)
Part 2: Types as Formulas [Slides]
 December 19 Deductive systems (3–4:30pm)
 December 20 Monads in programming languages (3–4:30pm)
 January 9 λcalculus (8–9:30am)
Exam
 January 16 Exam (3–5pm) E208
 January 17 Seminar. TBA (3–4:30pm)
Evaluation
The final score will be composed of two marks:
 A written exam: January 16, 3:00 – 4:30 AM
 All TDs will be marked.
Seminars

Samuel Mimram (Ecole Polytechnique, France)

Title: A typetheoretical definition of weak ωcategories [Slides]

Abstract: Higherdimensional categories generalize the traditional notion of category by taking in account higherdimensional cells. In the most general setting, we want their operations to be weak, in the sense that their coherence laws should hold up to higher dimensional cells, which should themselves be coherent. In this talk, I will present a type theory which describe those structures: it allows deriving precisely all the operations and coherence which are expected to be found in those. This work is based on a categorical definition due to Malstiniotis. Our main contribution is to provide a canonical typetheoretical characterization of pasting schemes as contexts which can be derived from inference rules. An implementation of a corresponding proof system will also be presented. This is joint work with Eric Finster and Thibaut Benjamin.

Previous Exams
Bibliography
 Books
 William Lawvere, Stephen H. Schanuel. Conceptual Mathematics: A First Introduction to Categories. 1997, 2009
 Tom Leinster. Basic Category Theory. 2016 pdf
 Joachim Lambek, Philip J. Scott. Introduction to HigherOrder Categorical Logic. 1986
 Saunders Mac Lane, Categories for the Working Mathematician. 1972
 Papers
Office Hours
 Thursdays, 2–4 PM (Inria)
 For any request/question, feel free to email me (see address below)