Also available at http://math.ucr.edu/home/baez/week209.html
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:
1) n-Categories: Foundations and Applications,
http://www.ima.umn.edu/categories/
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:
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.
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
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:
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: 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":
5) Andre Joyal, Peter May and Timothy Porter, Weak categories.
Notes available at http://www.ima.umn.edu/categories/#tues
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.
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:
7) Peter May, Model categories. Notes available at
http://www.ima.umn.edu/categories/#wed
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
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.
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:
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
It relies on some work by Dominic Verity which has finally been
written up after many years of unpublished limbo:
11) Dominic Verity, Complicial sets, available as 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
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:
13) David Corfield, n-Category theory as a catalyst for change in
philosophy. Notes available at http://www.ima.umn.edu/categories/#fri
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:
15) Bertrand Toen, n-Stacks and n-gerbes: algebraic geometry.
Notes by Joachim Kock available at http://www.ima.umn.edu/categories/#fri
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 math.AT/9808082.
Z. Fiedorowicz, Constructions of E_n operads, available at 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 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
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:
23) Marco Mackaay, Topological quantum field theories.
Notes available at http://www.ima.umn.edu/categories/#wed2
24) John Baez, Space and state, spacetime and process.
Notes available at http://www.ima.umn.edu/categories/#wed2
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:
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 Philippe Gaucher, Lisbeth Fajstrup and Eric Goubault spoke about
higher-dimensional automata and directed homotopy theory:
28) Philippe Gaucher, Towards a homotopy theory of higher dimensional
automata. Notes available at http://www.ima.umn.edu/categories/#thur2
Lisbeth Fajstrup, 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:
30) John Baez, IMA, http://math.ucr.edu/home/baez/IMA/
If you want to see what these crazy n-category people look like,
you can see most of them here.
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
M: R[n] -> R[m] is the space of n x m matrices with entries in R,
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 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.
Since as a groupoid with one object is a group, this weak 3-groupoid with
one object deserves to be called a "3-group".
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
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
-----------------------------------------------------------------------
Addendum: it turns out that the Picard-Brauer 3-group has a long
and illustrious history. Ross Street explained this to me:
Dear John
It is great that you jumped in and started writing that report on the
Minneapolis meeting. "A journey of a thousand miles . . . ".
[Carrying on the IMA Russian spirit, I just got back from
Christchurch NZ where I gave 11 hours (in 2 days) of lectures on
topos theory to a very patient group of physicists, philosophers,
mathematicians, and even one economist.]
It is also great that you promoted the work of the Granada School.
That subject is particularly close to my heart. So here goes another
personal history. Probably back at Tulane U in 1969-70, Jack Duskin
(who was a great source of inspiration to me and, I believe, to the
Granada School) would have pointed me to the papers
32) Grothendieck, Alexander Le groupe de Brauer. III. Exemples et
complements. (French) 1968 Dix Exposes sur la Cohomologie des Sch�mas
pp. 88--188 North-Holland, Amsterdam; Masson, Paris
33) Grothendieck, Alexander Le groupe de Brauer. II. Th�orie
cohomologique. (French) 1968 Dix Expos�s sur la Cohomologie des
Schemas pp. 67--87 North-Holland, Amsterdam; Masson, Paris
34) Grothendieck, Alexander Le groupe de Brauer. I. Algebres d'Azumaya et
interpretations diverses. (French) 1968 Dix Exposes sur la
Cohomologie des Sch�mas pp. 46-66 North-Holland, Amsterdam; Masson,
Paris
pushing the Brauer group concept of ring theorists (e.g. Azumaya)
into the scheme view of algebraic geometry. I later read papers by
category theorists, like
35) Lindner, Harald, Morita equivalences of enriched categories.
Conferences du Colloque sur l'Algebre des Categories (Amiens, 1973),
III. Cahiers Topologie Geom. Differentielle 15 (1974), no. 4,
377-397, 449-450.
36) Fisher-Palmquist, J.; Palmquist, P. H. Morita contexts of enriched
categories. Proc. Amer. Math. Soc. 50 (1975), 55--60.
which seemed to be the beginning of a simpler understanding. Somehow
(?) I obtained an original bound reprint of
37) Froehlich, A.; Wall, C. T. C. Graded monoidal categories. Compositio
Math. 28 (1974), 229-285.
which I have just looked at and realised I should read again (since
Turaev and Mueger have been using G-graded categories to understand
the G-equivariant version of Turaev's 3-manifold invariant work).
