ONT Re: Category Theory
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CAT. Note 19
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| 1.3. Functors (cont.)
|
| Functors arise naturally in algebra.
|
| To any commutative ring K the set of all non-singular
| n x n matrices with entries in K is the usual general
| linear group GL_n (K); moreover, each homomorphism
| f : K -> K' of rings produces in the evident way a
| homomorphism GL_n f : GL_n (K) -> GL_n (K') of groups.
| These data define for each natural number n a functor
| GL_n : CRng -> Grp.
|
| For any group G the set of all products of commutators x y x^(-1) y^(-1),
| (x, y in G), is a normal subgroup [G, G] of G, called the 'commutator'
| subgroup. Since any homomorphism G -> H of groups carries commutators
| to commutators, the assignment G ~> [G, G] defines an evident functor
| Grp -> Grp, while G ~> G/[G, G] defines a functor Grp -> Ab, the
| factor-commutator functor. Observe, however, that the center Z(G)
| of G (all a in G with ax = xa for all x) does not naturally define
| a functor Grp -> Grp, because a homomorphism G -> H may carry an
| element in the center of G to one not in the center of H.
|
| A functor which simply "forgets" some or all of the structure of an
| algebraic object is commonly called a 'forgetful' functor (or, an
| 'underlying' functor). Thus the forgetful functor U : Grp -> Set
| assigns to each group G the set UG of its elements ("forgetting"
| the multiplication and hence the group structure), and assigns
| to each morphism f : G -> G' of groups the same function f,
| regarded just as a function between sets. The forgetful
| functor U : Rng -> Ab assigns to each ring R the additive
| abelian group of R and to each morphism f : R -> R' of
| rings the same function, regarded just as a morphism
| of addition.
|
| Mac Lane, 'Cat Work Math', p. 14.
|
| Saunders Mac Lane,
|'Categories for the Working Mathematician',
| 2nd edition, Springer, New York, NY, 1997.
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