Quantum Gravity Seminar - Spring 2007

John Baez and Derek Wise

• Quantization and Cohomology (Tuesdays)
• Cohomology and Computation (Thursdays)

As usual, John Baez will lecture and Derek Wise will take beautiful notes which you will be able to read here. If you discover any errors in these notes, please email me, and we'll try to correct them. We'll keep a list of errors that haven't been fixed yet.

These courses are also available via my blog at the n-Category Café. Each week's notes comes with a blog entry where you can ask questions, make comments, and chat with other people following the course.

The week numbers for these notes continue where last quarter's notes leave off.

There may also be some LaTeX, encapsulated PostScript and xfig files to download if for some bizarre reason you want them. However, we reserve all rights to this work.

Quantization and Cohomology

• John Baez and Apoorva Khare, with figures by Christine Dantas, Course Notes on Quantization and Cohomology:
  • Fall 2006 notes, available in PDF and Postscript.
  • Winter 2007 notes, available in PDF and Postscript.

Here's a rough draft of this quarter's notes, without any figures so far:

• John Baez and Apoorva Khare, Course Notes on Quantization and Cohomology, Spring 2007, available in PDF and Postscript.

You can also see Derek's beautiful handwritten notes, with figures and extra reading material:

• Week 19 (Apr. 3) - Finding critical points of an action functor S: C → R. For this, C should be a "smooth category" and S should be something like a "smooth functor". How can we make these concepts precise? The example where C is the smooth path groupoid of a manifold equipped with a 1-form (for example, a cotangent bundle equipped with its canonical 1-form). The definition of "smooth category" - that is, a category internal to some category of "smooth spaces". (There are 2 errors in these notes.) Blog entry.
• Week 20 (Apr. 10) - Smooth spaces and smooth categories. The concept of a "category internal to K" where K is any category with pullbacks. The category of smooth manifolds does not have pullbacks. Grothendieck's dictum. Chen's category of smooth spaces. Examples: the discrete and indiscrete smooth structures on a set. Any convex set or smooth manifold is a smooth space. The product and coproduct of smooth spaces. Any subset of a smooth space becomes a smooth space. Blog entry.

  Homework: the category of smooth spaces has pullbacks.

  • Answers by Alex Hoffnung.
  • Answers by Christopher Walker.
• Week 21 (Apr. 17) - Any quotient of a smooth space becomes a smooth space. The category of smooth spaces has pushouts. The category of smooth spaces is cartesian closed. The path groupoid PX of a smooth space X. The path groupoid is a smooth category. Smooth functors. Theorem: a smooth functor S: PX → R is the same as a 1-form on X. Blog entry.

  Supplementary reading:

  • John Baez and Urs Schreiber, Higher gauge theory II: 2-connections, draft version. Section 6.1: proof that for any Lie group G, smooth functors S: PX → G are the same as Lie(G)-valued 1-forms on X.
• Week 22 (May 1) - Smooth functors and beyond. Review of what we've done so far. Principal U(1) bundles and phases in quantum mechanics. The need for nontrivial principal bundles in geometric quantization. Torsors. Blog entry.
• Week 23 (May 8) - Principal bundles. The transport groupoid Trans(P) of a principal G-bundle P over a smooth space M. Connections as smooth functors hol: PM → Trans(P), where PM is the path groupoid of M. Proof that connections are described locally by smooth functors hol: PU → G, where U is a neighborhood in M. Theorem: smooth functors hol: PU → G are in 1-1 correspondence with Lie(G)-valued 1-forms on U. Blog entry.

• Week 24 (May 15) - Three approaches to connections. Connections on the trivial principal G-bundle over M are smooth functors hol: PM → G; gauge transformations are smooth natural transformations between these. Connections on a fixed principal G-bundle P → M are smooth functors hol: PM → Trans(P); gauge transformations are smooth natural transformations between these. Connections on an arbitrary, or variable principal G-bundle over M are smooth anafunctors hol: PM → G; gauge transformations are smooth ananatural transformations between these. The definition of smooth anafunctor. Blog entry.

  Supplementary reading:

  • Toby Bartels, Higher gauge theory I: 2-bundles. Section 2.2.2, on "2-maps", describes smooth anafunctors between smooth categories. Section 2.2.3 describes what I'm calling smooth ananatural transformations.
  • Urs Schreiber and Konrad Waldorf, Parallel transport and functors. This develops some closely related ideas, including the notion of "π-local i-trivialization" for a functor.

• Week 25 (May 22) - Bundles, connections, cohomology and anafunctors. A simplified version of the claim made last week: principal G-bundles over M correspond to smooth anafunctors hol: Disc(M) → G, where Disc(M) is the smooth category with points of M as objects and only identity morphisms. Bundle isomorphisms correspond to smooth ananatural transformations between these. To prove this, use Cech 1-cocycles to describe principal G-bundles, and Cech 0-cochains to describe isomorphisms between these. Claim: the first Cech cohomology consists of smooth anafunctors modulo smooth ananatural transformations. Blog entry.

