# Is $Cat$, the category of all the small categories, a small category?

Does $$Cat$$, the category of all the small categories, contain $$Cat$$ itself? I.e. is $$Cat$$ a small category?

The following quote from Categories from the Working Mathematicians says $$\times : Cat \times Cat \to Cat$$ is a functor. Does it mean that the domain $$Cat \times Cat$$ is a (small) category, and $$Cat$$ is a (small) category?

The product $$\times$$ is thus a pair of functions: To each pair $$(B, C)$$ of categories, a new category $$B \times C$$; to each pair of functors $$(U, V)$$, a new functor $$U \times V$$. Moreover, when the composites $$U' U$$ and $$V' V$$ are defined, one clearly has $$(U' \times V') (U \times V) = U' U \times V'V$$. Hence the operation $$\times$$ itself is a functor; more exactly, on restricting to small categories, it is a functor $$\times : Cat \times Cat \to Cat$$.

Thanks.

No, elements of $$Cat$$ are small categories - $$Cat\times Cat$$ is the domain, and $$Cat$$ the codomain, of $$\times$$. Analogously, "a function on natural numbers" would be $$f:\mathbb{N}\rightarrow\mathbb{N}$$ but that wouldn't imply that $$\mathbb{N}$$ is a natural number.
Indeed, $$Cat$$ itself is not a small category, since there are a proper class of small categories (easy exercise: there is a category of cardinality $$\kappa$$ for every cardinal $$\kappa$$).
(Of course, $$Cat$$ is locally small - given any two small categories there are only set-many functors between them - but it's not small.)
• Thanks. According to the definition of a functor, its domain and codomain both are categories. $×:Cat×Cat→Cat$ is a functor. So are the domain $Cat×Cat$ and codomain $Cat$ (small) categories?