Category of small categories
Encyclopedia
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 morphism
s are functor
s between categories. Cat may actually be regarded as a 2-category
with natural transformation
s 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, specifically in category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, the category of small categories, denoted by Cat, is the category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
whose objects are all small categories and whose morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
s are functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
s between categories. Cat may actually be regarded as a 2-category
2-category
In category theory, a 2-category is a category with "morphisms between morphisms"; that is, where each hom set itself carries the structure of a category...
with natural transformation
Natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...
s 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
Russell's paradox
In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
one cannot form the “category of all categories”. But it is possible to form a quasicategory
Glossary of category theory
-Categories:A category A is said to be:* small provided that the class of all morphisms is a set ; otherwise large....
of all categories.