# April 26, 1997 {#week103}

As I segue over from the homotopy theory conference at Northwestern
University to the conference on higher-dimensional algebra and physics
that took place right after that, it's a good time to mention Ronnie
Brown's web page:

1) Ronald Brown, Higher-dimensional group theory, `http://www.bangor.ac.uk/~mas010/home.html`

Brown is the one who coined the phrase "higher-dimensional algebra",
and for many years he has been developing this subject, primarily as a
tool for doing homotopy theory. I wrote a bit about his ideas two years
ago, in ["Week 53"](#week53). A lot has happened in
higher-dimensional algebra since then, and the web page above is a good
place to get an overview of it. It includes a nice bibliography on the
subject. Also, if you find the math a bit strenuous, you can rest your
brain and delight your eyes at the related site:

2) Symbolic sculptures and mathematics, `http://www.bangor.ac.uk/SculMath/`

which opens with a striking image of rotating linked tori, and includes
pictures of the mathematical sculpture of John Robinson.

The Workshop on Higher Category Theory and Physics was exciting because
it pulled together a lot of people working on the interface between
these two subjects, many of whom had never before met. It was organized
by Ezra Getzler and Mikhail Kapranov. Getzler is probably most
well-known for his proof of the Atiyah-Singer index theorem. This
wonderful theorem captured the imagination of mathematical physicists
for many years starting in the 1960s. The reason is that it relates the
topology of manifolds to the the solutions of partial differential
equations on these manifolds, and thus ushered in a new age of
applications of topology to physics. In the 1980s, working with ideas
that Witten came up with, Getzler found a nice "supersymmetric proof"
of the Atiyah-Singer theorem. Later Getzler turned to other things, such
as the use of "operads" (see ["Week 42"](#week42)) to study
conformal field theory (which shows up naturally in string theory).
Kapranov has also done a lot of work with operads and conformal field
theory, and many other things, but I first learned about him through his
paper with Voevodsky on "braided monoidal $2$-categories" (see
["Week 4"](#week4)). This got me very excited since it turned me on
to many of the main themes of $n$-category theory.

Alas, my description of this fascinating conference will be terse and
dry in the extreme, since I am flying to Warsaw in 3 hours for a quantum
gravity workshop. I'll just mention a few papers that cover some of the
themes of this conference. Ross Street gave two talks on Batanin's
definition of weak $n$-categories (and even weak $\omega$-categories), which one
can get as follows:

3) Ross Street, _The role of Michael Batanin's monoidal globular categories._ "Lecture I: Globular categories and trees". "Lecture II: Higher operads and weak $\omega$-categories". Available in gzipped Postscript form at `http://www.math.mq.edu.au/~street/Publications.html`

Subsequently Batanin has written a more thorough paper on his
definition:

4) Michael Batanin, "Monoidal globular categories as a natural environment for the theory of weak $n$-categories", _Adv. Math_ **136** (1998), 39--103, preprint available at `http://www-math.mpce.mq.edu.au/~mbatanin/papers.html`

I gave a talk on Dolan's and my definition of weak $n$-categories, which
one can get as follows:

5) John Baez, "An introduction to $n$-categories", to appear in the proceedings of _Category Theory and Computer Science '97_, preprint available as [`q-alg/9705009`](https://arxiv.org/abs/q-alg/9705009) or in Postscript form at `http://math.ucr.edu/home/baez/ncat.ps`

Unfortunately Tamsamani was not there to present *his* definition of
weak $n$-categories, but at least I have learned how to get his papers
electronically:

6) Zouhair Tamsamani, "Sur des notions de $\infty$-categorie et $\infty$-groupoide non-strictes via des ensembles multi-simpliciaux", preprint available as [`alg-geom/9512006`](https://arxiv.org/abs/alg-geom/9512006).

    Zouhair Tamsamani, "Equivalence de la theorie homotopique des $n$-groupoides et celle des espaces topologiques $n$-tronques", preprint available as [`alg-geom/9607010`](https://arxiv.org/abs/alg-geom/9607010).

Also, Carlos Simpson has written an interesting paper using Tamsamani's
definition:

7) Carlos Simpson, "A closed model structure for $n$-categories, internal Hom, $n$-stacks and generalized Seifert-Van Kampen", preprint available as [`alg-geom/9704006`](https://arxiv.org/abs/alg-geom/9704006).

In a different but related direction, Masahico Saito discussed a paper
with Scott Carter and Joachim Rieger in which they come up with a nice
purely combinatorial description of all the ways to embed $2$-dimensional
surfaces in $4$-dimensional space:

8) J. Scott Carter, Joachim H. Rieger and Masahico Saito, "A
combinatorial description of knotted surfaces and their isotopies", to
appear in _Adv. Math._, preprint available at `http://www.math.usf.edu/~saito/home.html`

My student Laurel Langford has translated their work into $n$-category
theory and shown that "unframed unoriented 2-tangles form the free
braided monoidal $2$-category on one unframed self-dual object":

9) John Baez and Laurel Langford, "2-Tangles", preprint available as [`q-alg/9703033`](https://arxiv.org/abs/q-alg/9703033) and in Postscript form at `http://math.ucr.edu/home/baez/2tang.ps`

This paper summarizes the results; the proofs will appear later.

While I was there, Carter also gave me a very nice paper he'd done with
Saito and Louis Kauffman. This paper discusses 4-manifolds and also
2-dimensional surfaces in $3$-dimensional space, again getting a purely
combinatorial description which is begging to be translated into
$n$-category theory:

10) J. Scott Carter, Louis H. Kauffman and Masahico Saito, "Diagrammatics, singularities, and their algebraic interpretations", preprint available at `http://www.math.usf.edu/~saito/home.html`

I am sorry not to describe these papers in more detail, but I've been
painfully busy lately. (In fact, I am trying to figure out how to reform
my life to give myself more spare time. I think the key is to say "no"
more often.)

Thanks to Justin Roberts for pointing out an error in
["Week 102"](#week102). The phase ambiguity in conformal field
theories is not necessarily a 24th root of unity; it's $\exp(2\pi ic/24)$ where $c$ is the central charge of the associated Virasoro
representation. This is a big hint as far as my puzzle goes.

Also I thank Dan Christensen for helping me understand $\pi_4(S^2)$ in a
simpler way, and Scott Carter for a fascinating letter on the themes of
["Week 102"](#week102). Alas, I have been too busy to reply
adequately to these nice emails!

Gotta run....