It was forerunner to
38) Froehlich, A.; Wall, C. T. C. Equivariant Brauer groups. Quadratic
forms and their applications (Dublin, 1999), 57-71, Contemp. Math.,
272, Amer. Math. Soc., Providence, RI, 2000.
On my sabbatical at Wesleyan University (Middletown CT) in 1976-77, I
joined in the algebraists workshop on SLNM 181 on separable algebras
over commutative rings which was trying to do some of Grothendieck's
stuff without the cohomology and alg geom. Joyal taught me a bit
about Brauer too, motivating to some extent the work I did on stacks.
Anyway, out of all this, other stuff I've forgotten, and the
experience in module theory for enriched categories, it became clear
that Morita contexts were a bit silly and adjunctions of (bi)modules
were probably better and less ad hoc. The beginning point should be a
particular monoidal bicategory Alg(R-Mod) based on a commutative ring
R: objects are R-algebras, morphisms are bimodules, 2-cells are
module morphisms. The group of units, Picard group and Brauer group
all sat happily in there as homotopy groups of the monoidal
bicategory.
> 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!
After beginning the work with Joyal on braided monoidal categories
and learning of his work with Tierney on homotopy 3-types, I spoke at
the homotopy meeting in Bangor in 1986(?) on this monoidal bicategory
Alg(R-Mod) as a fundamental example. (It is discussed much later in
the last part of
39) R. Gordon, A.J. Power and R. Street, Coherence for tricategories,
Memoirs of the American Math. Society 117 (1995) Number 558.)
At the 1987 Meeting in Louvain-La-Neuve, Duskin (who loves simplicial
sets) found a simplicial set whose only non-trivial homotopy groups
were the three in question:
40) Duskin, John W. The Azumaya complex of a commutative ring.
Categorical algebra and its applications (Louvain-La-Neuve, 1987),
107-117, Lecture Notes in Math., 1348, Springer, Berlin, 1988.
I pointed out to Jack that this was the nerve of Alg(R-Mod) and he
included a remark about that in the published version. Also see
41) Duskin, J. An outline of a theory of higher-dimensional descent.
Actes du Colloque en l'Honneur du Soixantieme Anniversaire de Rene
Lavendhomme (Louvain-la-Neuve, 1989). Bull. Soc. Math. Belg. S�r. A
41 (1989), no. 2, 249-277.
The Brauer group section of
42) Categorical and combinatorial aspects of descent theory, Applied
Categorical Structures (to appear; March 2003 preprint available at
math.CT/0303175).
gives some more on this.
The article
43) K. K. Ulbrich, Group cohomology for Picard categories, J. Algebra 91
(1984) 464-498.
should also be mentioned. It is a great, to use your term,
"categorification" of usual cohomology with abelian group
coefficients: one step towards the grander goal of coefficients in a
general weak n-category.
The Spanish School (and the Belgian School) is continuing with nice
work in this area. For example there is the recent paper by
Carrasco/Martinez-Moreno. Here is the review I wrote yesterday.
-----------------------------------------------------------------
Carrasco/Martinez-Moreno: Simplicial cohomology with coefficients in
symmetric categorical groups
The full cohomology theory of simplicial sets with coefficients in a
general weak n-category is a long-term goal. The classical
cohomology revolves around the fact that an abelian group A can be
regarded as an n-category whose simplicial nerve is the
combinatorial Eilenberg-Mac Lane space K(A,n). Following Takeuchi
and Ulbrich [J. Pure Appl. Algebra 27 (1983) 61--73; MR84g:18025] and
Ulbrich [J. Algebra 91 (1984) 464--498; MR86h:18003], the present
authors develop cohomology where the coefficient object is a
symmetric categorical group A. In this important case too, A can be
regarded as a weak n-category whose simplicial nerve is here denoted
by K(A,n); it has non-vanishing homotopy groups only in dimensions n
and n+1, and represents the cohomology of simplicial sets in the
homotopy category. This functor K(-,n) essentially has a left-adjoint
left-inverse P_n so that homotopy classes of simplicial maps from
X to Y are classified by the cohomology of X with coefficients in P_n(Y).
----------------------------------------
Back to marking papers.
Best wishes,
Ross
This last paper is:
44) P. Carrasco and J. Martinez-Moreno, Simplicial cohomology with
coefficients in symmetric categorical groups, Applied Categorical
Structures 12 (2004), 257-286.
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