THIS WEEK`s FINDS

 November 21, 2004
This Week's Finds in Mathematical Physics (Week 209)
John Baez
~
Time flies! This June, Peter May and I organized a workshop on
n-categories at the Institute for Mathematics and its Applications:
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1) n-Categories: Foundations and Applications,
"http://www.ima.umn.edu/categories/"
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I've been meaning to write about it ever since, but I keep putting
it off because it would be so much work. The meeting lasted almost
two weeks. It was an intense, exhausting affair packed with talks,
conversations, and "Russian-style seminars" where the audience
interrupted the speakers with lots of questions. I took about 50
pages of notes. How am I supposed to describe all that?!
~
Oh well... I'll just dive in. I'll quickly list all the official
talks in this conference. I won't describe the many interesting
"impromptu talks", some of which you can see on the above webpage.
Nor will I explain what n-categories are, or what they're good for!
If you want to learn what they're good for, you should go back to
"week73" and read "The Tale of n-Categories". And if you want to
know what they are, try this brand-new book:
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2) Eugenia Cheng and Aaron Lauda, Higher-Dimensional Categories: an
Illustrated Guide Book, available free online at:
"http://www.dpmms.cam.ac.uk/~elgc2/guidebook/"
~
Eugenia and Aaron wrote it specially for the workshop! It's packed
with pictures and it's lots of fun.
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I'm just going to list the talks....
~
Throwing etiquette to the winds, I kicked off the conference myself
with two talks explaining some reasons why n-categories are
interesting and what they should be like:
~
3) John Baez, Why n-Categories? and What n-categories should be
like. Notes available at "http://www.ima.umn.edu/categories/#mon"
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If you're a long-time reader of This Week's Finds you'll know what
I said: n-categories give a new world of math in which equations
are always replaced by isomorphisms, and this world is incredibly
rich in structure. The n-categories called "n-groupoids" magically
know everything there is to know about homotopy theory, while those
called "n-categories with duals" know everything there is to know
about the topology of manifolds. There are, unfortunately, some
details that still need to be worked out!
~
After my talks there was a reception. Later, over dinner, Tom
Leinster gave a "Russian style seminar" outlining the different
approaches to n-categories:
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4) Tom Leinster, Survey and Taxonomy. Talk based on chapter 10 of
his book Higher Operads, Higher Categories, Cambridge U. Press,
Cambridge, 2004, also available free online at
"http://www.arXiv.org/abs/math.CT/0305049".
~
You'll notice these young n-category people are smart: they force
their publishers to keep their books available for free online! All
scientists should do this, since the only people who make serious
money from scientific monographs are the publishers. What
scientists get from writing technical books is not money but
attention. As George Franck said, "Attention is a mode of
payment... reputation is the asset into which the attention
received from colleagues crystallizes."
~
The next morning began with a triple-header talk on "weak
categories":
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5) Andre Joyal, Peter May and Timothy Porter, Weak categories.
Notes available at "http://www.ima.umn.edu/categories/#tues"
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Here a "weak category" means a category where the usual laws hold
only up to homotopy, where the homotopies satisfy laws of their own
up to homotopy, ad infinitum. If you know what weak
infinity-categories are, you can define a weak category to be one
of these where all the j-morphisms are equivalences for j > 1. But,
the nice thing is that there are ways to define weak categories
without the full machinery of infinity-categories! People have come
up with different approaches: "categories enriched over simplicial
sets", "Segal categories", "A_infinity categories" and also Joyal's
"quasicategories". The talk was a nice introduction to all these
approaches.
~
Then Michael Batanin explained his definition of
infinity-categories. This was a blackboard talk, so there are no
notes on the web, but you can try his original paper:
~
6) Michael Batanin, Monoidal globular categories as natural
environment for the theory of weak n-categories, Adv. Math. 136
(1998), 39-103, also available at
"http://www.ics.mq.edu.au/~mbatanin/papers.html"
~
and when you get stuck, try the books by Cheng-Lauda and Leinster.
