Here are some students’ notes from my Fall 2015 seminar on category theory. The goal was not to introduce technical concepts from category theory—I started that in the next quarter. Rather, I tried to explain how category theory unifies mathematics and makes it easier to learn. We began with a study of duality, and then got into a bit of Galois theory and Klein geometry:

If you discover any errors in the notes please email me, and I’ll add them to the list of errors.

You can get all 10 weeks of notes in a single file here. Take a look at them and choose your favorite student!

Or, you can look at individual topics:

Lecture 1 (Sept. 28) - Duality. How Descartes saw geometry as dual to commutative algebra, and how Grothendieck clarified this vision. The category of affine schemes as opposite to the category of commutative rings. Jordan Tousignant’s notes. Also see the errata. Christina Osborne’s notes. Jason Erbele’s notes.



Lecture 2 (Oct. 8) - How to see topology as dual to a special kind of commutative algebra. C*-algebras, the commutative C*-algebra of continuous functions on a compact Hausdorf space, and the spectrum of a commutative C*-algebra. The Gelfand-Naimark theorem: the category of compact Hausdorff space as opposite of the category of commutative C*-algebras. Jordan Tousignant’s notes. Christina Osborne’s notes. Jason Erbele’s notes.



Lecture 3 (Oct. 15) - How to see set theory as dual to propositional logic. The notion of a dualizing object. Replacing ℂ \mathbb{C} with 2 = { 0 , 1 } 2 = \{0,1\} . The category of sets as opposite of the category of complete atomic Boolean algebras. Jordan Tousignant’s notes. Christina Osborne’s notes. Jason Erbele’s notes.

with . The category of sets as opposite of the category of complete atomic Boolean algebras.

Lecture 6 (Nov. 2) - Groupoids. The core of a category. The translation groupoid coming from a group action. Moduli spaces and moduli stacks. Comparing the moduli space of line segments in the plane to the moduli stack. Jordan Tousignant’s notes. Christina Osborne’s notes. Jason Erbele’s notes.



Lecture 7 (Nov. 9) - Moduli spaces and moduli stacks. The moduli moduli stack of line segments in the plane. The moduli stack of triangles in the plane. The moduli space of elliptic curves. Jordan Tousignant’s notes. Also see the errata. Christina Osborne’s notes. Jason Erbele’s notes.



Lecture 8 (Nov. 16) - Klein geometry. How Euclidean plane geometry, spherical geometry and hyperbolic geometry are associated to different symmetry groups G G , with the ‘space of points’ and also the ‘space of lines’ being a homogeneous G G -space in each case. Projective geometry, and how duality lets us switch the concept of point and line in projective geometry. Klein’s general framework where a ‘geometry’ is just a group G G and a ‘type of figures’ is just a homogeneous G G -space. How to classify homogeneous G G -spaces in terms of subgroups of G G . Jordan Tousignant’s notes. Christina Osborne’s notes. Jason Erbele’s notes.

, with the ‘space of points’ and also the ‘space of lines’ being a homogeneous -space in each case. Projective geometry, and how duality lets us switch the concept of point and line in projective geometry. Klein’s general framework where a ‘geometry’ is just a group and a ‘type of figures’ is just a homogeneous -space. How to classify homogeneous -spaces in terms of subgroups of .

Lecture 9 (Nov. 23) - Klein geometry. A category G Rel G \mathrm{Rel} with G G -sets as objects and G G -invariant relations. The example of projective plane geometry: if G = PGL ( 3 , ℝ ) G = PGL(3,\mathbb{R}) , the set Y Y of ‘flags’ (point-line pairs, where the point lies on the line) is a homogeneous space, and there are 6 ‘atomic’ invariant relations between flags. Enriched categories. The category G Rel G \mathrm{Rel} is enriched over complete atomic Boolean algebras. Jordan Tousignant’s notes. Christina Osborne’s notes. Jason Erbele’s notes.

with -sets as objects and -invariant relations. The example of projective plane geometry: if , the set of ‘flags’ (point-line pairs, where the point lies on the line) is a homogeneous space, and there are 6 ‘atomic’ invariant relations between flags. Enriched categories. The category is enriched over complete atomic Boolean algebras.