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 initial object of Cat is the empty category0, which is the category of no objects and no morphisms.[1] The terminal object is the terminal category or trivial category1 with a single object and morphism.[2]
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 (meaning objects and morphisms merely form a conglomerate) of all categories.
^empty category at nLab
^terminal category at nLab
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