Category theory; homological algebra Special categories Example

Category of small categories

Example

A category is small when its objects and its morphisms form sets. The category of small categories $\mathbf{Cat}$ has the small categories as objects and the functors between them as morphisms. The composite of $F \colon \mathcal{A} \to \mathcal{B}$ and $G \colon \mathcal{B} \to \mathcal{C}$ is the functor $G \circ F$ , given on objects and on morphisms by composing the maps. The identity functor of $\mathcal{A}$ is the identity morphism. Composition of functors is associative because it is composition of maps on objects and on morphisms, so $\mathbf{Cat}$ is a category.

Remarks

The empty category is an initial object, and the category with one object and one morphism is a terminal object. The product of $\mathcal{A}$ and $\mathcal{B}$ is the product category $\mathcal{A} \times \mathcal{B}$ . $\mathbf{Cat}$ is Cartesian closed: the exponential object of $\mathcal{A}$ and $\mathcal{B}$ is the functor category $\mathcal{B}^{\mathcal{A}}$ . With natural transformations as morphisms between functors, $\mathbf{Cat}$ becomes a 2-category. The smallness condition avoids a paradox: $\mathbf{Cat}$ itself is not small, so it is not one of its own objects.

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2-category

Tag	`ZY6R4H65P8`
URI	`m2://ZY6R4H65P8`
Kind	example
Writing	W0 stub
MSC	18A05, 18N10
Instance of	Category
Uses	Category, Functor
Same as	Wikidata Q5051852 Wikipedia Category of small categories

Category of small categories

Example

Remarks

Reading path

Where this leads

Further reading

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