• Week 26 (May 29) - The definition of smooth anafunctor (review), and the definition of smooth ananatural transformation (new). Why the first Cech cohomology of a smooth space M with coefficients in a smooth group G is isomorphic to the set of smooth anafunctors F: Disc(M) → G mod smooth ananatural transformations. Generalization: the first Cech cohomology of a smooth category. Dreams of further generalizations. Blog entry.

• Week 27 (June 5) - Review and prospectus. Classical and quantum mechanics from categories equipped with action or phase functors. Subtleties: path integrals and anafunctors. Geometric quantization of symplectic manifolds. Categorification, to obtain the classical and quantum mechanics of strings. Categorifying the theory of connections as anafunctors. Blog entry.

  Supplementary reading:

  • John Baez and Urs Schreiber, Higher gauge theory.
  • John Baez, Higher gauge theory, higher categories.
  • Urs Schreiber, Talks at "Higher Categories and their Applications".

Cohomology and Computation

• Week 19 (Apr. 5) - The origin of cohomology in the study of "syzygies", or "relations between relations". Syzygies in the study of linear equations, and more generally in the study of any presentation of any algebraic gadget. Building a topological space from a presentation of an algebraic gadget. Euler characteristic. For a better explanation of the picture on page 4, see the blog entry.
• Week 20 (Apr. 12) - Cohomology and the category of simplices. Simplices as special categories: finite totally ordered sets, which are isomorphic to "ordinals". The algebraist's category of simplices, Δ_alg. Face and degeneracy maps. The functor from Δ_alg to Top sending the ordinal n to the standard (n-1)-simplex. Simplicial sets. Preview of the cohomology of spaces. (There is an error in these notes.) Blog entry.

• Week 21 (Apr. 19) - Simplicial sets and cohomology. Two sources of simplicial sets: topology and algebra. The topologist's category of simplices, Δ_top. How a topological space X gives a simplicial set called its "singular nerve" SX. How this gives a functor S: Top → SimpSet. Blog entry.

• Week 22 (May 3) - Cohomology and chain complexes. The functor from simplicial sets to simplicial abelian groups. The functor from simplicial abelian groups to chain complexes. The homology of a chain complex as a general method of "counting holes". Some examples: the hollow triangle has H₁ = Z because it has a "one-dimensional hole". The twice filled triangle (a triangulated 2-sphere) has H₁ = {0} but H₂ = Z because it has a "2-dimensional hole". Blog entry.
• Week 23 (May 10) - Simplicial sets from algebraic gadgets. Algebraic gadgets and adjoint functors. The unit and counit of an adjunction: the unit "includes the generators", while the counit "evaluates formal expressions". The canonical presentation of an algebraic gadget. Simplicial objects from adjunctions: the bar construction. 1-simplices as proofs. Blog entry.
• Week 24 (May 17) - The bar construction. Why do adjoint functors give simplicial objects? First, Δ_alg is the free monoidal category on a monoid object - or "the walking monoid", for short. Second, adjoint functors give certain monoids, called "monads". Blog entry.

  Supplementary reading:

  • Todd Trimble, On the bar construction.
  • Todd Trimble, Bar constructions.

• Week 25 (May 24) - The bar construction, continued. The zig-zag identities for the unit and counit of an adjunction. Monads and comonads from adjunctions. Simplicial objects from adjunctions. Blog entry.

• Week 26 (May 31) - The bar construction, continued. Comonads as comonoids. Given adjoint functors L: C → D and R: D → C, the bar construction turns an object d in D into a simplicial object d in D. Example: the cohomology of groups. Given a group G, the adjunction L: Set → G-Set, R: G-Set → Set lets us turn any G-set X into a simplicial G-set X. This is a "puffed-up" version of X in which all equations gx = y have been replaced by edges, all equations between equations (syzygies) have been replaced by triangles, and so on. When X is a single point, X is called EG. It's a contractible space on which G acts freely. The "group cohomology" of G is the cohomology of the space BG = EG/G. Blog entry.

• Week 27 (June 7) - Cohomology of algebraic gadgets. The bar construction "puffs up" any algebraic gadget, replacing equations by edges, syzygies by triangles and so on, with the result being a simplicial object with one contractible component for each element of the original gadget. Examples: Ext and Tor, group cohomology and homology, Lie algebra cohomology and homology. How Ext and Tor arises from the adjoint functors between the category of abelian groups and the category of modules of a ring. Free resolutions. Group cohomology as a special case of Ext. Group cohomology as the cohomology of the the classifying space BG = EG/G. Blog entry.

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