One of the most remarkable accomplishments of the early 20th century
was to formalize all of mathematics in terms of a language with a
deliberately impoverished vocabulary: the language of set theory. In
Zermelo-Fraenkel set theory, everything is a set, the only fundamental
relationships between sets are membership and equality, and two sets are
equal if and only if they have the same elements. If in Zermelo-Fraenkel
set theory you ask what sort of thing is the number , the
relationship 'less than', or the exponential function, the answer is
always the same: a set! Of course one must bend over backwards to think
of such varied entities as sets, so this formalization may seem almost
deliberately perverse. However, it represents the culmination of a
worldview in which things are regarded as more fundamental than processes
or relationships.
More recently, mathematicians have developed a somewhat more flexible
language, the language of category theory. Category theory is an attempt
to put processes and relationships on an equal status with things. A
category consists of a collection of 'objects', and for each pair of
objects and
, a collection of 'morphisms' from
to
. We
write a morphism from
to
as
. We demand
that for any morphisms
and
, we can
'compose' them to obtain a morphism
. We also demand
that composition be associative. Finally, we demand that for any object
there be a morphism
, called the 'identity' of
, such that
for any morphism
and
for any
morphism
.
Perhaps the most familiar example of a category is . Here the
objects are sets and the morphisms are functions between sets.
However, there are many other examples. Fundamental to quantum theory
is the category
. Here the objects are complex Hilbert spaces
and the morphisms are linear operators between Hilbert spaces. In
Section 3 we also met a category important in differential
topology, the category
. Here the objects are
-dimensional manifolds and the morphisms are cobordisms between
such manifolds. Note that in this example, the morphisms are not
functions! Nonetheless we can still think of them as 'processes' going
from one object to another.
An important part of learning category theory is breaking certain habits one may have acquired from set theory. For example, in category theory one must resist the temptation to 'peek into the objects'. Traditionally, the first thing one asks about a set is: what are its elements? A set is like a container, and the contents of this container are the most interesting thing about it. But in category theory, an object need not have 'elements' or any sort of internal structure. Even if it does, this is not what really matters! What really matters about an object is its morphisms to and from other objects. Thus category theory encourages a relational worldview in which things are described, not in terms of their constituents, but by their relationships to other things.
Category theory also downplays the importance of equality between
objects. Given two elements of a set, the first thing one asks about
them is: are they equal? But for objects in a category, we should ask
instead whether they are isomorphic. Technically, the objects and
are said to be 'isomorphic' if there is an morphism
that has an 'inverse': a morphism
for which
and
. A morphism with an inverse is
called an 'isomorphism'. An isomorphism between two objects lets us turn
any morphism to or from one of them into a morphism to or from the other
in a reversible sort of way. Since what matters about objects are their
morphisms to and from other objects, specifying an isomorphism between
two objects lets us treat them as 'the same' for all practical purposes.
Categories can be regarded as higher-dimensional analogs of sets. As shown in Figure 5, we may visualize a set as a bunch of points, namely its elements. Similarly, we may visualize a category as a bunch of points corresponding to its objects, together with a bunch of 1-dimensional arrows corresponding to its morphisms. (For simplicity, I have not drawn the identity morphisms in this figure.)
We may use the analogy between sets and categories to 'categorify'
almost any set-theoretic concept, obtaining a category-theoretic
counterpart [8]. For example, just as there are functions
between sets, there are 'functors' between categories. A function from
one set to another sends each element of the first to an element of the
second. Similarly, a functor from one category to another sends
each object
of the first to an object
of the second, and
also sends each morphism
of the first to a morphism
of the second. In addition, functors are
required to preserve composition and identities:
SET THEORY | CATEGORY THEORY |
elements | objects |
equations between elements | isomorphisms between objects |
sets | categories |
functions between sets | functors between categories |
equations between functions | natural isomorphisms between functors |
Table 2. Analogy between set theory and category theory
We summarize the analogy between set theory and category theory in Table 2. In addition to the terms already discussed there is a concept of 'natural isomorphism' between functors. This is the correct analog of an equation between functions, but we will not need it here -- I include it just for the sake of completeness.
