This is the page for my course/seminar on category theory, held in the Winter 2026 quarter at Caltech. The target audience is mathematically-inclined engineers. Undergrads, grad students, postdocs, and professors are all welcome.
Thursdays inGates-Thomas 235
1pm: Discussion of previous lecture
2pm: Lecture
Over-ambitious list of topics I’ll try to cover: category, commutative diagram, functor, natural transformation, poset, monoid, fundamental groupoid, co/product, initial/terminal object, co/limit, presheaf/C-set, database lens, monoidal category, string diagram, quantum computation, Petri net, adjunction, Galois connection, adjoint functor theorem, monad, functional programming, F-coalgebra, dynamical/control system, probability, Markov category, combinatorics, graph, combinatorial species, Yoneda’s lemma, enriched category, Lawvere metric space, bicategory, double category, infinity category…
- Jan 15: Definition and examples of categories: Set, Mat, Vect, FinVect
- Jan 22: More examples: path category of a graph, a monoid as a one-object category; definition and examples of isomorphisms
- Jan 29: More on monoids and groups as one-object categories, definition of functor, free vector space construction as a functor
- Feb 5: The category of categories, graphs as functors, the path functor induced by a graph map, the graph path category as a functor, databases as functors
- Feb 12: Hybrid systems as databases, natural transformations, products
- Feb 19: Monoidal categories, string diagrams, monoids Fox’s theorem, Eckmann-Hilton argument
- Feb 26: Categorical Probability
- Mar 5: Categorical Quantum Computing
- Mar 12: Categorical Lyapunov Stability
Resources:
Historically, pure mathematicians have been the most interested in category theory. Thus, the traditional approach to teaching category theory assumes the student is a math major or math grad student.
Here are some introductions to category theory aimed at a broader or just different audience:
- Seven Sketches in Compositionality: An Invitation to Applied Category Theory, Brendan Fong and David Spivak
- Notes on Category Theory with examples from basic mathematics, Paolo Perrone
- Applied Category Theory for Engineering, Andrea Censi, Jonathan Lorand, and Gioele Zardini
- Picturing Quantum Processes, Bob Coecke and Aleks Kissinger
- Category Theory for Computing Science, Michael Barr and Charles Wells
Here are some texts aimed at mathematicians:
- Category Theory in Context, Emily Riehl
- Categories for the Working Mathematician, Saunders Mac Lane
- Handbook of Categorical Algebra (three volumes), Francis Borceaux
- Basic Category Theory, Tom Leinster
- Category Theory, Steve Awodey
- Topology – A Categorical Approach, Tai-Danae Bradley, Tyler Bryson, John Terilla
- Algebra: Chapter 0, Paolo Aluffi
- Natural Operations in Differential Geometry, Ivan Kolář, Jan Slovák, and Peter W. Michor
Some more advanced texts:
- Basic Concepts of Enriched Category Theory, Max Kelly
- 2-Dimensional Categories, Niles Johnson and Donald Yau
- Higher Algebra, Jacob Lurie
Joe Moeller 