May 15, 1998

This Week's Finds in Mathematical Physics (Week 121)

John Baez

This time I want to talk about higher-dimensional algebra and its applications to topology. Marco Mackaay has just come out with a fascinating paper that gives a construction of 4-dimensional TQFTs from certain "monoidal 2-categories".

1) Marco Mackaay, Spherical 2-categories and 4-manifold invariants, preprint available as math.QA/9805030

Beautifully, this construction is just a categorified version of Barrett and Westbury's construction of 3-dimensional topological quantum field theories from "monoidal categories". Categorification - the process of replacing equations by isomorphisms - is supposed to take you up the ladder of dimensions. Here we are seeing it in action!

To prepare you understand Mackaay's paper, maybe I should explain the idea of categorification. Since I recently wrote something about this, I think I'll just paraphrase a bit of that. Some of this is already familiar to long-time customers, so if you know it all already, just skip it.

2) John Baez and James Dolan, Categorification, to appear in the Proceedings of the Workshop on Higher Category Theory and Mathematical Physics at Northwestern University, Evanston, Illinois, March 1997, eds. Ezra Getzler and Mikhail Kapranov. Preprint available as math.QA/9802029 or at

So, what's categorification? This tongue-twisting term, invented by Louis Crane, refers to the process of finding category-theoretic analogs of ideas phrased in the language of set theory, using the following analogy between set theory and category theory:

elements                            objects                       
equations between elements          isomorphisms between objects        
sets                                categories                    
functions                           functors                      
equations between functions         natural isomorphisms between functors  

Just as sets have elements, categories have objects. Just as there are functions between sets, there are functors between categories. Interestingly, the proper analog of an equation between elements is not an equation between objects, but an isomorphism. More generally, the analog of an equation between functions is a natural isomorphism between functors.

For example, the category FinSet, whose objects are finite sets and whose morphisms are functions, is a categorification of the set N of natural numbers. The disjoint union and Cartesian product of finite sets correspond to the sum and product in N, respectively. Note that while addition and multiplication in N satisfy various equational laws such as commutativity, associativity and distributivity, disjoint union and Cartesian product satisfy such laws only up to natural isomorphism. This is a good example of how equations between functions get replaced by natural isomorphisms when we categorify.

If one studies categorification one soon discovers an amazing fact: many deep-sounding results in mathematics are just categorifications of facts we learned in high school! There is a good reason for this. All along, we have been unwittingly "decategorifying" mathematics by pretending that categories are just sets. We "decategorify" a category by forgetting about the morphisms and pretending that isomorphic objects are equal. We are left with a mere set: the set of isomorphism classes of objects.

To understand this, the following parable may be useful. Long ago, when shepherds wanted to see if two herds of sheep were isomorphic, they would look for an explicit isomorphism. In other words, they would line up both herds and try to match each sheep in one herd with a sheep in the other. But one day, along came a shepherd who invented decategorification. She realized one could take each herd and "count" it, setting up an isomorphism between it and some set of "numbers", which were nonsense words like "one, two, three,..." specially designed for this purpose. By comparing the resulting numbers, she could show that two herds were isomorphic without explicitly establishing an isomorphism! In short, by decategorifying the category of finite sets, the set of natural numbers was invented.

According to this parable, decategorification started out as a stroke of mathematical genius. Only later did it become a matter of dumb habit, which we are now struggling to overcome by means of categorification. While the historical reality is far more complicated, categorification really has led to tremendous progress in mathematics during the 20th century. For example, Noether revolutionized algebraic topology by emphasizing the importance of homology groups. Previous work had focused on Betti numbers, which are just the dimensions of the rational homology groups. As with taking the cardinality of a set, taking the dimension of a vector space is a process of decategorification, since two vector spaces are isomorphic if and only if they have the same dimension. Noether noted that if we work with homology groups rather than Betti numbers, we can solve more problems, because we obtain invariants not only of spaces, but also of maps.

