Happy New Year!
I'm making some changes in my life. For many years I've dreamt of writing a book on higher-dimensional algebra that will explain n-categories and their applications to homotopy theory, representation theory, quantum physics, combinatorics, logic - you name it! It's an intimidating goal, because every time I learn something new about these subjects I want to put it in this imaginary book, so it keeps getting longer and longer in my mind! Actually writing it will require heroic acts of pruning. But, I want to get started.
It'll be freely available online, and it'll show up here as it materializes - but so far I've just got a tentative outline:
1) John Baez, Higher-Dimensional Algebra, http://math.ucr.edu/home/baez/hda/
Unfortunately, I'm very busy these days. As you get older, duties accumulate like barnacles on a whale if you're not careful! When I started writing This Week's Finds a bit more than ten years ago, I was lonely and bored with plenty of time to spare. My life is very different now: I've got someone to live with, a house and a garden that seem to need constant attention, a gaggle of grad students, and too many invitations to give talks all over the place.
In short, the good news is I'm never bored and there's always something fun to do. The bad news is there's always TOO MUCH to do! So, a while ago I decided to shed some duties and make more time for things I consider really important: thinking, playing the piano, writing this book... and yes, writing This Week's Finds.
First I quit working for all the journals I helped edit. Then I started refusing most requests to referee articles. Both these are the sort of job it's really fun to quit. But doing so didn't free up nearly enough time.
So now I've also decided to stop moderating the newsgroup sci.physics.research - and stop posting so many articles there. This is painful, because I've learned so much from this newsgroup over the last 10 years, met so many interesting people, and had such fun. I thank everyone on the group. I'll miss you! I'll probably be back whenever I get lonely or bored.
Ahem. Before I get weepy and nostalgic, I should talk about some math.
This November in Florence there was a conference in honor of the 40th anniversary of Bill Lawvere's Ph.D. thesis - a famous thesis called "Functorial Semantics of Algebraic Theories", which explored the applications of category theory to algebra, logic and physics. There are videos of all the talks on the conference website:
2) Ramifications of Category Theory, http://ramcat.scform.unifi.it/
The conference was organized and funded by Michael Wright, a businessman with a great love of mathematics and philosophy, so it was appropriate that it was held in the old city of Cosimo de Medici, Renaissance banker and patron of scholars. And since there were talks both by mathematicians and philosophers - especially Alberto Peruzzi, a philosopher at the University of Florence who helped run the show - I couldn't help but remember Cosimo's "Platonic Academy", which spearheaded the rebirth of classical learning in Renaissance Italy. When not attending talks, I spent a lot of time roaming around twisty old streets, talking category theory at wonderful restaurants, reading The Rise and Fall of the House of Medici, and desperately trying to soak up the overabundance of incredible art and architecture: the Ponte Vecchio, the Piazza del Duomo, the Santa Croce where everyone from Galileo to Dante to Machiavelli is buried....
What was Lawvere's thesis about? It's never been published, so I've never read it - though I hear it's going to be. So, my impression of its contents comes from gossip, rumors and later research that refers to his work.
Lawvere started out as a student of Clifford Truesdell, working on "continuum mechanics", which is the very practical branch of field theory that deals with fluids, elastic bodies and the like. In the process, Lawvere got very interested in the foundations of physics, particularly the notions of "continuum" and "physical theory". Somehow he decided that only category theory could give him the tools to really make progress in understanding these notions. After all, this was the 1960s, and revolution was in the air. So, he somehow got himself sent to Columbia University to learn category theory from Sam Eilenberg, one of the two founders of the subject. He later wrote:
In my own education I was fortunate to have two teachers who used the term "foundations" in a common-sense way (rather than in the speculative way of the Bolzano-Frege-Peano-Russell tradition). This way is exemplified by their work in Foundations of Algebraic Topology, published in 1952 by Eilenberg (with Steenrod), and The Mechanical Foundations of Elasticity and Fluid Mechanics, published in the same year by Truesdell. The orientation of these works seemed to be "concentrate the essence of practice and in turn use the result to guide practice".It may seem like a big jump from the down-to-earth world of continuum mechanics to category theory, but to Lawvere the connection made perfect sense - and while I've always found his writings inpenetrable, after hearing him give four long lectures in Florence I think it makes sense to me too! Let's see if I can explain it.
Lawvere first observes that in the traditional approach to physical theories, there are two key players. First, there are "concrete particulars" - like specific ways for a violin string to oscillate, or specific ways for the planets to move around the sun. Second, there are "abstract generals": the physical laws that govern the motion of the violin string or the planets.
