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39 3. Functors and Natural Transformations Chapter 3 FUNCTORS AND NATURAL TRANSFORMATIONS The starting point of Category Theory is the premise that every kind of mathematically structured object comes equipped with a notion of ÒacceptableÓ transformation or construction, that is, a morphism that preserves the structure of the object. This premise holds for categories themselves: a functor is the ÒnaturalÓ notion of morphisms between categories. By a further step, the question about what a morphism between functors should look like suggests the notion of natural transformation.

It is usual practice to omit the subscripts ÒobÓ and ÒmorÓ as it is always clear from the context whether the functor is meant to operate on objects or on morphisms. 40 3. 2 Examples 1. If C and D are preorders, then a functor F: C®D is just a monotone function from the objects of C to the objects of D, indeed a

4). There is a broad literature on categories with partial maps. We partly followed the approach in DiPaola and Heller (1984), and in Longo and Moggi (1984), where the notions of Òcomplete objectÓ and partial Cartesian Closed Category were introduced (and applied in Asperti and Longo (1987) ). The properties of complete objects carry on also when using a more topos-theoretic perspective, as shown in Moggi (1988), which is devoted to a deeper insight into the concepts just sketched here. New results and surveys on categories with partial morphisms may be also found in Rosolini (1986), Moggi (1988a), Robinson and Rosolini (1988) and Curien and Obtulowicz (1988).

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