The full impact of category-theoretic thinking has taken a while to be
felt. Categories were invented in the 1940s by Eilenberg and Mac Lane
for the purpose of clarifying relationships between algebra and
topology. As time passed they became increasingly recognized as a
powerful tool for exploiting analogies throughout mathematics
[21]. In the early 1960s they led to revolutionary -- and
still controversial -- developments in mathematical logic
[17]. It gradually became clear that category theory
was a part of a deeper subject, 'higher-dimensional algebra', in which
the concept of a category is generalized to that of an '-category'.
But only by the 1990s did the real importance of categories for physics
become evident, with the discovery that higher-dimensional algebra is
the perfect language for topological quantum field theory [14,20].
Why are categories important in topological quantum field theory?
The most obvious answer is that a TQFT is a functor. Recall from Section
3 that a TQFT maps each manifold representing space to a
Hilbert space
and each cobordism
representing
spacetime to an operator
, in such a
way that composition and identities are preserved. We may summarize all
this by saying that a TQFT is a functor
But the role of category theory goes far beyond this. The real surprise
comes when one examines the details of specific TQFTs. In Section
4
I sketched the construction of 3-dimensional quantum gravity,
but I left out the recipe for computing amplitudes for spacetime
geometries. Thus the most interesting features of the whole business
were left as unexplained 'miracles': the background-independence of the
Hilbert spaces and operators
, and the fact that they
satisfy Atiyah's axioms for a TQFT. In fact, the recipe for amplitudes
and the verification of these facts make heavy use of category theory.
The same is true for all other theories for which Atiyah's axioms have
been verified. For some strange reason, it seems that category theory is
precisely suited to explaining what makes a TQFT tick.
For the last 10 years or so, various researchers have been trying to
understand this more deeply. Much remains mysterious, but it now seems
that TQFTs are intimately related to category theory because of special
properties of the category . While
is defined using
concepts from differential topology, a great deal of evidence suggests
that it admits a simple description in terms of '
-categories'.
I have already alluded to the concept of 'categorification' -- the
process of replacing sets by categories, functions by functors and so
on, as indicated in Table 2. The concept of '-category' is obtained
from the concept of 'set' by categorifying it
times! An
-category has objects, morphisms between objects, 2-morphisms between
morphisms, and so on up to
-morphisms, together with various
composition operations satisfying various reasonable laws [5].
Increasing the value of
allows an ever more nuanced treatment of
the notion of 'sameness'. A 0-category is just a set, and in a set
the elements are simply equal or unequal. A 1-category is a category,
and in this context we may speak not only of equal but also of isomorphic
objects. Unfortunately, this careful distinction between equality and
isomorphism breaks down when we study the morphisms. Morphisms in a
category are either the same or different; there is no concept of
isomorphic morphisms. In a 2-category this is remedied by introducing
2-morphisms between morphisms. Unfortunately, in a 2-category we cannot
speak of isomorphic 2-morphisms. To remedy this we must introduce the
notion of 3-category, and so on.
We may visualize the objects of an -category as points, the morphisms
as arrows going between these points, the 2-morphisms as 2-dimensional
surfaces going between these arrows, and so on. There is thus a natural
link between
-categories and
-dimensional topology. Indeed, one
reason why
-categories are a bit formidable is that calculations with
them are most naturally done using
-dimensional diagrams. But this
link between
-categories and
-dimensional topology is precisely
why there may be a nice description of
in the language of
-categories.
Dolan and I have proposed such a description, which we call the 'cobordism hypothesis' [7]. Much work remains to be done to make this hypothesis precise and prove or disprove it. Proving it would lay the groundwork for understanding topological quantum field theories in a systematic way. But beyond this, it would help us towards a purely algebraic understanding of 'space' and 'spacetime' -- which is precisely what we need to marry them to the quantum-mechanical notions of 'state' and 'process'.
© 1999 John Baez
baez@math.removethis.ucr.andthis.edu