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Over dinner, Eugenia Cheng and Tom Leinster explained the concepts
of "operad" and "multicategory" which play such an important role
in so much work on n-categories. Again there are no notes, so try
their books.
~
I forget when it happened, but sometime around the second or third
day of the conference people decided it was too much of a nuisance
listening to math lectures while eating dinner - mainly because
there wasn't enough room in the dining hall to take notes, and the
blackboards weren't big enough. So at that point, we switched to
having lectures after dinner. As I said, this workshop was not for
wimps!
~
The morning of the third day began with a no-holds-barred
minicourse on model categories by Peter May:
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7) Peter May, Model categories. Notes available at
"http://www.ima.umn.edu/categories/#wed"
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Model categories are a wonderful framework for relating different
approaches to homotopy theory, and a bunch of people hope they can
also be used to relate different approaches to n-categories.
~
Then Clemens Berger explained Andre Joyal's approach to weak
n-categories:
~
8) Clemens Berger, Cellular definitions. Notes available at
"http://www.ima.umn.edu/categories/#wed"
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Then, either during or after dinner, Eugenia Cheng explained
various "opetopic" approaches to weak n-categories. Again, the best
way to learn about these is to read the book she wrote with Lauda,
or else the book by Leinster.
~
On the morning of the fourth day, Andre Joyal explained his work on
quasicategories - an approach to weak categories in which they are
simplicial sets satisfying a restricted version of the Kan
condition. They've been around a long time, but Joyal is redoing
all of category theory in this context! He's been writing a book
about this, which deserves to be called "Quasicategories for the
Working Mathematician". Since Joyal is a perfectionist, this will
take forever to finish. However, we're hoping to extract a
preliminary version from him for the proceedings of this
conference. For now, you can read a bit about quasicategories in
Tim Porter's notes mentioned in item 5) above.
~
Then Tom Leinster and Nick Gurski spoke about Ross Street's
definition to weak infinity-categories, where they are simplicial
sets satisfying an even more subtly restricted version of the Kan
condition.
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9) Nick Gurski and Tom Leinster, Simplicial definition. Notes
available at "http://www.ima.umn.edu/categories/#thur"
~
Street's definition is tough to understand at first, but it should
eventually include Joyal's quasicategories as a special case, which
is nice. For Street's own discussion, see:
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10) Ross Street, Weak omega-categories, in Diagrammatic Morphisms
and Applications, eds. David Radford, Fernando Souza, and David
Yetter, Contemp. Math. 318, AMS, Providence, Rhode Island, 2003,
pp. 207-213. Also available as
www.maths.mq.edu.au/~street/Womcats.pdf
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It relies on some work by Dominic Verity which has finally been
written up after many years of unpublished limbo:
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11) Dominic Verity, Complicial sets, available as
"http://www.arXiv.org/abs/math.CT/0410412"
~
After dinner we took a turn towards applications, and Larry Breen
explained his work on n-stacks and n-gerbes. An n-stack is like a
sheaf that has an (n-1)-category of sections, while an n-gerbe has
an (n-1)-groupoid of sections. Such things show up a lot in
algebraic geometry, and more recently in mathematical physics
inspired by string theory. Alas, the audience was rather tired this
evening, so Larry only got to 1-stacks and 1-gerbes! But he gave an
impromptu talk later where he reached n = 2, and the notes for both
talks are available in combined form here:
~
12) Larry Breen, n-Stacks and n-gerbes: homotopy theory. Notes
available at "http://www.ima.umn.edu/categories/#thur"
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You've heard about David Corfield's quest for a philosophy of real
mathematics in "week198". He's one of the few philosophers who
understands enough math to realize how cool n-categories are -
which may explain why he's having trouble getting a job. On the
morning of the fourth day, he gave a talk on the impact
n-categories could have in philosophy:
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13) David Corfield, n-Category theory as a catalyst for change in
philosophy. Notes available at
"http://www.ima.umn.edu/categories/#fri"
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Later that day, Bertrand Toen explained Segal categories, which are
another popular approach to weak categories:
~
14) Bertrand Toen, Segal categories. Notes by Joachim Kock
available at "http://www.ima.umn.edu/categories/#fri"
~
After dinner, he spoke about n-stacks and n-gerbes:
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15) Bertrand Toen, n-Stacks and n-gerbes: algebraic geometry. Notes
by Joachim Kock available at
"http://www.ima.umn.edu/categories/#fri"
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Everyone slept all weekend long. Then on Monday of the second week,
the homotopy theorist Zbigniew Fiedorowicz spoke about his work on
a kind of n-fold monoidal category that has an n-fold loop space as
its nerve. He has some good papers on the web about this, too:
~
16) Zbigniew Fiedorowicz, n-Fold categories. Notes available at
"http://www.ima.umn.edu/categories/#mon2"
~
C.Balteanu, Z.Fiedorowicz, R.Schwaenzl and R.Vogt, Iterated
monoidal categories, available at
"http://www.arXiv.org/abs/math.AT/9808082"
~
Z.Fiedorowicz, Constructions of E_{n} operads, available at
"http://www.arXiv.org/abs/math.AT/9808089".