In traditional logic, an abstract general is called a "theory", while a concrete particular is called a "model" of this theory. A theory is usually presented by giving some mathematical language, some rules of deduction, and then some axioms. A model is typically some sort of map that sends everything in the theory to something in the world of sets and truth values, in such a way that all the axioms get mapped to "true".
Since theories involve playing around with symbols according to fixed rules, the study of theories is often called "syntax". Since the meaning of a theory is revealed when you look at its models, the study of models is called "semantics". The details vary a lot depending on what you want to do, and physicists rarely bother to formulate their theories axiomatically, but this general setup has been regarded as the ideal of rigor ever since the work of Bolzano, Frege, Peano and Russell around the turn of the 20th century.
And this is what Lawvere wanted to overthrow!
Actually, I'm sort of kidding. He didn't really want to "overthrow" this setup: he wanted to radically build on it. First, he wanted to free the notion of "model" from the chains of set theory. In other words, he wanted to consider models not just in the category of sets, but in other categories as well. And to do this, he wanted a new way of describing theories, which is less tied up in the nitty-gritty details of syntax.
To see what Lawvere did, we need to look at an example. But there are so many examples that first I should give you a vague sense of the range of examples.
You see, in logic there are many levels of what you might call "strength" or "expressive power", ranging from wimpy languages that don't let you say very much and deduction rules that don't let you prove very much, to ultra-powerful ones that let you do all sorts of marvelous things. Near the bottom of this hierarchy there's the "propositional calculus" where we only get to say things like
((P implies Q) and (not Q)) implies (not P)Further up there's the "first-order predicate calculus", where we get to say things like
for all x (for all y ((x = y and P(x)) implies P(y)))Even further up, there's the "second-order predicate calculus" where we get to quantify over predicates and say things like
for all x (for all y (for all P (P(x) iff P(y)) implies x = y))Etcetera...
And, while you might think it's always best to use the most powerful form of logic you can afford, this turns out not to be true!
One reason is that the more powerful your logic is, the fewer categories can contain models of theories expressed in this logic. This point may sound esoteric, but the underlying principle should be familiar. Which is better: a hand-operated drill, an electric drill, or a drill press? A drill press is the most powerful. But I forgot to mention: you're using it to board up broken windows after a storm. You can't carry a drill press around, so now the electric drill sounds best. But another thing: this is in rural Ghana! With no electricity, now the hand-operated drill is your tool of choice.
In short, there's a tradeoff between power and flexibility. Specialized tools can be powerful, but they only operate in a limited context. These days we're all painfully aware of this from using computers: fancy software only works in a fancy environment!
Lawvere has even come up with a general theory of how this tradeoff works in mathematical logic... he called this the theory of "doctrines". But I'm getting way ahead of myself! He came up with "doctrines" in 1969, and I'm still trying to explain his 1963 thesis.
Just like traditional logic, Lawvere's new approach to logic has been studied at many different levels in the hierarchy of strength. He began fairly near the bottom, in a realm traditionally occupied by something called "universal algebra", developed by Garrett Birkhoff in 1935. The idea here was that a bunch of basic mathematical gadgets can be defined using very simple axioms that only involve n-ary operations on some set and equations between different ways of composing these operations. A theory like this is called an "algebraic theory". The axioms for an algebraic theory aren't even allowed to use words like "and", "or", "not" or "implies". Just equations.
Okay, now for an example.
A good example is the algebraic theory of "groups". A group is a set equipped with a binary operation called "multiplication", a unary operation called "inverse", and a nullary operation (that is, a constant) called the "unit", satisfying these equational laws:
(gh)k = g(hk) ASSOCIATIVITY 1g = g LEFT UNIT LAW g1 = g RIGHT UNIT LAW g-1 g = 1 LEFT INVERSE LAW gg-1 = 1 RIGHT INVERSE LAWSuch a primitive gadget is robust enough to survive in very rugged environments... it's more like a stone tool than a drill press!
Lawvere noticed that we can talk about models of these axioms not just in the category of sets, but in any "category with finite products". The point is that to talk about an n-ary operation, we just need to be able to take the product of an object G with itself n times and consider a morphism
f: G x ... x G → G |- n times -|For example, the category of smooth manifolds has finite products, so we can talk about a "group object" in this category, which is just a Lie group. The category of topological spaces has finite products, so we can talk about a group object in this category too: it's a topological group. And so on.