In modern lingo, the nth rational homology is a functor defined on the category of topological spaces, while the nth Betti number is a mere function, defined on the set of isomorphism classes of topological spaces. Of course, this way of stating Noether's insight is anachronistic, since it came before category theory. Indeed, it was in Eilenberg and Mac Lane's subsequent work on homology that category theory was born!

Decategorification is a straightforward process which typically destroys information about the situation at hand. Categorification, being an attempt to recover this lost information, is inevitably fraught with difficulties. One reason is that when categorifying, one does not merely replace equations by isomorphisms. One also demands that these isomorphisms satisfy some new equations of their own, called "coherence laws". Finding the right coherence laws for a given situation is perhaps the trickiest aspect of categorification.

For example, a monoid is a set with a product satisfying the associative law and a unit element satisfying the left and right unit laws. The categorified version of a monoid is a "monoidal category". This is a category C with a product

⊗: C × C → C

and unit object 1. If we naively impose associativity and the left and right unit laws as equational laws, we obtain the definition of a "strict" monoidal category. However, the philosophy of categorification suggests instead that we impose them only up to natural isomorphism. Thus, as part of the structure of a "weak" monoidal category, we specify a natural isomorphism

ax,y,z: (x ⊗ y) ⊗ z → x ⊗ (y ⊗ z)

called the "associator", together with natural isomorphisms

lx: 1 ⊗ x → x,

rx: x ⊗ 1 → x.

Using the associator one can construct isomorphisms between any two parenthesized versions of the tensor product of several objects. However, we really want a unique isomorphism. For example, there are 5 ways to parenthesize the tensor product of 4 objects, which are related by the associator as follows:

((x ⊗ y) ⊗ z) ⊗ w ----------------> (x ⊗ (y ⊗ z)) ⊗ w
         |                                   |
         |                                   |
         |                                   |
         V                                   |
(x ⊗ y) ⊗ (z ⊗ w)                            |
         |                                   |
         |                                   |
         |                                   |
         V                                   V
{x ⊗ (y ⊗ (z ⊗ w)) <---------------- x ⊗ ((y ⊗ z) ⊗ w)

In the definition of a weak monoidal category we impose a coherence law, called the "pentagon identity", saying that this diagram commutes. Similarly, we impose a coherence law saying that the following diagram built using a, l and r commutes:

(1 ⊗ x) ⊗ 1  ----------> 1 ⊗ (x ⊗ 1)
      |                       |
      |                       | 
      V                       V
   x ⊗ 1 ------> x <------ 1 ⊗ x

This definition raises an obvious question: how do we know we have found all the right coherence laws? Indeed, what does "right" even mean in this context? Mac Lane's coherence theorem gives one answer to this question: the above coherence laws imply that any two isomorphisms built using a, l and r and having the same source and target must be equal.

Further work along these lines allow us to make more precise the sense in which N is a decategorification of FinSet. For example, just as N forms a monoid under either addition or multiplication, FinSet becomes a monoidal category under either disjoint union or Cartesian product if we choose the isomorphisms a, l, and r sensibly. In fact, just as N is a "rig", satisfying all the ring axioms except those involving additive inverses, FinSet is what one might call a "rig category". In other words, it satisfies the rig axioms up to natural isomorphisms satisfying the coherence laws discovered by Kelly and Laplaza, who proved a coherence theorem in this context.

Just as the decategorification of a monoidal category is a monoid, the decategorification of any rig category is a rig. In particular, decategorifying the rig category FinSet gives the rig N. This idea is especially important in combinatorics, where the best proof of an identity involving natural numbers is often a "bijective proof": one that actually establishes an isomorphism between finite sets.

