FOREWORD Categories. A concrete category & is defined by defining a class © of sets, called objects of 16\ and for each ordered pair of objects (X, Y) a set Map(X, Y) of functions / : X-*Y, called mappings, such that (a) The identity function on each object is a mapping (b) Every function which is a composition of mappings is a mapping. The analysis of this definition presents some peculiar difficulties, because the class © may be larger than any.cardinal number, and is larger in all cases arising in this book. But the difficulties need not concern us here. All we need is an indication of what is superfluous in the definition, i.e., of when two con- crete categories determine the same abstract category. A covariant functor F: 5£— & is given when we are given two functions, F0 and Fu as follows. FQ assigns to each object X of & an object F0(X) of &'. F{ assigns to each mapping /: X-^Y of & a mapping Fx(f)\ F0(X)—F0(Y) of 3 . Further, (A) For each identity mapping lx of &, FI(IX) = IF0(X)', (B) For every composed mapping gf of&yFl(gf) = Fl(g)Fl(f). Having noted the distinction between F0 and Fu we can ignore it for applications, writing F 0 (X) as F(X) , F^f) as F(f). The short notation de- fines a composition of covariant functors F: 5f— f2?, G: 2)— Sf for us: GF(X) = G(F(X)), GF(/) = G(F(/)). Then an isomorphism is a covariant functor F: &—* St for which there exists a covariant functor F" 1 : ^ — ^ such that both FF~l and F~lF are identity functors. A categorical property is a predicate of categories ^3 such that if *$(&) and ^ i s isomorphic with 2 then %(2#). Similarly we speak of categorical def- initions, ideas, and so on. The notion of a mapping /: X— Y having an inverse f"1: Y—X is categori- cal. The defining conditions are just ff~1 = lY, f~1f=lx- A mapping having an inverse m$£ is called an isomorphism \n$g. The notion of a mapping /: X—Y being one-to-one is not categorical. However, it has an important categorical consequence. If /: X—*Y is one- to-one then for any two mappings d: W—X, e: W—X, fd=fe implies d=e . A mapping having this left cancellation property is called a monomorphism. Similarly a mapping /such thatgf=hf impliesg= h is called an epimorphism. A mapping / : X—X satisfying / / = / is a retraction. REMARK. If a retraction is either a monomorphism or an epimorphism then it is an identity. A contravariant functor F: 5f—Q assigns to each object X of 5^ an object XI

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