But Lawvere's really big idea was that there's a certain category with finite products whose only goal in life is to contain a group object. To build this category, first we put in an object
GSince our category has finite products this automatically means it gets objects 1, G, G x G, G x G x G, and so on. Next, we put in a binary operation called "multiplication", namely a morphism
m: G x G → GWe also put in a unary operation called "inverse":
inv: G → Gand a nullary operation called the "unit":
i: 1 → GAnd then we say a bunch of diagrams commute, which express all the axioms for a group listed above.
Lawvere calls this category the "theory of groups", Th(Grp). The object G is just like a group - but not any particular group, since its operations only satisfy those equations that hold in every group!
By calling this category a "theory", Lawvere is suggesting that like a theory of the traditional sort, it can have models - and indeed it can! A "model" of theory of groups in some category X with finite products is just a product-preserving functor
F: Th(Grp) → XBy the way things are set up, this gives us an object
F(G)in C, together with morphisms
F(m): F(G) x F(G) → F(G) F(inv): F(G) → F(G) F(i): F(1) → F(G)that serve as the multiplication, inverse and identity element for F(G)... all making a bunch of diagrams commute, that express the axioms for a group!
So, a model of the theory of groups in X is just a group object in X.
Whew. So far I've just explained the title of Lawvere's PhD thesis: "Functorial Semantics of Algebraic Theories". In Lawvere's approach, an "algebraic theory" is given not by writing down a list of axioms, but by specifying a category C with finite products. And the semantics of such theories is all about product-preserving functors F: C → X. Hence the term "functorial semantics".
Lawvere did a lot starting with these ideas. Let me just briefly summarize, and then move on to his work on topos theory and mathematical physics.
Wise mathematicians are interested not just in models, but also the homomorphisms between these. So, given an algebraic theory C, Lawvere defined its category of models in X, say Mod(C,X), to have product-preserving functors F: C → X as objects and natural transformations between these as morphisms. For example, taking C to be the theory of groups and X to be the category of sets, we get the usual category of groups:
Mod(Th(Grp),Set) = GrpThat's reassuring, and that's how it always works. What's less obvious, though, is that one can always recover C from Mod(C,Set) together with its forgetful functor to the category of sets.
In other words: not only can we get the models from the theory, but we can also get back the theory from its category of models!
I explained how this works in "week136" so I won't do so again here. This result actually generalizes an old theorem of Birkhoff on universal algebra. But fans of the Tannaka-Krein reconstruction theorem for quantum groups will recognize this duality between "theories and their category of models" as just another face of the duality between "algebras and their category of representations" - the classic example being the Fourier transform and inverse Fourier transform!
And this gives me an excuse to explain another bit of Lawvere's jargon: while a theory is an "abstract general", and particular model of it is a "concrete particular", he calls the category of all its models in some category a "concrete general". For example, Th(Grp) is an abstract general, and any particular group is a concrete particular, but Grp is a concrete general. I mention this mainly because Lawvere flings around this trio of terms quite a bit, and some people find them off-putting. There are lots of reasons to find his work daunting, but this need not be one.
In short, we have this kind of setup:
ABSTRACT GENERAL CONCRETE GENERAL theory models syntax semanticsand a precise duality between the two columns!
I would love to dig deeper in this direction - I've really just scratched the surface so far, and I'm afraid the experts will be disappointed... but I'm even more afraid that if I went further, the rest of you readers would drop like flies. So instead, let me say a bit about Lawvere's work on topos theory and physics.
Most practical physics makes use of logic that's considerably stronger than that of "algebraic theories", but still considerably weaker than what most of us have been brainwashed into accepting as our default setting, namely Zermelo-Fraenkel set theory with the axiom of choice. So if we want, we can do physics in a context less general than an arbitrary category with finite products, while still not restricting ourselves to the category of sets. This is where "topoi" come in - they're a lot like the category of sets, but vastly more general.
Topos theory was born when Grothendieck decided to completely rewrite algebraic geometry as part of a massive plan to prove the Weil conjectures. Grothendieck was another revolutionary of the early 1960s, and he arrived at his concept of "topos" sometime around 1962. In 1969-70, Lawvere and Myles Tierney took this concept - now called a "Grothendieck topos" - and made it both simpler and more general, arriving at the present definition. Briefly put, a topos is a category with finite limits, exponentials, and a subobject classifier. But instead of saying what these words mean, I'll just say that this lets you do most of what you normally want to do in mathematics, but without the law of excluded middle or the axiom of choice.
