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Near the end of April I was invited by Ronnie Brown to Bangor, Wales for a very exciting get-together. Readers of "This Week's Finds" will know I'm interested in n-categories and higher-dimensional algebra these days. Brown is the originator of the term "higher-dimensional algebra" and has been sort of a prophet of the subject for many years. Tim Porter at Bangor also works on this subject; I'll try to say a bit more about his stuff next week. And visiting Bangor at the time were John Power and Ross Street, two category theorists who do a lot of work on n-categories. So I had a chance to learn some more higher-dimensional algebra and category theory and see what these folks thought of my crazy ideas.
1) Ronald Brown, Out of line, Royal Institution Proceedings 64, 207-243.
Brown is very interested in explaining mathematics to the public, and this paper is based on a talk he gave to a general audience. It is a very accessible introduction to what mathematics is really all about, and what higher-dimensional algebra is about in particular. "Out of line" is a pun, of course, because not only does higher-dimensional algebra seek to burst free of certain habits of "linear thinking" that tend to go along with symbol string manipulation, it also has been a bit outside the mainstream of mathematics until recently.
Now, when I speak of "linear thinking" I am not indulging in some vague new-agey complaint against rationality. I mean something very precise: the tendency to focus ones energy on operations that are easily modelled by the juxtaposition of symbols in a line. The primordial example is addition: we have a bunch of sticks in a row:
|||||
and we say there are "5" sticks and write
1+1+1+1+1 = 5.
Fine. But when we have a 2-dimensional array of sticks:
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we are in a hurry to bring the situation to linear form by making up a new operation, "multiplication", and saying we have 2 x 4 sticks. This isn't so bad for plenty of purposes; it's not as if I'm against times tables! But certain things, particular in topology, can get obscured by neglecting operations that correspond most naturally to higher-dimensional forms of juxtaposition, and Brown's paper explains some of these, and how to deal with these problems. The point is not to avoid linear notation; it's to avoid falling into certain mental traps it can lead you into if you're not careful!
2) A. J. Power, Why tricategories?, preprint available as ECS-LFCS-94-289 from Laboratory for Foundations of Computer Science, University of Edinburgh. Also available at http://www.ima.umn.edu/talks/workshops/SP6.7-18.04/power/power.pdf
When I mentioned this paper to a friend, she puzzledly asked "`Why try categories?'?" And indeed, one must have tried categories and enjoyed them before moving on to bicategories, tricategories and that great beckoning terra incognita of mathematics, n-category theory.
In a sense I already know "why tricategories". I think they're great, and in a paper with James Dolan --- summarized in "week49" --- I did my best to get everyone else interested in general n-categories. For me, the great thing about n-category theory is that it strives to formalize the notion of "process" in all its recursive splendor. An n-category is a mathematical structure containing not only objects, which one might think of as "things", and morphisms, which one might think of as "processes leading from one thing to another", but also 2-morphisms, which are "processes leading from one process to another", and then 3-morphisms, etc., on up to n-morphisms.
In physics and topology applications, the k-morphisms can often be thought of as k-dimensional geometrical objects, since (as the Greeks knew) the process of motion of a point traces out a 1-dimensional figure, and similarly the motion of a 1-dimensional figure traces out a 2-dimensional surface... and n-dimensional spacetime is in some rough sense "traced out" by the motion of (n-1)-dimensional spacelike slices through time. If you think this is vague, you're right --- and that's why one needs n-category theory, to make it precise! When one understands n-categories (which so far we really do only up to n = 3) one sees that the possibilities inherent in n-dimensional topology match up very nicely with one might have stumbled on not knowing topology at all, but just playing around with this iterated notion of processes between processes between processes... This "natural correspondence" between purely algebraic concepts and topological ones is what makes topological quantum field theory tick, and I can't help but feel that it will have quite a bit to say about physics eventually.
Power, however, gives a quite different set of reasons for being interested in tricategories. These are drawn from computer science and logic, and they make me realize yet again how poor and outdated my education in logic was, and how much interesting stuff there is going on in the subject!
