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6 for the properties of Top. T h e properties of T o p * arise analogously from the properties of S*. I n Chapter 3 we shall deal with S*, G r , A b , R i , and M o d i n more detail. 11 PRODUCTS AND COPRODUCTS L10 Sections and Retractions 29 A morphism / : A —> B i n a category ? is called a section if there is a morphism gin^ such that gf = 1^ . / is called a retraction if there is a morphism £ in ? such that fg = \ . If / is a section with g / = 1^ , then, by definition, g is a retraction and conversely.
W e define on the objects of 88 by ^ , since $ is bijective on the objects. F o r the morphisms of 88 we define #f by the maps induced by <&' and 3>~ . F is füll and faithful. W i t h this definition of the map one verifies easily that Jf is a functor and that ^ satisfies the required commutativities. x 7 Let ^ be a small category. L e t M be a small füll subcategory of S containing the images of all representable functors from ^ to S. x#). Correspondingly, we define a representation functor H from Funct(^,^#) which is again a small category, into F u n c t ( F u n c t ( ^ , S).
I L13 Images, Coimages, and Counterimages Let / : A —* B be a morphism in a category ? T h e image of / is the smallest subobject g : B' —> B of B to which there exists a morphism h : A —>- B' with gh — / Since g is a monomorphism, h is uniquely determined. If h is an epimorphism, then // is called the epimorphic image of / . T h e image o f / i s often denoted by I m ( / ) , where we assume that the morphism £ is known and consider I m ( / ) as an object. If there are [epimorphic] images for all morphisms in ?