ONT Re: Category Theory
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CAT. Note 18
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NB. When necessary to embolden characters,
I will use percent brackets, for example:
%R% = the real numbers, %Z% = the integers.
| 1.3. Functors (cont.)
|
| Functors were first explicitly recognized in algebraic topology,
| where they arise naturally when geometric properties are described
| by means of algebraic invariants.
|
| For example, singular homology in a given dimension n (n a natural number)
| assigns to each topological space X an abelian group H_n (X), the n^th
| homology group of X, and also to each continuous map f : X -> Y of
| spaces a corresponding homomorphism H_n (f) : H_n (X) -> H_n (Y)
| of groups, and this in such a way that H_n becomes a functor
| Top -> Ab.
|
| For example, if X = Y = S^1 is the circle, H_1 (S^1) = %Z%, so
| the group homomorphism H_1 (f) : %Z% -> %Z% is determined by
| an integer d (the image of 1); this integer is the usual
| "degree" of the continuous map f : S^1 -> S^1. In this
| case and in general, homotopic maps f, g : X -> Y yield
| the same homomorphism H_n (X) -> H_n (Y), so H_n can
| actually be regarded as a functor Toph -> Grp,
| defined on the homotopy category.
|
| The Eilenberg-Steenrod axioms for homology start with the axioms
| that H_n, for each natural number n, is a functor on Toph, and
| continue with certain additional properties of these functors.
| The more recently developed extraordinary homology and
| cohomology theories are also functors on Toph.
|
| The homotopy groups !p!_n (X) of a space X can also be regarded as
| functors; since they depend on the choice of a base point in X,
| they are functors Top_* -> Grp.
|
| The leading idea in the use of functors in topology is that H_n or !p!_n
| gives an algebraic picture or image not just of the topological spaces,
| but also of all the continuous maps between them.
|
| Mac Lane, 'Cat Work Math', p. 13.
|
| Saunders Mac Lane,
|'Categories for the Working Mathematician',
| 2nd edition, Springer, New York, NY, 1997.
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