While coherence laws can sometimes be justified retrospectively by coherence theorems, certain puzzles point to the need for a deeper understanding of the origin of coherence laws. For example, suppose we want to categorify the notion of "commutative monoid". The strictest possible approach, where we take a strict monoidal category and impose an equational law of the form x ⊗ y = y ⊗ x, is almost completely uninteresting. It is much better to start with a weak monoidal category equipped with a natural isomorphism

Bx,y: x ⊗ y → y ⊗ x

called the "braiding" and then impose coherence laws called "hexagon identities" saying that the following two diagrams built from the braiding and the associator commute:

x ⊗ (y ⊗ z)  ------------>  (y ⊗ z) ⊗ x 
     |                            ^
     |                            |
     V                            |

(x ⊗ y) ⊗ z                 y ⊗ (z ⊗ x)
     |                            ^
     |                            |
     V                            |

(y ⊗ x) ⊗ z  ------------>  y ⊗ (x ⊗ z)

(x ⊗ y) ⊗ z  ------------>  z ⊗ (x ⊗ y)
     |                            ^
     |                            |
     V                            |

x ⊗ (y ⊗ z)                 (z ⊗ z) ⊗ y
     |                            ^
     |                            |
     V                            |

x ⊗ (z ⊗ y)  ------------>  (x ⊗ z) ⊗ y

This gives the definition of a weak "braided monoidal category". If we impose an additional coherence law saying that Bx,y is the inverse of By,x, we obtain the definition of a "symmetric monoidal category". Both of these concepts are very important; which one is "right" depends on the context. However, neither implies that every pair of parallel morphisms built using the braiding are equal. A good theory of coherence laws must naturally account for these facts.

The deepest insights into such puzzles have traditionally come from topology. In homotopy theory it causes problems to work with spaces equipped with algebraic structures satisfying equational laws, because one cannot transport such structures along homotopy equivalences. It is better to impose laws only up to homotopy, with these homotopies satisfying certain coherence laws, but again only up to homotopy, with these higher homotopies satisfying their own higher coherence laws, and so on. Coherence laws thus arise naturally in infinite sequences. For example, Stasheff discovered the pentagon identity and a sequence of higher coherence laws for associativity when studying the algebraic structure possessed by a space that is homotopy equivalent to a loop space. Similarly, the hexagon identities arise as part of a sequence of coherence laws for spaces homotopy equivalent to double loop spaces, while the extra coherence law for symmetric monoidal categories arises as part of a sequence for spaces homotopy equivalent to triple loop spaces. The higher coherence laws in these sequences turn out to be crucial when we try to iterate the process of categorification.

To iterate the process of categorification, we need a concept of "n-category" - roughly, an algebraic structure consisting of a collection of objects (or "0-morphisms"), morphisms between objects (or "1-morphisms"), 2-morphisms between morphisms, and so on up to n-morphisms. There are various ways of making this precise, and right now there is a lot of work going on devoted to relating these different approaches. But the basic thing to keep in mind is that the concept of "(n+1)-category" is a categorification of the concept of "n-category". What were equational laws between n-morphisms in an n-category are replaced by natural (n+1)-isomorphisms, which need to satisfy certain coherence laws of their own.

To get a feeling for how these coherence laws are related to homotopy theory, it's good to think about certain special kinds of n-category. If we have an (n+k)-category that's trivial up to but not including the k-morphism level, we can turn it into an n-category by a simple reindexing trick: just think of its j-morphisms as (j-k)-morphisms! We call the n-categories we get this way "k-tuply monoidal n-categories". Here is a little chart of what they amount to for various low values of n and k:

                   k-tuply monoidal n-categories 

              n = 0           n = 1             n = 2

k = 0         sets          categories         2-categories

k = 1        monoids         monoidal           monoidal
                            categories        2-categories

k = 2       commutative      braided            braided
             monoids         monoidal           monoidal
                            categories        2-categories 

k = 3         " "           symmetric            weakly
                             monoidal          involutory
                            categories          monoidal

k = 4         " "             " "               strongly 

k = 5         " "             " "                "  "

One reason James Dolan and I got so interested in this chart is the "tangle hypothesis". Roughly speaking, this says that n-dimensional surfaces embedded in (n+k)-dimensional space can be described purely algebraically using the a certain special "k-tuply monoidal n-category with duals". If true, this reduces lots of differential topology to pure algebra! It also helps you understand the parameters n and k: you should think of n as "dimension" and k as "codimension".

