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, ca ...
, the category of small categories, denoted by Cat, is the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
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 ...
s are
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
s between categories. Cat may actually be regarded as a
2-category with
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 (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
s serving as
2-morphisms.
The
initial object of Cat is the ''empty category'' 0, which is the category of no objects and no morphisms. The
terminal object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
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
** ...
) of all categories.
Free category
The category Cat has a forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given sign ...
''U'' into the quiver category
In graph theory, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed, i.e. a multidigraph. They are commonly used in representation theory: a representation of a quiver assigns a vector space&nb ...
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
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
''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
In category theory, a discipline within mathematics, the nerve ''N''(''C'') of a small category ''C'' is a simplicial set constructed from the objects and morphisms of ''C''. The geometric realization of this simplicial set is a topological space ...
* Universal set, the notion of a 'set of all sets'
References
*
External links
*
Small categories
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