A companion to S.Lang's Algebra 4ed. by Bergman G.M.

By Bergman G.M.

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Example text

In the above matters I shall follow the choices Lang has made. On the other hand, as mentioned earlier, I will not follow Lang where he has introduced various pieces of very nonstandard terminology and notation; thus, what he calls ‘‘universal repelling’’ and ‘‘universal attracting’’ objects of a category we shall call ‘‘initial’’ and ‘‘terminal’’ objects, what he calls ‘‘stripping’’ functors we shall call ‘‘forgetful G. M. 41 functors’’, and the covariant and contravariant functions that he writes MA and M A we will denote hA and h A.

Which is small as a set. Within our larger set theory, there exists a set of all small sets (namely ), hence also a set of all small groups, and similarly for other sorts of mathematical objects. ), and can be handled mathematically without kid gloves. (Even the collection of all categories of the above sort – categories having object-sets and morphism-sets which are subsets of – will form a genuine set in our set-theory. ) Since ZFC was apparently sufficient for studying group theory before categories came along, one can regard the category of small groups (with respect to an arbitrary fixed universe ) as embracing the subject of traditional group theory, and likewise for other sorts of mathematical objects; and these categories can be treated conveniently within set theory.

The fact that θ A is a homomorphism, and the equation in the last statement (which intuitively says that the maps θ A and θ B identify the finite abelian groups A and B with their double duals in a way ‘‘consistent with’’ the construction f → f ^^ on group homomorphisms), are straightforward verifications, which you should write out; the nontrivial result is that these maps are isomorphisms. The most direct way to get this is to write A as a direct sum of cyclic subgroups, and verify that θ A carries each of these summands isomorphically to the corresponding summand in A^^.

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