Quantum Gravity Seminar  Spring 2007
John Baez and Derek Wise
This quarter's seminar will be a continuation
of this year's Fall
and Winter seminars, but
instead of discussing "classical versus quantum computation",
we'll mainly focus on the role of cohomology in computation.
So, the two parts of the seminar are:
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
nCategory
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.
This quarter we're finally going to get into cohomology in a more
explicit way. Notes from the last two quarters are here:

John Baez and Apoorva Khare, with figures by Christine Dantas,
Course Notes on Quantization and Cohomology:
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. 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 (n1)simplex.
Simplicial sets. Preview of the cohomology of spaces.
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 H_{1} =
Z because it has a "onedimensional hole". The
twice filled triangle (a triangulated 2sphere) has H_{1} =
{0} but H_{2} = Z because it has a "2dimensional 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. 1simplices 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:

Week 25 (May 24) 
The bar construction, continued. The zigzag 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 → GSet, R: GSet → Set lets us turn any
Gset X into a simplicial Gset X. This is a
"puffedup" 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.
© 2007 John Baez and Derek Wise
baez@math.removethis.ucr.andthis.edu