For example, take n = 1 and k = 2. Knots, links and tangles in 3-dimensional space can be described algebraically using a certain "braided monoidal categories with duals". This was the first interesting piece of evidence for the tangle hypothesis. It has spawned a whole branch of math called "quantum topology", which people are trying to generalize to higher dimensions.

More recently, Laurel Langford tackled the case n = 2, k = 2. She proved that 2-dimensional knotted surfaces in 4-dimensional space can be described algebraically using a certain "braided monoidal 2-category with duals". These so-called "2-tangles" are particularly interesting to me because of their relation to spin foam models of quantum gravity, which are also all about surfaces in 4-space. For references, see "week103". But if you want to learn about more about this, you couldn't do better than to start with:

3) J. S. Carter and M. Saito, Knotted Surfaces and Their Diagrams, American Mathematical Society, Providence, 1998.

This is a magnificently illustrated book which will really get you able to see 2-dimensional surfaces knotted in 4d space. At the end it sketches the statement of Langford's result.

Another interesting thing about the above chart is that k-tuply monoidal n-categories keep getting "more commutative" as k increases, until one reaches k = n+2, at which point things stabilize. There is a lot of evidence suggesting that this "stabilization hypothesis" is true for all n. Assuming it's true, it makes sense to call a k-tuply monoidal n-category with k ³ n+2 a "stable n-category".

Now, where does homotopy theory come in? Well, here you need to look at n-categories where all the j-morphisms are invertible for all j. These are called "n-groupoids". Using these, one can develop a translation dictionary between n-category theory and homotopy theory, which looks like this:

ω-groupoids                    homotopy types 
n-groupoids                        homotopy n-types
k-tuply groupal ω-groupoids    homotopy types of k-fold loop spaces
k-tuply groupal n-groupoids        homotopy n-types of k-fold loop spaces 
k-tuply monoidal ω-groupoids   homotopy types of Ek spaces 
k-tuply monoidal n-groupoids       homotopy n-types of Ek spaces
stable ω-groupoids             homotopy types of infinite loop spaces
stable n-groupoids                 homotopy n-types of infinite loop spaces 
Z-groupoids                        homotopy types of spectra  

The entries on the left-hand side are very natural from an algebraic viewpoint; the entries on the right-hand side are things topologists already study. We explain what all these terms mean in the paper, but maybe I should say something about the first two rows, which are the most basic in a way. A homotopy type is roughly a topological space "up to homotopy equivalence", and an ω-groupoid is a kind of limiting case of an n-groupoid as n goes to infinity. If infinity is too scary, you can work with homotopy n-types, which are basically homotopy types with no interesting topology above dimension n. These should correspond to n-groupoids.

Using these basic correspondences we can then relate various special kinds of homotopy types to various special kinds of ω-groupoids, giving the rest of the rows of the chart. Homotopy theorists know a lot about the right-hand column, so we can use this to get a lot of information about the left-hand column. In particular, we can work out the coherence laws for n-groupoids, and - this is the best part, but the least understood - we can then guess a lot of stuff about the coherence laws for general n-categories. In short, we are using homotopy theory to get our foot in the door of n-category theory.

I should emphasize, though, that this translation dictionary is partially conjectural. It gets pretty technical to say what exactly is and is not known, especially since there's pretty rapid progress going on. Even in the last few months there have been some interesting developments. For example, Breen has come out with a paper relating k-tuply monoidal n-categories to Postnikov towers and various far-out kinds of homological algebra:

4) Lawrence Breen, Braided n-categories and Σ-structures, Prepublications Matematiques de l'Universite Paris 13, 98-06, January 1998, to appear in the Proceedings of the Workshop on Higher Category Theory and Mathematical Physics at Northwestern University, Evanston, Illinois, March 1997, eds. Ezra Getzler and Mikhail Kapranov.

