n-categories: foundations and applications

I thought some of you might like this.... ....................................................................... 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 ones 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 Phillipe Gaucher, Lisbeth Falstrup and Eric Goubault spoke about higher-dimensional automata and directed homotopy theory: 28) Phillipe Gaucher, Towards a homotopy theory of higher dimensional automata. Notes available at http://www.ima.umn.edu/categories/#thur2 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: 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. 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 commuative 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 ----------------------------------------------------------------------- Previous issues of "This Week's Finds" and other expository articles on mathematics and physics, as well as some of my research papers, can be obtained at http://math.ucr.edu/home/baez/ For a table of contents of all the issues of This Week's Finds, try http://math.ucr.edu/home/baez/twf.html A simple jumping-off point to the old issues is available at http://math.ucr.edu/home/baez/twfshort.html If you just want the latest issue, go to http://math.ucr.edu/home/baez/this.week.html 23-Nov-2004 20:40:13 -0400,1305;000000000000-00000000