Category Of Small Categories
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In mathematics, specifically in category theory, 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, morphis ...
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, an ...
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 na ...
s serving as 2-morphisms. The
initial 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) ...
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 In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contain ...
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 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&n ...
Quiv: :''U'' : Cat → Quiv This functor forgets the identity morphisms of a given category, and it forgets morphism compositions. The
left adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are k ...
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, an ...
''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 In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in ...
, with exponential D^C given by the
functor category In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object i ...
\mathrm(C, D). * 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 spa ...
* Universal set, the notion of a 'set of all sets'


References

*


External links

* Small categories {{categorytheory-stub