One of the many reasons this middle ground is so attractive is that it lets you do calculus with infinitesimals the way physicists enjoy doing it! Lawvere started doing this in 1967 - he called it "synthetic differential geometry". Basically, he cooked up some axioms on a topos that let you do calculus and differential geometry with infinitesimals. The most famous topos like this is the topos of "schemes" - algebraic geometers use this one a lot. The usual category of smooth manifolds is not even a topos, but there are topoi that can serve as a substitute, which have infinitesimals.
I won't list the axioms of synthetic differential geometry, but the main idea is that our topos needs to contain an object T called the "infinitesimal arrow". This is a rigorous version of those little arrows physicists like to draw when talking about vectors:
----->The usual problem with these "little arrows" is that they need to be really tiny, but still point somewhere. In other words, the head can't be at a finite distance from the tail - but they can't be at the same place, either! This seems like a paradox, but one can neatly sidestep it by dropping the law of excluded middle - or in technical jargon, working with a "non-Boolean topos".
That sounds like a drastic solution - a cure worse than the disease, perhaps! - but it's really not so bad. Indeed, algebraic geometers are perfectly comfortable with the topos of schemes, and they don't even raise an eyebrow over the fact that this topos is non-Boolean - mainly because you're allowed to use ordinary logic to reason about a topos, even if its internal logic is funny.
But enough logic! Let's do some geometry! Let's say we're in some topos with an infinitesimal arrow object, T. I'll call the objects of this topos "smooth spaces" and the morphisms "smooth maps". How does geometry work in here?
It's very nice. The first nice thing is that given any smooth space X, a "tangent vector in X" is just a smooth map
f: T → Xthat is, a way of drawing an infinitesimal arrow in X. In general, the maps from any object A of a topos to any other object B form an object called BA - this is part of what we mean when we say a topos has exponentials. So, the space of all tangent vectors in X is XT.
And this is what people usually call the "tangent bundle" of X!
So, the tangent bundle is pathetically simple in this setup: it's just a space of maps. This means we can compose a tangent vector f: T -> X with any smooth map g: X → Y to get a tangent vector gf: T -> Y. This is what people usually call "pushing forward tangent vectors". This trick gives a smooth map between tangent bundles, the "differential of g", which it makes sense to call
gT: XT → YTMoreover, it's pathetically easy to check the chain rule:
(gh)T = gT hTAnd so far we haven't used any axioms about the object T - just basic stuff about how maps work!
We can also define higher derivatives using T. For second derivatives we start with T x T, which looks like an "infinitesimal square". Then we mod out by the map
ST,T: T x T → T x Tthat switches the two factors. You should visualize this map as "reflection across the diagonal". When we mod out by it, we get a quotient space that deserves the name
T2/2!and if we now use some axioms about T, it turns out that a smooth map
f: T2/2! → Xpicks out what's called a "second-order jet" in X. This is a concept familiar from traditional geometry, but not as familiar as it should be. The information in a second-order jet consists of a point in X, the first derivative of a curve through X, and also the second derivative of a curve through X. Or in physics lingo: position, velocity and acceleration!
We can go ahead and define nth-order jets using Tn/n! in a perfectly analogous way, and the visual resemblance to Taylor's theorem is by no means an accident... but let me stick to second derivatives, since I'm trying to get to Newton's good old F = ma.
Just as the space of all tangent vectors in X is the tangent bundle XT, the space of all 2nd-order jets in X is the "2nd-order jet bundle"
XT2/2!There's a map called the "diagonal":
diag: T → T2/2!and composing this with any 2nd-order jet turns it into a tangent vector. This defines a smooth map
pX: XT2/2! → XTfrom the 2nd-order jet bundle to the tangent bundle. Intuitively you can think of this as sending any position-velocity-acceleration triple, say (q,q',q"), to the pair (q,q').
Now for the fun part: Lawvere defines a "dynamical law" to be a smooth map going the other way:
sX: XT → XT2/2!such that sX followed by pX is the identity. In other words, it's a way of mapping any position-velocity pair (q,q') to a triple (q,q',q"). So, it's a formula for acceleration in terms of position and velocity!
