A History of Mathematics- From Mesopotamia to Modernity

250 A History ofMathematics

Fig. 5The ‘Smale horseshoe map’ is often quoted as an example of chaotic behaviour. The mapftakes the square, stretches it, and

folds it over into the horseshoe shape. (Think of making puff pastry, if that helps). If the mapfis repeated over and over again, most

ponts in the square end up outside; those that remain inside (forever) wander around chaotically.

The reason why the idea came from topology is interesting in itself. As we have seen, Alexandrov,

(with some help from Noether) came to understand that many basic topological ‘invariants’ asso-

ciated to a spaceXagroup G(X). It was also realized—indeed it had been implicit for some time—

that if you had a continuous mapffromXtoY, then you would be given a homomorphismG(f)

fromG(X)toG(Y). This is true^10 of the fundamental groupπ 1 (X), defined by looking at loops

inX.fmaps loops inXto loops inY, and so—with appropriate care! (see a textbook)—defines

a homomorphism of groups fromπ 1 (X)toπ 1 (Y).

Even this observation was probably too trivial to deserve a formal language. However, let us give

it one, following Eilenberg and MacLane in their path-breaking paper of 1942. We say that we have

acategorywhen we have a set of ‘objects’ (e.g. spaces or groups), and maps (‘morphisms’) between

the objects (e.g. continuous maps or homomorphisms); and obvious rules for how the maps should

behave:

1. givenX, it has an ‘identity morphism’ 1X;


2. given morphismsffromXtoYandgfromYtoZ, the compositiong◦ffromXtoZis defined,
   and (of course);


3. (h◦g)◦f=h◦(g◦f)if the compositions of morphisms are defined.

So, topological spaces (and continuous maps) form a category usually calledTop; and groups

(and homomorphisms of groups) form a second category, which we callGp.^11 AfunctorfromTop

toGpis a machineGwhich

(i) to any objectXinTopassigns an objectG(X)inGp;

(ii) to anyTop-morphismffromXtoYassigns aGp-morphismG(f)fromG(X)toG(Y);

(iii) such that identity maps go to identity maps, and compositions to compositions.

10. Strictly, it is not. The fundamental group needs a ‘basepoint’ to define it, and so is defined on the category of ‘spaces with
    a basepoint’.


11. The quotation from Kant, which opens the chapter, and which places what he called ‘categories’ at the centre of thought, is
    probably irrelevant—there is no evidence that Eilenberg and MacLane had Kant in mind. And yet one wonders where the word came
    from.