John Baez
August 22-26, 2005
Higher Gauge Theory,
Homotopy Theory
and n-Categories
These are rough notes for four lectures on higher gauge
theory, aimed at explaining how this theory is related to
some classic themes from homotopy theory, such as
Eilenberg-Mac Lane spaces. After a brief introduction
to connections on principal bundles, with a heavy emphasis
on the concept of `torsor', we describe how to build the
classifying space BG of a topological group G starting from
the topological category of its torsors. In the case of an
abelian topological group A, we explain how this construction can
be iterated, with points of BnA corresponding to `finite
collections of A-charged particles on the n-sphere'. Finally, we
explain how BnA can be constructed from the n-category
of n-torsors of A. In the process, these notes give
a quick introduction to the most basic notions of enriched
category and strict n-categories. References provide avenues
for further study.
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Higher Gauge Theory, Homotopy Theory and n-Categories, in
PDF and
Postscript.
Unfortunately the above notes are missing some pictures which I
have drawn by hand. Someday I'll rectify this - but right now,
if you're desperate, you can download a ridiculously large
(30 megabyte) file that includes scanned-in versions of the pictures:
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Higher Gauge Theory, Homotopy Theory and n-Categories, with
pictures, in PDF.
Here's some extra reading material to get ready for the
course:
I also suggest reading parts of this book:
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John Milnor and James D. Stasheff, Characteristic Classes,
Princeton U. Press, Princeton, 1974.
especially the parts where they describe the classifying space of
a topological group, and the appendix where they construct characteristic
classes using deRham cohomology by picking a connection on a vector
bundle and constructing closed forms from its curvature. This assumes
you know about deRham cohomology, connections, and
their curvature! If you don't know this stuff, you should definitely
learn it, and one simple way is to look in here:
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John Baez and Javier Muniain, Gauge Fields, Knots and Gravity,
World Scientific Press, Singapore, 1994.
© 2005 John Baez
baez@math.removethis.ucr.andthis.edu