There is a category where an object is a smooth space equipped with a dynamical law and a morphism is a "lawful motion": that is, a smooth map
f: X → Ythat makes the obvious diagram commute:
sX XT -------------> XT2/2! | | | | | | fT | | fT2/2! | | | | | | V sY V YT -------------> YT2/2!In particular, if we take R to be the real numbers - "time" - and equip it with the law saying
q" = 0meaning that "time ticks at an unchanging rate", then a lawful motion
f: R → Xis precisely a trajectory in X that "follows the law", meaning that the acceleration of the trajectory is the desired function of position and velocity. This example is a setup for the classical mechanics of a point particle; it's easy to generalize to classical field theory by replacing R by a higher-dimensional space.
In fact, under some mild conditions this category whose objects are spaces equipped with dynamical law and whose morphisms are lawful motions is a topos! As Lawvere notes, "all the usual smooth dynamical systems, including the infinite-dimensional ones (elasticity, fluid mechanics, and Maxwellian electrodynamics) are included as special objects." This topos is an example of what Lawvere calls a "concrete general". Even better, there is also a corresponding "abstract general".
I'm sure many of you have the same impression that I had when seeing this stuff, namely that it's a bit quixotic for a high-powered mathematician to be reformulating the foundations of classical mechanics here at the turn of the 21st century, instead of working on something "cutting-edge" like string theory. Even if Lawvere's approach is better, one can't help but wonder if it gives truly new insights, or just a clearer formulation of existing ones. And either way, one can't help wonder: does he actually expect enough people to learn this stuff to make a difference? Does he really think topos theory can break the Microsoft-like grip that ordinary set theory has on mathematics?
(Note the software analogy raising its ugly head again. Zermelo-Fraenkel set theory is a bit like the Windows operating system: once you're locked into it, it's hard to imagine breaking out. You use it because everyone else does and you're too lazy to do anything about it. Topos theory is more like the "open source" movement: you're welcome and even expected to keep tinkering with the code.)
I have some sense of the answer to these questions. First of all, Lawvere wants to do math the right way regardless of whether it's popular. But secondly, he's been hard at work trying to make the subject accessible to beginners. He's recently written a couple of textbooks you don't need a degree in math to read:
3) F. William Lawvere and Steve Schanuel, Conceptual Mathematics: A First Introduction to Categories, Cambridge U. Press, Cambridge, 1997.
4) F. William Lawvere and Robert Rosebrugh, Sets for Mathematics, Cambridge U. Press, Cambridge, 2002.
And third, the great thing about topos theory is that you don't need to "accept it" to profit from it. In math, what really matters is not "believing the axioms" but coming up with good ideas. Topos theory is full of good ideas, and these are bound to propagate.
I'll finish off with some references to help you learn more about this stuff.
Alas, I believe Lawvere's thesis is still lurking in the stacks at Columbia University:
5) F. W. Lawvere, Functorial semantics of algebraic theories, Dissertation, Columbia University, 1963.
and so far he's only gotten around to publishing a brief summary:
6) F. William Lawvere, Functorial semantics of algebraic theories, Proceedings, National Academy of Sciences, U.S.A. 50 (1963), 869-872.
But, you can find expositions of his work on algebraic theories here and there. Here's a gentle one geared towards computer scientists:
7) Roy L. Crole, Categories for Types, Cambridge U. Press, Cambridge, 1993.
A considerably more macho one is available free online:
8) Michael Barr and Charles Wells, Toposes, Triples and Theories, Springer-Verlag, New York, 1983. Available for free electronically at http://www.cwru.edu/artsci/math/wells/pub/ttt.html
This book also talks about "sketches", which are a way of syntactically presenting a category with finite products. It also serves as an introduction to topoi... umm, or at least toposes. I used to find it fearsomely difficult and dry. Now I don't, which is sort of scary.
By the way, a "triple" is just another name for a monad.
A really beautiful more advanced treatment of algebraic theories and also "essentially algebraic theories" can be found here:
9) Maria Cristina Pedicchio, Algebraic Theories, in Textos de Matematica: School on Category Theory and Applications, Coimbra, July 13-17, 1999, pp. 101-159.
Someone should urge her to make this available online - it's already in TeX, and it deserves to be easier to get!
Shortly after his thesis, Lawvere tackled topoi in this paper:
10) F. William Lawvere, Elementary theory of the category of sets, Proceedings of the National Academy of Science 52 (1964), 1506-1511.
He then wrote a number of other papers on algebraic theories and the like:
11) F. William Lawvere, Algebraic theories, algebraic categories, and algebraic functors, in Theory of Models, North-Holland, Amsterdam (1965), 413-418.
12) F. William Lawvere, Functorial semantics of elementary theories, Journal of Symbolic Logic, Abstract, 31 (1966), 294-295.