~
Stefan Forcey continued this theme by discussing enrichment over
n-fold monoidal categories. He also has a number of papers about
this on the arXiv, of which I'll just mention one:
~
17) Stefan Forcey, Higher enrichment: n-fold Operads and enriched
n-categories, delooping and weakening. Notes available at
"http://www.ima.umn.edu/categories/#mon2"
~
Stefan Forcey, Enrichment over iterated monoidal categories,
Algebraic and Geometric Topology, 4 (2004), 95-119, available
online at
"http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-7.abs.html" Also
available as "http://www.arXiv.org/abs/math.CT/0403152".
~
After dinner we discussed how to relate different definitions of
weak n-category.
~
On Tuesday of the second week, the logician Michael Makkai
presented his astounding project of redoing logic in a way that
completely eliminates the concept of "equality". This forces you to
do all of mathematics using weak infinity-categories. I thought
this stuff was great, in part because I finally understood it, and
in part because it leads naturally to the "opetopic" definition of
n-categories that James Dolan and I introduced. The idea of
eliminating equality was very much on our mind in inventing this
definition, but we didn't create a system of logic that
systematizes this idea.
~
There are no notes for Makkai's talk online, but you can get a lot
of good stuff from his website, including:
~
18) Michael Makkai, On comparing definitions of weak n-category,
available at "http://www.math.mcgill.ca/makkai/"
~
and this more technical paper which works out the details of his
vision:
~
19) Michael Makkai, The multitopic omega-category of all multitopic
omega-categories, available at "http://www.math.mcgill.ca/makkai/"
~
After Makkai's talk, Mark Weber spoke on n-categorical
generalizations of the concept of "monad", which is a nice way of
describing mathematical gadgets. There are no notes for this talk,
but his work on higher operads is at least morally related:
~
20) Mark Weber, Operads within monoidal pseudo algebras, available
as "http://www.arXiv.org/abs/math.CT/0410230".
~
Again, after dinner we talked about how to relate different
definitions of weak n-category.
