AChaoticEnd? 251Fig. 6This object (also called a ‘string worldsheet’) is (a) a surface, (b) the story of two strings which (reading time upwards)
coalesce and then divide again, (c) a morphism in the category of strings from the two loops at the bottom to the two loops at the top.
(See Baez, n.d).You may (if you have done any of this kind of mathematics before) find that this is an amazingly
enlightening idea; or you may sympathize with P. A. Smith, who allegedly said of Eilenberg and
MacLane’s work that he had never read a more trivial paper in his life (Corry 2004, p. 361).
The ideas of category, functor and (still more important, but there is no space for it) natural
transformation were in any case unexpectedly useful for the algebraic topologists of the 1940s and
1950s as they tried to pin down the ways in which one structure (group, or family of groups, etc.)
carried information about another (space).
However, equally obviously, the way in which they are defined above has nothing to do with
topology; and the ideas of category theory, besides becoming (naturally) a rapidly burgeoning
field on their own account, were taken over in algebra, algebraic geometry, number theory, even
analysis in the 1950s and 1960s. As with the ‘old’ abstract Bourbaki viewpoint, there were those
who warned that the new categorical viewpoint was doing nothing but turn out trivialities (‘general
nonsense’ was a term much in use among those who needed the theory but did not wish it to be
thought that that was all they were doing). And, if we remarked with alarm in the last chapter that
mathematics had moved from its traditional concerns to become centred on sets, category theory
(as Dieudonné points out in his early celebration of the idea) is capable of finding sets too restrictive.
Where next?
The attentive reader will notice that a category by no means escapes the set theoretic problems
of the last chapter, and ‘all groups’ is much too large a set to be dealing with if one wants to avoid
paradoxes. Sometimes this worries those who use categories; sometimes they assume that they,
or the reader, is taking care in some specified way. Everything connects; and in the last 30 years,
we have learned to see a category as defining a topological space; and to construct categories in
which the sets of morphisms themselves have some extra structure. And (as a guard against the
accusation of excessive abstraction) we could exhibit the category, crucial for string theory, whose
objects(simplifying again) are sets of strings (circles); while themorphismsfrom one set of strings
S 1 to another setS 2 aresurfaces(see Fig. 6) whose upper boundary isS 1 , and lower boundaryS 2.
8. Physics
He now pushedaway the paper, covered with formulae and symbols, on which the last thing he had written was
an equation of state of water, as a physical example, in order to apply a new mathematical operation that he was
describing. (Musil 1953 vol. 1, p. 128.)