13) F. William Lawvere, The category of categories as a foundation for mathematics, in La Jolla Conference on Categorical Algebra, Springer, Berlin 1966, pp. 1-20.
14) F. William Lawvere, Some algebraic problems in the context of functorial semantics of algebraic theories, in Reports of the Midwest Category Seminar, eds. Jean Benabou et al, Springer Lecture Notes in Mathematics No. 61, Springer, Berlin 1968, pp. 41-61.
Then came his work on "doctrines", which I vaguely alluded to a while back:
15) F. William Lawvere, Ordinal sums and equational doctrines, Springer Lecture Notes in Mathematics No. 80, Springer, Berlin, 1969, pp. 141-155.
Lawvere started publishing his ideas on mathematical physics in the late 1970s, though he must have been thinking about them all along:
16) F. William Lawvere, Categorical dynamics, in Proceedings of Aarhus May 1978 Open House on Topos Theoretic Methods in Geometry, Aarhus/Denmark (1979).
17) F. William Lawvere, Toward the description in a smooth topos of the dynamically possible motions and deformations of a continuous body, Cahiers de Topologie et Geometrie Differentielle Categorique 21 (1980), 337-392.
In 1981, Anders Kock came out with a textbook on synthetic differential geometry:
18) Anders Kock, Synthetic Differential Geometry, Cambridge U. Press, Cambridge, 1981.
More recently, Lawvere came out with a book on applications of category theory to physics:
19) F. William Lawvere and S. Schanuel, editors, Categories in Continuum Physics, Springer Lecture Notes in Mathematics No. 1174, Springer, Berlin, 1986.
The quote about Lawvere's teachers is from:
20) F. William Lawvere, Foundations and applications: axiomatization and education, Bulletin of Symbolic Logic 9 (2003), 213-224. Also available at http://www.math.ucla.edu/~asl/bsl/0902/0902-006.ps
and this gives a good overview of his ideas, though not easy to read! He also has some other papers online summarizing his ideas on differential geometry and physics:
21) F. William Lawvere, Outline of synthetic differential geometry, available at http://www.acsu.buffalo.edu/~wlawvere/downloadlist.html.
22) F. William Lawvere, Toposes of laws of motion, available at http://www.acsu.buffalo.edu/~wlawvere/downloadlist.html.
Finally, Colin McLarty - whom I was delighted to meet in Florence - has a nice quick introduction to synthetic differential geometry in his textbook on categories and topos theory:
23) Colin McLarty, Elementary Categories, Elementary Toposes, Clarendon Press, Oxford, 1995.
Along with Lawvere's books "Conceptual Mathematics" and "Sets for Mathematics", this is the one reference that's really good for beginners!
Okay... now that everyone is gone except the people who are absolutely nuts about category theory, let me say a bit more about doctrines and theory-model duality. The nuts who are still reading are probably disappointed that I kept everything very gentle and expository and didn't drop any mind-blowing bombshells of abstraction, which is what they like about category theory! So, let's turn up the abstraction a few notches.
What's a "doctrine"?
Well, in "week89" I described a "monad" in an arbitrary 2-category. But most of the time when people talk about monads they mean monads in Cat, the 2-category of all categories. These are the most important monads - but I've never really said what they're good for! I need to come clean and explain this now, since a doctrine is a categorified version of a monad.
What monads are good for is to describe how objects in one category can be regarded as objects of some other category "equipped with extra structure". This theme pervades mathematics, and is of the utmost importance. For example: groups are sets equipped with extra structure, abelian groups are groups equipped with extra structure, rings are abelian groups equipped with extra structure, and so on. We keep building up fancier gadgets from simpler ones. And pretty much whenever we do, there's a monad lurking in the background, running the show!
Suppose we've got two categories C and D, and the objects of D are objects of C equipped with extra structure. Then we get a pair of adjoint functors:
R: D → C L: C → DThe right adjoint R sends each D-object to its "underlying" C-object, and the left adjoint L sends each C-object to the "free" D-object on it. Often R is called a "forgetful" functor. For example, if
C = Setand
D = Grpthen we can take the underlying set of any group, and the free group on any set.
We get a "monad on C" by letting
T = LR: C → CThen, we can use facts about adjoint functors to get natural transformations called "multiplication"
m: TT => Tand the "unit"
i: 1C => TUsing more facts about adjoint functors, we can check that these satisfy associativity and the left and right unit laws. I did all this in "week92" so I won't do it again here. The upshot is that T is a lot like a monoid - which is why Benabou dubbed it a "monad".
