Héctor Bland́in and Rafael Díaz March 6 , 2008
نویسنده
چکیده
Let Cat be the category whose objects are small categories (categories whose collection of objects are sets) and whose morphisms Cat(C,D) from category C to category D are functors F : C → D. Let Set be the category whose objects are sets and given x, y ∈ Ob(Set) morphisms in Set from x to y are maps f : x → y. We define an equivalence relation IsoC on Ob(C) as follows: two objects x and y are equivalent if and only if there exists an isomorphism f ∈ C(x, y). There is a natural functor D : Cat → Set called decategorification given by D(C) = Ob(C)/IsoC for all C ∈ Ob(Cat), see [2] for details. Given functor F : C → D then D(F ) : Ob(C)/IsoC → Ob(D)/IsoD is the induced map. If x is a set and C is a category such that D(C) = x, we say that C is a categorification of x. For example a categorification of the natural numbers is the category B whose objects are finite sets and whose morphisms are bijections. We are interested in the categorification of sets with additional properties. For example see [2], a partial categorification of a ring R is a symmetric monoidal category with negative objects C together with a map | | : Ob(C) → R, such that |x⊕ y| = |x|+ |y|, |x⊗ y| = |x||y|, |1| = 1 and |0| = 0. The main goal of this paper is to introduce a categorification of the ring Q of rational numbers.
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تاریخ انتشار 2008