FOREWORD F(X) of Q) or to each mapping /: X—Y of & a mapping in the opposite direction F(f) : F(Y)-*F(X) of^satisfying condition (A) above and (B*) For every composed mapping gf of if, F(gf) = F(f)F(g). Contra- variant functors can be composed but the composition of two contravar- iant functors is a covariant functor. In fact, functors of mixed variances can be composed, with an obvious rule for the variance of the composition. A duality F: ( &—Q)\s a contravariant functor admitting an inverse, which is a contravariant functor F"1: Q)-*^ such that both FF~l and F~lF are identity functors. It is a theorem tnat: Every concrete category is the domain of a duality. An interested reader may prove this, letting F(X) be the set of all subsets o f X a n d F ( / ) = / - \ The principle of duality says roughly that any categorical theorem 0 for arbitrary categories implies another theorem 0* for arbitrary categories. For example, if 0 is a theorem about a single category # , the statement of 0 for# is equivalent to a statement about a category^ related to SS by a duality F: 5£—*Si. that statement about D is 0*, and it is true for arbitrary categories because every category is the range of a duality. The theorem 0 that a retraction / which is a monomorphism is an identity can illustrate duality. The statement 0* is that if / is a mapping in 5£, F:5£—*3t is a duality, and F(f) is a retraction and a monomorphism, then F(f) is an identity. Using several translation lemmas we can simplify 0* to the equivalent form: if / is a retraction and an epimorphism then / is an identity. Note that we may have "dual problems" which are not equivalent to each other. A typical problem in a category S? is, does Sf have the property ^? If ^P'is categorical, there is a dual property*®*, and the given problem is equiv- alent to the problem does a category dual to^have the property ^J*? By the dual problem we mean: Does ^have the property ^J*? Finally, we need definitions of subcategory, full subcategory, and functor of several variables. A subcategory £} of Sg is a category such that every object of QJ is an object of & and every mapping of ^ is a mapping of ^. 2} is a full subcategory if further, every mapping of whose domain and range are objects of Q) is a mapping of Q). For several variables we want the notion of a product of finitely many cate- gories S£\, -^n. The product is a category whose objects may be described as n-tuples (Xi, -,X„), each X, an object of SS{. To represent these as sets the union Xx\j UXn would serve, if some care is taken about disjointness. Theset Map((Xj , . .-,X„), (Yi, ••-, YJ) is the product set Map(XltYi) xii
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