In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically in
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, the category of small categories, denoted by Cat, is the
category whose objects are all
small categories and whose
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s are
functors between categories. Cat may actually be regarded as a
2-category
In category theory, a strict 2-category is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category enriched over Cat (the category of catego ...
with
natural transformations serving as
2-morphism
In category theory, a strict 2-category is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category enriched over Cat (the category of catego ...
s.
The
initial object of Cat is the ''empty category'' 0, which is the category of no objects and no morphisms. The
terminal object is the ''terminal category'' or ''trivial category'' 1 with a single object and morphism.
terminal category
at nLab
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
Conglomerate or conglomeration may refer to:
* Conglomerate (company)
* Conglomerate (geology)
* Conglomerate (mathematics)
In popular culture:
* The Conglomerate (American group), a production crew and musical group founded by Busta Rhymes
** Co ...
) of all categories.
Free category
The category Cat has a forgetful functor ''U'' into the quiver category Quiv:
:''U'' : Cat → Quiv
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'' : Quiv → Cat
1-Categorical properties
* Cat has all small limits and colimits.
* Cat is a Cartesian closed category, with exponential given by the functor category .
* Cat is ''not'' locally Cartesian closed.
* Cat is locally finitely presentable.
See also
* Nerve of a category
* Universal set, the notion of a 'set of all sets'
References
*
External links
*
Small categories
{{categorytheory-stub