Now, monoids like to act on things, and the same is true for monads. It turns out that a monad T on C can act on any object of C. When this happens, we call that object an "algebra" of T, or a "T-algebra" for short. And when our monad comes from a pair of adjoint functors as above, the main way we get T-algebras is from objects of D. And in nice cases, T-algebras are the same as objects of D.
So, for example, we can describe groups as T-algebras where T is some monad on the category of sets. And we can describe abelian groups as T-algebras where T is some monad on the category of groups. And we can describe rings as T-algebras where T is some monad on the category of abelian groups. And so on!
To really see how this works, we'd need to look at a few examples. I remember when James Dolan was first teaching me this stuff in a little coffeeshop here in Riverside, which has since gone out of business. I considered monads "too abstract" and dug my heels in like a stubborn mule, refusing to learn about them - until I went through a bunch of examples and saw that yes, this monad business really does capture the essence of what it means to build up fancy gadgets from simple ones by adding extra structure! And by now I'm completely sold on it. One reason is the relation to topology, which I explained in part N of "week118", and also "week174".
But alas, I'm too eager to get to the really cool stuff to work through examples right now. So if you're a complete novice at monads, you'll have to work out some examples yourself. Right now, I'll just say a bit of fancier stuff to fill in a couple gaps for the semi-experts.
First, when I said "in nice cases", I really meant that the category of T-algebras is equivalent to D when the forgetful functor R: D → C is "monadic". A bit more precisely: for any monad T on C there's a category of T-algebras, which is usually called CT for some silly reason. And, whenever we have a pair of adjoint functors R: D → C and L: C → D, we get a monad T = LR and a functor from D to CT. This is just a careful way of saying that any D-object gives us a T-algebra. And finally, we say that R is "monadic" if this functor from D to CT is an equivalence of categories. There's a theorem by Beck that says how to tell when a functor is monadic, just by looking at it.
Second, to make the analogy between monoids and monads precise, we just need to realize that a monad on C is a monoid object in the monoidal category hom(C,C). I already explained this in "week92", in even greater generality than we need here, but we need this now because I'm about to categorify monads and get "doctrines".
Okay: so, monads are good for describing "objects equipped with extra structure and properties". But now suppose we want to describe categories equipped with extra structure and properties! For example, the "categories with finite products" that I was talking about earlier, or "topoi". There are LOTS of different interesting kinds of categories equipped with extra structure and properties, and each of them gives a different kind of logic: the logic that works inside this kind of category! The more structure and properties our category has, the more powerful logic we can use inside it. This is what gives the "hierarchy of expressive power" I was talking about. So, it pays to have a good general way to describe categories equipped with extra structure and properties.
And this is what Lawvere's "doctrines" do!
I've said how monads on a category C are good for describing "objects of C equipped with extra structure and properties". But there's a certain category called Cat whose objects are categories! So, let's take C = Cat! A monad on Cat will describe categories equipped with extra structure and properties.
And this is the simplest definition of "doctrine": a monad on Cat.
However, those of you familiar with n-categories will realize that it's odd to talk about "the category of all categories". Not because of Russell's paradox - though that's a problem too, forcing us to talk about the category of small categories - but because what's really important is the 2-CATEGORY of all categories. It's best to think of Cat as a 2-category. But this suggests that we should work with a categorified, weakened version of monad when defining doctrines.
For this, we need to categorify and weaken the concept of monad. People have done this, and the result is sometimes called a "pseudomonad", but I prefer to call it a weak 2-monad, since I have dreams of categorifying further, and I don't want my notation to become ridiculous. I'd rather talk about "weak 3-monads" than "pseudopseudomonads", wouldn't you? Furthermore, if you look up "pseudomonad" in the dictionary you'll get this:
PSEUDOMONAD: bacteria usually producing greenish fluorescent water-soluble pigment; some pathogenic for plants and animals.Yuck! So, let's be very general and sketch how to define a weak 2-monad in any weak 3-category (aka tricategory).
Given a weak 3-category C and an object c of C, a "weak 2-monad on c" is just a weak monoidal category object in hom(c,c).
Huh? Well, hom(c,c) is a weak monoidal 2-category, which is precisely the right environment in which to define a "weak monoidal category object", and that's what we're doing here. Start with the usual definition of a weak monoidal category, which is a gadget living in Cat. Cat is an example of a weak monoidal 2-category, and we can write down the same definition in any weak monoidal 2-category X, getting the concept of "weak monoidal category object in X". Then, take X = hom(c,c).
