By Robert L. Knighten
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0 then we speak of a zero object and write it as 0. Of course in a category with a zero object all initial objects, final objects and zero objects are then uniquely isomorphic. 42: If a category has a zero object, then for any objects A and ! G ! , there is a unique morphism which “factors through” the zero object. All such morphisms are called zero morphisms and are denoted by 0. Although the category of sets does not have a zero object, there are very many familiar categories that do. In the categories of monoids, of groups, of Abelian groups, of vector spaces, and many others, there is a trivial object that is the zero object in the category.
10. Prove that if a morphism f has a section, then f is an epimorphism. This result can be restated as “every retract is an epimorphism” which partly explains why retracts are also called split epimorphisms . G G to indicate an epimorphism. 2. SPECIAL MORPHISMS 15 As with sections and retracts, let’s also connect this with the Hom sets. 11. Show that f : A ∗ C, the functions fC are all injective. 12. Find an example where f : A for some object C the function f∗C is not surjective. ] The next three exercises ask you to work out the meaning of epimorphisms in a few special cases where the categories are monoids.
3 we’ll see the good reasons why we would like to write the morphism from a sum into another object as a column vector, but space considerations demands a more compact notation, so we will use [f1 , · · · , fn ] as a synonym for f1 .. . 1, for each result about products, there is a corresponding result about sums. All of the following results in this section, including the exercises, are examples. 2 If S with ιj : Ai j G S is an isomorphism with both sums of A1 , · · · , An , then [ι1 , · · · , ιn ] : S G S as inverse.