Category of small categories

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In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category whose objects are all small categories and whose morphisms are functors between categories. Cat may actually be regarded as a 2-category with natural transformations serving as 2-morphisms.

The category Cat is itself a large category, 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 quasicategory of all categories.


The category Cat has a forgetful functor U into the quiver category Quiv:

U : CatQuiv

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 free categories:

F : QuivCat