## Seminar - Fall 2015

### Category Theory

#### John Baez

Here are some student's notes from a seminar on category theory. The goal this quarter was not to introduce technical concepts of category theory. 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 moved on to a very general approach to Galois theory and Klein geometry, getting a taste of 'stacks'.

For a more systematic course starting from scratch, see the Winter 2016 notes.

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; then pick your favorite student:

Or, you can read notes of individual lectures here:

• 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.

• 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.

• Lecture 3 (Oct. 15) - How to see set theory as dual to propositional logic. The notion of a dualizing object. Replacing $$\mathbb{C}$$ with $$2 = \{0,1\}$$. The category of sets as opposite of the category of complete atomic Boolean algebras.

• Lecture 4 (Oct. 20) - Other examples of categories and their opposites. The opposite of the category of Boolean algebras is the category of 'Stone spaces'. The opposite of the category of finite-dimensional vector spaces, or finite abelian groups, is itself! Galois theory. Partially ordered sets as categories, order-preserving maps as functors, and Galois connections as adjoint functors.

Also try this:

• Lecture 5 (Oct. 27) - Galois theory: how to classify 'subgadgets' of an algebraic gadget using group theory. The Galois connection between the poset of subgadgets and the poset of subgroups of the Galois group.

To see this worked out in the most familiar example, see:

• 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.

• 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.

• Lecture 8 (Nov. 16) - Klein geometry. How Euclidean plane geometry, spherical geometry and hyperbolic geometry are associated to different symmetry groups $$G$$, with the 'space of points' and also the 'space of lines' being a homogeneous $$G$$-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 $$G$$ and a 'type of figures' is just a homogeneous $$G$$-space. How to classify homogeneous $$G$$-spaces in terms of subgroups of $$G$$.

• Lecture 9 (Nov. 23) - Klein geometry. A category $$G \mathrm{Rel}$$ with $$G$$-sets as objects and $$G$$-invariant relations. The example of projective plane geometry: if $$G = PGL(3,\mathbb{R})$$, the set $$Y$$ 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 $$G \mathrm{Rel}$$ is enriched over complete atomic Boolean algebras.

• Lecture 10 (Nov. 30) - Klein geometry. Enriched categories and internal monoids. The example of projective plane geometry: if $$G = PGL(3,\mathbb{R})$$ and $$\mathrm{CABA}$$ is the monoidal category of complete atomic Boolean algebras, $$G \mathrm{Rel}$$ is a CABA-enriched category. Taking $$Y$$ to be the set of flags, $$\mathrm{hom}(Y,Y)$$ is a monoid in CABA. We can work out the multiplication table for the atoms in this CABA, and the result is closely related to the 3-strand braid group and the symmetric group $$S_3$$.

baez@math.removethis.ucr.andthis.edu