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13 image of 1 IMAGES, COIMAGES, AND COUNTERIMAGES 37 1 ( C ) . T h e monomorphisms from (gf)~ (C ) and from A equivalent, thus the corresponding subobjects 1 m t 1 0 a r e are equal. (g) W e Start with the commutative diagram A >B •C h is a monomorphism because (gf)(A ) and g(f(A )) are subobjects of C . h is an epimorphism becausef (A-^ and g(f(A )) are epimorphic images. T h u s h is an isomorphism, since is balanced. x x 1 (h) W e have the commutative diagram a i -/(A) / f(A ) fulfills the property of an image for A .
Similarly, we get e' = £**. F o r a e M o r ^ e ' , e) b G Mor^(e**, e*) we have (a, 6) if and only i f (ae', (a, e'), (e*, 6), (e', **) e f i f and only i f (e', e*) G f i f and only i f e' = £*. I n this case we have (e, ab), ( ß j , ß * * ) G f , thus ab G Mor^(e**, e). N o w it is easy to verify the associativity and the properties of the identities. T o get the connection with set theory as discussed i n the appendix, we now define the category as a special class. A class 2 is called a category if it satisfies the following axioms: y y (a) (b) (c) (d) S C H x H x M T>(2) QSßß(2) x Wß(2) 2 is a map F o r Jt = 3&{2), *T = T>(2) and 2 : f (1), (2), and (3) given above are satisfied.
T h u s ? is said to be a pointed category. I n ? the distinguished points of Mor^(^4, B) are uniquely determined by the condition that M o r ^ ( / , —) and Mor^(-yg) are pointed set maps. I n the category S an initial object is 0 and a final object is {0}. Zero objects do not exist. T h e only zero morphisms have the form 0 —> A. I n the category S* each set with one point is a zero object. T h u s there are zero morphisms between all objects. Similarly, the set with one point y c 24 1. PRELIMINARY NOTIONS with the corresponding structure is a zero object i n the categories A b , and Top*.