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Then we call ? 13 IMAGES, COIMAGES, AND COUNTERIMAGES 35 with [epimorphic] images. Dually, we define [monomorphic] coimages and denote them by C o i m ( / ) . I f A' is a subobject of A, then we denote the image of the morphism A' —* A —* B by f(A'). L E M M A 1. If^isa locally small category with intersections, then ? is a category with images. Proof. F o r m the intersection of all those subobjects of B through which / : A —* B may be factored. T h i s intersection exists and is the smallest subobject with the property that / may be factored through it.

Categories with difference kernels and difference cokernels are S, S*, Top, Top*, G r , A b , R i , and ^Mod. W e want to give the construction of a difference cokernel i n S. L e t two maps f g : A ^ B be given. Take the smallest equivalence relation on the set B under which f(a) and g(a) are equivalent for all aeA. T h e equivalence classes of this equivalence relation form a set C, onto which B is mapped i n the obvious way. T h i s map is a difference cokernel of (fg) as may easily be verified.

2 we mentioned that a category may be considered as a special class. N o w we want to speeify this. First, we deal with the definition of a category that describes only the properties of the morphisms, but does not define the objects. T h i s definition w i l l be slightly narrower than the one given before. 1. A category is a class Ji together with a subclass VC map *V B(a,b)v-+ J£ X Jl and a abeJ? such that (1) F o r all a b, c E Ji the following are equivalent y (i) (ii) (iii) (iv) (2) (a b) (b c)E^ (a b) {ab, c)Ei (a bc) (b c)Ei (a b) (b c), (a bc) (ab c) e y y y r y y r y y y y y y F o r each ae and y y there are e y r b'e = b' t l ec — c y r x (3) y t y r y y r E y y r c'e = c' y for all (e , b), (b' e ) (e , c) (c' e ) T h e n e and e are called units.

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