The n-Category Café

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!

• Jordan Tousignant’s notes. Also see the errata.
• Christina Osborne’s notes.
• Jason Erbele’s notes.
• Samuel Britton’s notes.

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 $\mathbb{C}$ with $2 = \{0,1\}$. 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.

• 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.
  • Jordan Tousignant’s notes.
  • Christina Osborne’s notes.
  • Jason Erbele’s notes.
  Also try this:
  • Drew Armstrong, Introduction to Galois connections, with solved problems, from his abstract algebra course notes.

• 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.
  • Jordan Tousignant’s notes.
  • Christina Osborne’s notes.
  • Jason Erbele’s notes.
  To see this worked out in the most familiar example, see:
  • Amanda Bower, Category theory and Galois theory, Rose-Hulman Undergraduate Mathematics Journal 14 (2013), 134–142.

• 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 $G$, 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 $G$.
  • Jordan Tousignant’s notes.
  • Christina Osborne’s notes.
  • Jason Erbele’s notes.

• 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.
  • Jordan Tousignant’s notes.
  • Christina Osborne’s notes.
  • Jason Erbele’s notes.

• 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 $S_3$.
  • Jordan Tousignant’s notes.
  • Christina Osborne’s notes.
  • Jason Erbele’s notes.