(Of course I'm lying slightly here: Cat is more strict than your average weak monoidal 2-category, so it may not be immediately obvious how to generalize the concept of "weak monoidal category" as I'm suggesting. Still, I claim it's not hard if you know about this stuff.)
Now that you know how to define a weak 2-monad on any object c of a 3-category C, you can take c to be Cat and C to be 2Cat... and this is what we really should call a "doctrine".
Unsurprisingly, people often consider stricter versions of the concept of "2-monad" and "doctrine". For example, most people define their "pseudomonads" not in a weak 3-category but just a semistrict one, also known as a "Gray-category" - since 2Cat is one of these. For more details, try these papers:
24) R. Blackwell, G. M. Kelly, and A. J. Power, Two-dimensional monad theory, Jour. Pure Appl. Algebra 59 (1989), 1-41.
25) Brian Day and Ross Street, Monoidal bicategories and Hopf algebroids, Adv. Math. 129 (1997) 99-157.
26) F. Marmolejo, Doctrines whose structure forms a fully faithful adjoint string, Theory and Applications of Categories 3 (1997), 23-44. Available at http://www.tac.mta.ca/tac/volumes/1997/n2/3-02abs.html
27) S. Lack, A coherent approach to pseudomonads, Adv. Math. 152 (2000), 179-202. Also available at http://www.maths.usyd.edu.au:8000/u/stevel/papers/psm.ps.gz
Anyway, suppose T is a doctrine. Then we get a 2-category of T-algebras CatT, whose objects we should think of as "categories equipped with extra structure of type T". The classic example would be "categories with finite products". Just as Lawvere thought of these as algebraic theories, we can think of any T-algebra as a "theory of type T", and define its category of models: given T-algebras C and D, the category of models of C in D is hom(C,D), where the hom is taken in CatT.
Depending on what doctrine T we consider, we get many different forms of logic, and I'll just list a few to whet your appetite:
The typed λ-calculus is very popular in theoretical computer science, and I recommend Crole's book cited above for more about how it's related to cartesian closed categories. A good introduction to topos logic is McLarty's book cited above. For an exhaustive study of many other sorts of logic that should be on this list but aren't, I recommend part D of this book by Peter Johnstone:
28) Peter Johnstone, Sketches of an Elephant: a Topos Theory Compendium, Oxford U. Press, Oxford. Volume 1, comprising Part A: Toposes as Categories, and Part B: 2-Categorical Aspects of Topos Theory, 720 pages, 2002. Volume 2, comprising Part C: Toposes as Spaces, and Part D: Toposes as Theories, 880 pages, 2002.
We can do a lot of fun stuff with all these different forms of logic, and people have indeed done so... but I think I'll stop here. My point is merely that higher category theory and logic go hand-in-glove, and there is plenty of room for exploration here, especially if we keep categorifying - and also keep trying to craft our logic to real-world applications, especially in physics and computer science.
I wish you all a Happy New Year, and good luck on all your adventures.
Addendum: Micheal Barr wrote me the following email, correcting some errors in a previous version of this Week's Finds.
Now that I have read it, a few more comments and nit-picks. Lawvere and Tierney did elementary toposes in 69-70. True Bill had looked at toposes earlier, but had not stated the elementary axioms until he and Myles came together in Halifax during the years 69-71.
The reason Truesdell sent Bill to Columbia was because he and Eilenberg (and Mac Lane) were all working in the same office in NY doing ballistic trajectories (or some foolish thing like that) during the years 42-45. When he realized that Bill was really more of a mathematician than physicist, he thought about what mathematician he knew and came up with Eilenberg. I heard this version from Truesdell himself.
Mac Lane did not come up with the name "monad". It was Jean Benabou and it was in the summer of 1966 when there was a category meeting at Oberwohlfach. We were all trying to come up with something better than "triple". My contribution was Standard Natural Algebraic Functor with Unit, but for some reason it was not accepted. Jean was sitting next to me at lunch one day and came up with that name. I actually liked it, believe it or not, but Jon Beck disliked it and I was his close friend and felt obligated to go along. After that it became something of a fetish with me. Besides TTT was such a nice title.
As for toposes vs. topoi, there I do feel strongly. Whenever we use a classical plural in English, that plural seems eventually to become a singular. Need I mention "data" and "media", but I have also heard "phenomenas". And even "topois" (that from Andre Joyal).
© 2003 John Baez