What
An introductory course on category theory and its applications; runs at Tallinn University of Technology during the second semester (January to May) of each academic year.
When / Where
| Thursday | 14.00—15.30 & 16.00—17.30 | SOC-313 |
First lecture: February 5, 2026 / Last lecture: May 21, 2026
Lectures' Log
- Notes for a computer-science friendly version of the course (link)
- 0: Introductions, introduction, organization (link)
- 1: Monoids, posets, categories I (link)
- 2: Monoids, posets, categories II (link)
- 3: Functors and natural transformations I (link)
- 4: Functors and natural transformations II (link)
- 5: Universal properties I (link)
- 6: Universal properties II (link)
- 7: Limits and colimits I (link)
- 8: Limits and colimits II (link)
- 9: Adjunctions I (link)
- 10: Adjunctions II (link)
- 11: Monads I (link)
- 12: Monads II (link)
Exercises
For the mind
Warning: this set of exercises is not meant as part of the course, its content is extremely volatile in some parts, and it is usually addressed to people who already have a decent mathematical exposition.
The chapter may cause serious damage to the unwary and/or unprepared reader (implying such damage has not already been caused by the lectures…).
For the exam
Through the course (more or less every other week) I will publish a document with 3-4 exercises; you have until the next sheet to solve them. They are part of you final grade. Deadlines are flexible; if the only problem is that you need more time, just tell me.
Grading
Your final grade will be determined based on how well you perform on the exercise sheets handed through the course, and a final oral exam. No one stops you from using a robot to learn; embrace the future. But: we are going to have a problem if you come at the whiteboard clueless on how to solve the exercises “you” did.
For the final exam, you can choose a topic from a list, and give a (~30 min+questions) presentation about it. It can relate category theory to whatever you like (one year the link was “monads in functional programming” + supercollider, I dare you to do something cooler than that).
⚠️NEW⚠️ A list of possible exam topics: (legend: BCT = Leinster’s book; CTCtx = Riehl; CTCS = Barr-Wells; ACC = Adámek-Herrlich-Strecker; CT = Awodey).
- Basic results:
- Free objects as left adjoints, with applications to algebra: free monoids, free groups, free modules. (BCT: pp 19-20 and 43-50; CTCtx: pp 129-136; CTCS: pp 347-350; ACC: pp 135-139; CT: pp 18-28)
- Cartesian closed categories: products, exponentials, and proofs-as-morphisms. (BCT: pp. 164–167; CTCtx: pp. 129–131; CTCS: pp. 195–218; ACC: pp. 428–441; CT: pp. 122–146)
- Monads and Kleisli categories: partiality, exceptions, nondeterminism, state, input/output. (BCT: p. 173; CTCtx: pp. 153–188; CTCS: pp. 372–376; ACC: pp. 316–348; CT: pp. 253–277)
- The Yoneda lemma, with applications to algebraic structures and representation problems. (BCT: pp. 93–99; CTCtx: pp. 55–61; CTCS: pp. 121–126; ACC: p. 84; CT: pp. 188–193)
- Limits and colimits in algebra: products, coproducts, equalizers, quotients, tensor products, presentations by generators and relations. (BCT: pp. 107–140; CTCtx: pp. 73–114; CTCS: pp. 153–194, 265–298; ACC: pp. 188–205; CT: pp. 89–118)
- Intermediate/hard results:
- Monadicity and universal algebra: algebraic categories as categories of algebras for monads. (CTCtx: pp. 177–188; CTCS: pp. 372–382; ACC: pp. 316–348, 403–406; CT: pp. 253–277)
- Stone duality and classical propositional logic: Boolean algebras, ultrafilters, and Stone spaces. (ACC: pp. 29,35; CT: pp. 129–140)
- Initial algebras and final coalgebras: datatypes, recursion, streams, automata, and coinduction. (BCT: pp …; CTCtx: p. 44; CTCS: section 4.7; ACC: pp. 117-120; CT: section 10.5)
- Freyd’s adjoint functor theorem. (CTCtx: Theorem 4.6,3, what follows and what precedes; ACC: pp. 299–; CT: section 9.8)
- Grothendieck constructions and fibrations: indexed categories, families of structures, and dependent types. (CTCtx: section 2.4; CTCS: pp. 327–346; ACC: p. 349-380; CT: section 8.4)
- A list of interesting papers doing interesting things:
- Philip Wadler, “Theorems for Free!”
- Eugenio Moggi, “Notions of Computation and Monads” (LINK)
- Bart Jacobs, “Bases as coalgebras” (LINK)
- RFC Walters, “A note on context-free languages”, (LINK)
- Marcelo Fiore, Tom Leinster, “Objects of Categories as Complex Numbers” [(LINK)]((LINK)
- Bill Lawvere, “Quantifiers and Sheaves” (LINK)
- Bill Lawvere, “Diagonal Arguments and Cartesian Closed Categories” (LINK)
- Bill Lawvere, “An elementary theory of the category of sets” (LINK)
- Bill Lawvere, “Categorical Dynamics” (LINK)
- Bill Lawvere, “Categories in continuum physics” (LINK)
- Bill Lawvere, “Display of Graphics and Their Applications Exemplified by 2-Categories and the Hegelian «Taco»” (LINK)
- Bill Lawvere, “Unity and identity of opposites in calculus and physics” (LINK)
- Bill Lawvere, “Kinship and mathematical categories” (LINK)
- Bill Lawvere, “Introduction to Toposes, Algebraic Geometry and Logic” (LINK)
- More in general, all that appears in the Lawvere Archives.
- the wikipedia page “timeline of category theory and related mathematics” https://en.wikipedia.org/wiki/Timeline_of_category_theory_and_related_mathematics
- You can also consult my page on projects for students
