Category of small categories

In mathematics, specifically in , the category of small categories, denoted by Cat, is the whose objects are all and whose morphisms are functors between categories. Cat may actually be regarded as a with natural transformations serving as 2-morphisms. The initial object of Cat is the empty category 0, which is the category of no objects and no morphisms. The terminal object is the terminal category or trivial category 1 with a single object and morphism. The category Cat is itself a , and therefore not an object of itself. In order to avoid problems analogous to Russell's paradox one cannot form the “category of all categories”. But it is possible to form a (meaning objects and morphisms merely form a conglomerate) of all categories. Free category The category Cat has a forgetful functor U into the Quiv: U : Cat → Quiv This functor forgets the identity morphisms of a given category, and it forgets morphism compositions. The left adjoint of this functor is a functor F taking Quiv to the corresponding : F : Quiv → Cat Cat has . Cat is a , with exponential given by the . Cat is not locally Cartesian closed. Cat is .