At a completely naive level, one might already expect that relation between "processes" and "things" is subtle and interesting in computer science. For after all, we can think of a program either as a process that turns one thing into another, or as data, a sort of thing, which other programs can act on. Power does not really emphasize this issue explicitly, but I can't help remarking on it, especially because it reminds me of the curious fact that in mathematical physics, "time is the last dimension".
That is, in topological quantum field theory, the n-morphisms in an n-category, which are the processes having no further processes going between them, represent the passage of time. And indeed, for many practical purposes it appears that n = 4 is where things leave off, since spacetime appears 4-dimensional. On the other hand, nobody knows any mathematical reason why one has to stop at any given n. So ultimately we should try to understand "ω-categories", having n-morphisms for all n greater than or equal to zero (0-morphisms being simply objects, and 1-morphisms being morphisms). This corresponds philosophically to the notion that every "process" can also be regarded as a "thing" which other processes can transform. Moreover, we should also try to understand "Z-categories", having n-morphisms for all integers n, even negative ones! In this world, where there is no bottom as well as no top, every "thing" can also be regarded as a "process".
But I digress. Power is actually more interested in a different (though perhaps related) hierarchy. Sometimes people like to say computers just do stuff with bunches of numbers, but that's pretty misleading. Of course computers can do things with numbers, but that's one of the simpler mathematical things they can do. A number is an element of a set (the set of real numbers, or some set of more computer-manageable numbers.) And computers have no problem dealing with elements of sets. But computers can also deal with sets themselves --- and more fancy mathematical objects.
Many mathematical objects are sets, or bunches of sets, equipped with operations satisfying equational laws. For example, a group is a set equipped with a product and inverse operation satisfying various laws. Sometimes these operations are only defined if certain conditions hold, of course. For example, a category is a set of "objects" and a set of "morphisms", together with various operations like composition of morphisms, but one i can only compose two morphisms f: x → y and g: w → z if y = w. Other examples might include graphs, partially ordered sets... and all sorts of things computer scientists know and love.
We could call all of these "sets with essentially algebraic structure." Mathematically sophisticated computer scientists want to be able define data types corresponding to arbitrary sorts of sets with essentially algebraic structure, and to play around with them easily. So they need to ponder such things in considerable generality.
Note that in all cases, there is not just a bunch of objects to play with --- like "groups" or "partially ordered sets" --- but a category in which the morphisms are structure-preserving maps between the objects in question. For example, there is a category Grp whose objects are groups and whose morphisms are group homomorphisms.
The categories one gets this way are of a certain sort. Power calls them "categories of models of finite limit theories". To define this requires a bit of know-how, but it's basically simple. For example, suppose I were trying to explain the definition of a category to a computer scientist; I might say, every category has a set ob of objects and a set mor of morphisms; every morphism has an object called its source (or domain), so there is a function
source: mor → ob
and similarly every morphism has an object called its target (or codomain) so there is a function
target: mor → ob
Now, we can compose a morphism f and a morphism g to get fg if target(f) = source(g), so we have a composition function
composition: C → mor
defined only on the subset C of mor x mor that is the biggest subset making the following diagram commute:
C ------p1----> mor
| | p1(f,g) = f
p2 target p2(f,g) = g
| |
V V
mor ----source---> ob
Now category theorists have a slick way of dealing with these functions defined only a subset satisfying equational conditions; instead of talking about the "biggest subset" C they would say that C is the "limit" of the diagram
mor
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target
|
V
mor -source---> ob
If you don't get this, don't worry; in a sense it's just another way of talking about the same thing, with the advantage of being infinitely more general, since one can talk about the limit of any diagram, though here we will only need limits of finite diagrams.
Then, after having lined up these ingredients (and I have left some out!), I could go ahead and say what equational laws they need to satisfy, like associativity of composition; and if I wanted I could write all these laws out using commutative diagrams, too! Then I would have laid out the "theory of categories" --- a complete description of the operations in a category and the laws they obey.