~
On Wednesday of the second week, Michael Batanin spoke about his
recent work relating n-categories to n-fold loop spaces. Again no
notes, but you can read these papers:
~
21) Michael Batanin, The Eckmann-Hilton argument, higher operads
and E_{n}-spaces, available at
"http://www.ics.mq.edu.au/~mbatanin/papers.html"
~
Michael Batanin, The combinatorics of iterated loop spaces,
available at "http://www.ics.mq.edu.au/~mbatanin/papers.html"
~
Then Joachim Kock laid the ground for a discussion of n-categories
and topological quantum field theories, or "TQFTs", by explaining
the definition of a TQFT and the classification of 2d TQFTs:
~
22) Joachim Kock, Topological quantum field theory primer. Notes
available at "http://www.ima.umn.edu/categories/#wed2"
~
In the evening, Marco Mackaay and I said more about the relation
between TQFTs and n-categories:
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23) Marco Mackaay, Topological quantum field theories. Notes
available at "http://www.ima.umn.edu/categories/#wed2"
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24) John Baez, Space and state, spacetime and process. Notes
available at "http://www.ima.umn.edu/categories/#wed2"
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On Thursday, Ross Street started the day in a pleasantly different
way - he gave a historical account of work on categories and
n-categories in Australia! Australia is home to much of the best
work on these subjects, so if you can understand his history you'll
wind up understanding these subjects pretty well:
~
25) Ross Street, An Australian conspectus of higher category
theory. Notes available at
"http://www.ima.umn.edu/categories/#thur2"
~
As a younger exponent of the Australian tradition, it was then
nicely appropriate for Steve Lack to speak about ways of building a
model category of 2-categories:
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26) Steve Lack, Higher model categories. Notes available at
"http://www.ima.umn.edu/categories/#thur2"
~
In the afternoon we had a blast of computer science. First John
Power gave a hilarious talk phrased in terms of how one should
convince computer theorists to embrace categories, then
2-categories, and then maybe higher categories:
~
27) John Power, Why tricategories? Notes available at
"http://www.ima.umn.edu/categories/#thur2"
~
I spoke about Power's paper with this title back in "week53"; now
you can get it online!
~
Then Phillipe Gaucher, Lisbeth Falstrup and Eric Goubault spoke
about higher-dimensional automata and directed homotopy theory:
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28) Phillipe Gaucher, Towards a homotopy theory of higher
dimensional automata. Notes available at
"http://www.ima.umn.edu/categories/#thur2"
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Lisbeth Falstrup, More on directed topology and concurrency, Notes
available at "http://www.ima.umn.edu/categories/#thur2"
~
Eric Goubault, Directed homotopy theory and higher-dimensional
automata, Notes available at
"http://www.ima.umn.edu/categories/#thur2"
~
On Friday, Martin Hyland and Tony Elmendorf gave a double-header
talk on higher-dimensional linear algebra and how some concepts in
this subject can be simplified using symmetric multicategories.
There are, alas, no notes for this talk. You just had to be there.
~
Finally, my student Alissa Crans gave a talk on higher-dimensional
linear algebra, with an emphasis on categorified Lie algebras:
~
29) Alissa Crans, Higher linear algebra. Notes available at Notes
available at "http://www.ima.umn.edu/categories/#fri2"
~
Hers was the last talk in the workshop! I would like to say more
about it, but I'm exhausted... and her talk fits naturally into a
discussion of "higher gauge theory", which deserves a Week of its
own.
~
By the way, you can see pictures of this workshop here:
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30) John Baez, IMA, "http://math.ucr.edu/home/baez/IMA/"
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If you want to see what these crazy n-category people look like,
you can see most of them here.
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Hmm. If you wanted me to actually explain something this week, I'm
afraid you'll be rather disappointed - so far everything has just
been pointers to other material.
~
Luckily, while I was at this workshop I wrote a little explanation
of some material on Picard groups and Brauer groups. There's a
Spanish school of higher-dimensional algebra, centered in Granada,
and this spring Aurora del Rio Cabeza came from Granada to visit
UCR. She and James Dolan spent a lot of time talking about
categorical groups (also known as "2-groups") and cohomology
theory. I was, alas, too busy to keep up with their conversations,
but I learned a little from listening in... and here's my writeup!
~
Higher categories show up quite naturally in the study of
commutative rings and associative algebras over commutative rings.
I'd heard of things called "Brauer groups" and "Picard groups" of
rings, and something called "Morita equivalence", but I only
understood how these fit together when I learned they were part of
a marvelous thing: a weak 3-groupoid!
~
Here's how it goes. You don't need to know much about higher
categories for this to make some sense... at least, I hope not.
~
Starting with a commutative ring R, we can form a weak 2-category
Alg(R) where:
~
an object A is an associative algebra over R, a 1-morphism M: A ->
B is an (A,B)-bimodule, a 2-morphism f: M -> N is a homomorphism
between (A,B)-bimodules.