Also, the following folks have also developed a notion of "iterated monoidal category" whose nerve gives the homotopy type of a k-fold loop space, just as the nerve of a category gives an arbitrary homotopy type:

5) C. Balteanu, Z. Fiedorowicz, R. Schwaenzl, and R. Vogt, Iterated monoidal categories, available at math.AT/9808082.

Anyway, in addition to explaining the relationship between n-category theory and homotopy theory, Dolan's and my paper discusses iterated categorifications of the very simplest algebraic structures: the natural numbers and the integers. The natural numbers are the free monoid on one generator; the integers are the free group on one generator. We believe this is just the tip of the following iceberg:

  algebraic strutures and the free such structure on one generator

    sets                               the one-element set
   monoids                             the natural numbers
    groups                                the integers
 k-tuply monoidal                      the braid n-groupoid 
   n-categories                           in codimension k
  k-tuply monoidal                    the braid ω-groupoid 
  ω-categories                        in codimension k
 stable n-categories                  the braid n-groupoid 
                                     in infinite codimension
stable ω-categories               the braid ω-groupoid 
                                      in infinite codimension
  k-tuply monoidal                 the n-category of framed n-tangles
n-categories with duals                  in n+k dimensions
 stable n-categories                the framed cobordism n-category
    with duals
 k-tuply groupal                         the homotopy n-type 
  n-groupoids                        of the kth loop space of Sk
 k-tuply groupal                          the homotopy type  
 ω-groupoids                    of the kth loop space of Sk
stable ω-groupoids                   the homotopy type 
                                   of the infinite loop space of S
   Z-groupoids                          the sphere spectrum       

You may or may not know the guys on the right-hand side, but some of them are very interesting and complicated, so it's really exciting that they are all in some sense categorified and/or stabilized versions of the integers and natural numbers.

Whew! There is more to say, but I'll just mention a few related papers and then quit. If you're interested in n-categories you could also check out "the tale of n-categories", starting in week73.

6) Representation theory of Hopf categories, Martin Neuchl, Ph.D. dissertation, Department of Mathematics, University of Munich, 1997. Available at

Just as the category of representations of a Hopf algebra gives a nice monoidal category, the 2-category of representations of a Hopf category gives a nice monoidal 2-category! Categorification strikes again - and this is perhaps our best hopes for getting our hands on the data needed to stick into Mackaay's machine and get concrete examples of a 4d topological quantum field theories!

7) Jim Stasheff, Grafting Boardman's cherry trees to quantum field theory, preprint available as math.AT/9803156.

Starting with Boardman and Vogt's work, and shortly thereafter that of May, operads have become really important in homotopy theory, string theory, and now n-category theory; this review article sketches some of the connections.

8) Masoud Khalkhali, On cyclic homology of A algebras, preprint available as math.QA/9805051.

Masoud Khalkhali, Homology of L algebras and cyclic homology, preprint available as math.QA/9805052.

An A algebra is an algebra that is associative up to an associator which satisfies the pentagon identity up to a pentagonator which satisfies it's own coherence law up to something, ad infinitum. The concept goes back to Stasheff's work on A spaces - spaces with a homotopy equivalence to a space equipped with an associative product. (These are the same thing as what I called E1 spaces in the translation dictionary between n-groupoid theory and homotopy theory.) But here it's been transported from Top over to Vect. Similarly, an L algebra is a Lie algebra "up to an infinity of higher coherence laws". Loday-Quillen and Tsygan showed that that the Lie algebra homology of the algebra of stable matrices over an associative algebra is isomorphic, as a Hopf algebra, to the exterior algebra of the cyclic homology of the algebra. In the second paper above, Khalkali gets the tools set up to extend this result to the category of L algebras.

© 1998 John Baez