The theory of categories is a typical example of a "finite limit theory", because what I really did above, in describing the "theory of categories", is describe a CATEGORY, say Th, having objects called ob and mor, and morphisms called source, target, composition, and so on, such that various diagrams commute! Moreover, we should think of Th as a category with all finite limits, that is, one in which all finite diagrams have limits. That allows us to deal with things like the object C above, which are defined as limits of finite diagrams.
So we have this thing Th, the "theory of categories". And then, any particular category is a "model" of this theory Th. A "model" assigns to each object in Th a particular set --- for example, "mor" above gets assigned the set of morphisms in some particular category C --- and assigns to each morphism in Th a particular function --- for example, "composition" above gets assigned the function representing composition in C. Moreover, this assignment satisfies a bunch of utterly obvious consistency conditions which one summarizes by saying that a "model of the theory Th is a functor from Th to Set that preserves finite limits". In logic, you know, a model of a theory is something that assigns to each thingie in the theory an actual thingie, in such a way that all the stuff the theory SAYS is true about these thingies, IS true!
Now if you are with me thus far you either know this stuff better than I do, or else I congratulate you, because the example I picked was deliberately self-referential and confusing --- I was using category theory to describe the theory of categories, and also, the theory Th itself was a category! But the world of thought does have a funny way of wrapping back on itself like that... so it's good to get used to it.
In fact there is a big literature on "sets with essentially algebraic structure" and "categories of models of finite limit theories"... this is a branch of logic they never taught me about in school, but it definitely exists, and Power gives some references to it:
3) P. Gabriel and F. Ulmer, Lokal praesentierbare Kategorien, in Springer Lecture Notes in Math 221 (1971).
G. Kelly, Structures defined by finite limits in the enriched context I, Cahiers de Top. et. Geom. Diff. 23 (1982), 3-41.
Michael Makkai and Robert Pare, Accessible categories: the foundations of categorical model theory, in Contemp. Math. 104 (1989).
But let's dig in a bit further, since really the fun is just starting. Now, I told you what a model of one of these finite limit theories Th was, but not what a morphism between models is! Well, if a model is a sort of functor, a morphism between them should be a sort of natural transformation between functors; that's how it usually goes. So there is really a category Mod(Th) of models of one of these theories Th. If Th were the theory of categories as above, Mod(Th) would be the category of (small) categories, which is called Cat. To take a less fiendish example, if Th were the theory of groups, Mod(Th) would the category Grp.
But now suppose one wanted to build a computer language that could not only deal with all sorts of data types corresponding to different "sets with essentially algebraic structure", but also various "categories with essentially algebraic structure". For if one likes category theory well enough to do a lot of computer science using it, it makes sense to let the computer itself join the fun, by creating a language in which it's easy to talk about categories. After all, our eventual goal with computers is to have them completely replace computer scientists, right?
Well, in a way "categories with essentially algebraic structure" aren't terribly different from sets with essentially algebraic structure. Roughly, the idea is that instead of having a data type that consists of a bunch of sets with functions between them satisfying some equational laws, we have a data type consisting of a bunch of categories, functors between them, and natural transformations between THEM satisfying equational laws. What this means is that if we try to copy the above stuff, instead of a "theory" we will have a "2-theory" Th, which is some sort of 2-category, and then a model of this would be a 2-functor from Th to Cat. We want to wind up getting a 2-category Mod(Th) of models of Th.
But actually carrying this out is a bit tricky, and much of Power's paper goes into the details of various proposed schemes. Of course there is no reason in principle to stop here, other than our limited understanding of n-categories, sheer bewilderment, or boredom. Reasoning about n-categories always tends to drag in (n+1)-categories, because the collection of all n-categories with some particular structure (such as the "essentially algebraic structures" I've focussed on here, but also other sorts) typically forms an (n+1)-category. This is how Power motivates tricategories. Right now we are stuck at n = 3, but there are good reasons to expect that pretty soon we'll go beyond that. In fact, Power and Street showed me a sketch of their ideas on tetracategories....
© 1995 John Baez
baez@math.removethis.ucr.andthis.edu
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