~
This has all the structure you need to get a 2-category. In
particular, we can "compose" an (A,B)-bimodule and a (B,C)-bimodule
by tensoring them over B, getting an (A,C) bimodule. But since
tensor products are only associative up to isomorphism, we only get
a weak 2-category, not a strict one.
~
This weak 2-category has a tensor product, since we can tensor two
associative algebras over R and get another one. All the stuff
listed above gets along with this process! When an n-category has a
well-behaved tensor product we call it "monoidal", so Alg(R) is a
weak monoidal 2-category. But using a standard trick we can
reinterpret this as a weak 3-category with one object, as follows:
~
there's only one object, R a 1-morphism A: R -> R is an associative
algebra over R a 2-morphism M: A -> B is an (A,B)-bimodule a
3-morphism f: M -> N is a homomorphism between (A,B)-bimodules.
~
Note how all the morphisms have shifted up a notch. What used to be
called objects, the associative algebras over R, are now called
1-morphisms. We "compose" them by tensoring them over R.
~
Next, recall a bit of n-category theory from "week35". In an
n-category we define a j-morphism to be an "equivalence" iff it's
invertible... up to equivalence! This definition may sound
circular, but really just recursive. To start it off we just need
to add that an n-morphism is an equivalence iff it's invertible.
~
What does equivalence amount to in the 3-category Alg(R)? It's
easiest to figure this out from the top down:
~
A 3-morphism f: M -> N is an equivalence iff it's invertible, so
it's an isomorphism between (A,B)-bimodules.
~
A 2-morphism M: A -> B is an equivalence iff it's invertible up to
isomorphism, meaning there exists N: B -> A such that:
~
M tensor_{B} N is isomorphic to A as an (A,A)-bimodule, N
tensor_{A} M is isomorphic to B as a (B,B)-bimodule.
~
In this situation people say M is a "Morita equivalence" from A to
B.
~
A 1-morphism A: R -> R is an equivalence iff it's invertible up to
Morita equivalence, meaning there exists a 1-morphism B: x -> x
such that:
~
A tensor_{R} B is Morita equivalent to R as an associative algebra
over R, B tensor_{R} A is Morita equivalent to R as an associative
algebra over R.
~
In this situation people say A is an "Azumaya algebra".
~
Here's a nice example of how Morita equivalence works. Over any
commutative ring R there's an algebra R[n] consisting of n x n
matrices with entries in R. R[n] isn't usually isomorphic to R[m],
but they're always Morita equivalent! To see this, suppose
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M: R[n] -> R[m] is the space of n x m matrices with entries in R,
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N: R[m] -> R[n] is the space of m x n matrices with entries in R.
~
These become bimodules in an obvious way via matrix multiplication,
and a little calculation shows that they're inverses up to
isomorphism!
~
So, all the algebras R[n] are Morita equivalent. In particular this
means that they're all Morita equivalent to R, so they are Azumaya
algebras of a rather trivial sort.
~
If we take R to be the real numbers there is also a more
interesting Azumaya algebra over R, namely the quaternions H. This
follows from the fact that
~
H tensor_{R} H = R[4]
~
This says H tensor_{R} H is Morita equivalent to R as an
associative algebra over R, which implies (by the definition above)
that H is an Azumaya algebra.
~
Morita equivalence is really important in the theory of
C*-algebras, Clifford algebras, and things like that. Someday I
want to explain how it's connected to Bott periodicity! Oh, there's
so much I want to explain....
~
But right now I want to take our 3-category Alg(R), massage it a
bit, and turn it into a topological space! Then I'll look at the
homotopy groups of this space and see what they have to say about
our ring R.
~
To do this, we need a bit more n-category theory. A weak n-category
where all the 1-morphisms, 2-morphisms and so on are equivalences
is called a "n-groupoid". For example, given any weak n-category,
we can form a weak n-groupoid called its "core" by throwing out all
the morphisms that aren't equivalences.
~
So, let's take the core of Alg(R) and get a weak 3-groupoid. Here's
what it's like:
~
There's one object, R. The 1-morphisms A: x -> x are Azumaya
algebras over R. The 2-morphisms M: A -> B are Morita equivalences.
