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By Bergman G.M.

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A good rule for interpreting his results precisely is that whenever finitely many groups are mentioned, m denotes some common exponent of these groups, and duals are then taken with respect to some fixed cyclic groups Zm of order m. 1 [>]: This theorem makes two statements of isomorphism. I claim that the first of these is a ‘‘natural’’ isomorphism, and the second an ‘‘unnatural’’ one. Indeed, the theorem gives one a precise prescription for constructing an isomorphism (B × C)^ ∼ = B^ × C^; but the isomorphism A^ ∼ = A for A a finite abelian group depends on one’s choice of a family of cyclic subgroups having A as their direct sum: pick a different decomposition, and you will get a different isomorphism.

Lang has chosen to write morphism-sets as Mor(X, Y ); a more common notation is Hom(X, Y ) (even though one says ‘‘morphism’’ rather than ‘‘homomorphism’’). When one wants to specify the category unambiguously, one writes this morphism-set Hom (X, Y ); a simpler notation, also common, is (X, Y ). , some authors use one or another of these symbols to denote the set of morphisms from Y to X rather than the set of morphisms from X to Y ! Independent of this choice is the choice of whether to write gf (as we do in this course) or fg for the composite of a morphism f : X → Y and a morphism g : Y → Z.

Here, let it suffice that you see that each of these examples illustrates both concepts. Remark on the category of categories (optional). Though an ordinary category involves two sorts of entities, ‘‘objects’’ and ‘‘morphisms’’, when one studies the category of categories, one has three levels: categories, functors, and morphisms of functors. One can write down various laws relating composition of functors with composition of morphisms between functors, and abstract the resulting structure on the category of categories.

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