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This is called a cancellation law, and we say that e is an epimorphism iff it can be canceled on the right or has right cancellation. That “right cancellation” is a weakened form of “having a right inverse” is the content of this next exercise. 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.

Note that implicitly there are n triangles in this diagram featuring (π1 , f1 ), . . ] Similar situations occur constantly in the study of categories, under the name Universal Mapping Property. The Universal Mapping Property for a G Ai uniquely product of A1 , · · · , An is that every family of morphisms fi : C G Ai of projections. We will mention exfactors through the family πi : P amples of other Universal Mapping Properties as they occur, and then discuss the many ramifications in Chapter V (Universal Mapping Properties).

Verify that there is a unique fashion in which this is a category with three objects and six morphisms. This category is named 3. Although these examples may seem strange they will actually recur in various application later in these notes. 2. If each arrow in the above diagram represents a distinct morphism, can this be a category with two objects and five morphisms? In the previous section we saw the example of Iso(Set) as a somewhat unusual category. Let’s look at a few more. , f : X x1 = x2 .

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