The 3-morphisms f: M -> N are bimodule isomorphisms.
~
Next, given a weak n-groupoid with one object, it's very nice to
compute its "homotopy groups". These are easy to define in general,
but I'll just do it for the core of Alg(R) and let you guess the
general pattern. First, notice that:
~
The identity 1-morphism 1_{R}: R -> R is just R, regarded as an
associative algebra over itself in the obvious way.
~
The identity 2-morphism 1_{1_{R}}: 1_{R} -> 1_{R} is just R,
regarded as an (R,R)-bimodule in the obvious way.
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The identity 3-morphism 1_{1_{1_{R}}}: 1_{1_{R}} -> 1_{1_{R}} is
just the identity function on R, regarded as an isomorphism of
(R,R)-bimodules.
~
At this point we let out a cackle of n-categorical glee. Then, we
define the homotopy groups of the core of Alg(R) as follows:
~
The 1st homotopy group consists of equivalence classes of
1-morphisms from R to itself.
~
The 2nd homotopy group consists of equivalence classes of
2-morphisms from 1_{R} to itself
~
The 3rd homotopy group consists of equivalence classes of
3-morphisms from 1_{1_{R}} to itself.
~
Here we say two morphisms in an n-category are "equivalent" if
there is an equivalence from one to the other (or if they're equal,
in the case of n-morphisms).
~
I hope the pattern in this definition of homotopy groups is
obvious. In fact, n-groupoids are secretly "the same" - in a subtle
sense I'd rather not explain - as spaces whose homotopy groups
vanish above dimension n. Using this, the homotopy groups as
defined above turn out to be same as the homotopy groups of a
certain space associated with the ring R! So, we're doing something
very funny: we're using algebraic topology to study algebra.
~
But, we don't need to know this to figure out what these homotopy
groups are like. Unraveling the definitions a bit, one sees they
amount to this:
~
The 1st homotopy group consists of Morita equivalence classes of
Azumaya algebras over R. This is also called the BRAUER GROUP of R.
~
The 2nd homotopy group consists of isomorphism classes of Morita
equivalences from R to R. This is also called the PICARD GROUP of R.
~
The 3rd homotopy group consists of invertible elements of R. This
is also called the UNIT GROUP of R.
~
People had been quite happily studying these groups for a long time
without knowing they were the homotopy groups of the core of a weak
3-category associated to the commutative ring R! But, the
relationships between these groups are easier to explain if you use
the n-categorical picture. It's a great example of how n-categories
unify mathematics.
~
For example, everything we've done is functorial. So, if you have a
homomorphism between commutative rings, say
~
f: R -> S
~
then you get a weak 3-functor
~
Alg(f): Alg(R) -> Alg(S)
~
This gives a weak 3-functor from the core of Alg(R) to the core of
Alg(S), and thus a map between spaces... which in turn gives a long
exact sequence of homotopy groups! So, we get interesting maps
going from the unit, Picard and groups of R to those of S - and
these fit into an interesting long exact sequence.
~
For more, try the following papers. The first paper is actually
about a generalization of Azumaya algebras called "Azumaya
categories", but it starts with a nice quick review of Azumaya
algebras and Brauer groups:
~
31) Francis Borceux and Enrico Vitale, Azumaya categories,
available at "http://www.math.ucl.ac.be/AGEL/Azumaya_categories.pdf"
~
Category theorists will enjoy the generalization: since algebras
are just one-object categories enriched over Vect, the concept of
Azumaya algebra really wants to generalize to that of an Azumaya
category. I'm sure most of the Brauer-Picard-Morita stuff
generalizes too, but I haven't checked that out yet.
~
This second paper makes the connection between Picard and Brauer
groups explicit using categorical groups:
~
32) Enrico Vitale, A Picard-Brauer exact sequence of categorical
groups, Journal of Pure and Applied Algebra 175 (2002) 383-408.
Also available as
"http://www.math.ucl.ac.be/membres/vitale/cat-gruppi2.pdf"
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2004 John Baez
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baez@math.removethis.ucr.andthis.edu home