The following outline is provided as an overview of and guide to category theory:

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...

– area of study in mathematics

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...

 that examines in an abstract

Abstraction

Abstraction is a process by which higher concepts are derived from the usage and classification of literal concepts, first principles, or other methods....

 way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows (also called 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, although this term also has a specific, non category-theoretical sense), where these collections satisfy certain basic conditions. Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.

Essence of category theory

• 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...
  
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• 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....
  
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• 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...
  
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Branches of category theory

• Homological algebra
  Homological algebra
  Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
  
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• Diagram chasing –
• Topos theory
  Topos
  In mathematics, a topos is a type of category that behaves like the category of sheaves of sets on a topological space...
  
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• Enriched category
  Enriched category
  In category theory and its applications to mathematics, enriched category is a generalization of category that abstracts the set of morphisms associated with every pair of objects to an opaque object in some fixed monoidal category of "hom-objects" and then defines composition and identity solely...
  
   theory –

Specific categories

• Category of sets
  Category of sets
  In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...
  
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• Concrete category
  Concrete category
  In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets. This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions...
  
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• Category of vector spaces
  Category of vector spaces
  In mathematics, especially category theory, the category K-Vect has all vector spaces over a fixed field K as objects and K-linear transformations as morphisms...
  
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• Category of graded vector spaces –
• Category of finite dimensional Hilbert spaces
  Category of finite dimensional Hilbert spaces
  In mathematics, the category FdHilb has all finite dimensional Hilbert spaces for objects and linear transformations between them.-Properties:This category* is monoidal,* possesses finite biproducts, and* is dagger compact....
  
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• Category of sets and relations –
• Category of topological spaces
  Category of topological spaces
  In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous...
  
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• Category of metric spaces
  Category of metric spaces
  In category-theoretic mathematics, Met is a category that has metric spaces as its objects and metric maps as its morphisms. This is a category because the composition of two metric maps is again a metric map...
  
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• Category of preordered sets
  Category of preordered sets
  The category Ord has preordered sets as objects and monotonic functions as morphisms. This is a category because the composition of two monotonic functions is monotonic and the identity map is monotonic....
  
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• Category of groups
  Category of groups
  In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category...
  
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• Category of abelian groups
  Category of abelian groups
  In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category....
  
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• Category of rings
  Category of rings
  In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings and whose morphisms are ring homomorphisms...
  
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• Category of magmas
  Category of magmas
  In mathematics, the category of magmas , denoted by Mag, has as objects sets with a binary operation, and morphisms given by homomorphisms of operations ....
  
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• Category of medial magmas
  Category of medial magmas
  In mathematics, the medial category Med, that is, the category of medial magmas has as objects sets with a medial binary operation, and morphisms given by homomorphisms of operations ....
  
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Objects

• Initial object
  Initial object
  In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X...
  
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• Terminal object –
• Zero object –
• Subobject
  Subobject
  In category theory, a branch of mathematics, a subobject is, roughly speaking, an object which sits inside another object in the same category. The notion is a generalization of the older concepts of subset from set theory and subgroup from group theory...
  
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• Group object
  Group object
  In category theory, a branch of mathematics, group objects are certain generalizations of groups which are built on more complicated structures than sets...
  
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• Magma object –
• Natural number object
  Natural number object
  In category theory, a natural number object is an object endowed with a recursive structure similar to natural numbers. More precisely, in a category E with a terminal object 1 , an NNO N is given by:...
  
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• Exponential object
  Exponential object
  In mathematics, specifically in category theory, an exponential object is the categorical equivalent of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories...
  
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Morphisms

• Epimorphism
  Epimorphism
  In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...
  
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• Monomorphism
  Monomorphism
  In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X \hookrightarrow Y....
  
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• Zero morphism
  Zero morphism
  In category theory, a zero morphism is a special kind of morphism exhibiting properties like those to and from a zero object.Suppose C is a category, and f : X → Y is a morphism in C...
  
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• Normal morphism
  Normal morphism
  In category theory and its applications to mathematics, a normal monomorphism or conormal epimorphism is a particularly well-behaved type of morphism.A normal category is a category in which every monomorphism is normal...
  
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• Dual (category theory)
  Dual (category theory)
  In category theory, a branch of mathematics, duality is a correspondence between properties of a category C and so-called dual properties of the opposite category Cop...
  
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• Groupoid
  Groupoid
  In mathematics, especially in category theory and homotopy theory, a groupoid generalises the notion of group in several equivalent ways. A groupoid can be seen as a:...
  
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• Image (category theory)
  Image (category theory)
  Given a category C and a morphismf\colon X\to Y in C, the image of f is a monomorphism h\colon I\to Y satisfying the following universal property:#There exists a morphism g\colon X\to I such that f = hg....
  
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• Coimage
  Coimage
  In algebra, the coimage of a homomorphismis the quotientof domain and kernel.The coimage is canonically isomorphic to the image by the first isomorphism theorem, when that theorem applies....
  
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• Commutative diagram
  Commutative diagram
  In mathematics, and especially in category theory, a commutative diagram is a diagram of objects and morphisms such that all directed paths in the diagram with the same start and endpoints lead to the same result by composition...
  
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• Cartesian morphism –
• Slice category –

Functors

• Isomorphism of categories
  Isomorphism of categories
  In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C which are mutually inverse to each other, i.e. FG = 1D and GF = 1C. This means that both the objects and the morphisms of C and D stand in a one to one correspondence to each other...
  
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• 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...
  
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• Equivalence of categories
  Equivalence of categories
  In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics...
  
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• Subcategory
  Subcategory
  In mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in C with the same identities and composition of morphisms. Intuitively, a subcategory of C is a category obtained from C by "removing" some of its objects and...
  
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• Faithful functor –
• Full functor –
• Forgetful functor
  Forgetful functor
  In mathematics, in the area of category theory, a forgetful functor is a type of functor. The nomenclature is suggestive of such a functor's behaviour: given some object with structure as input, some or all of the object's structure or properties is 'forgotten' in the output...
  
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• Yoneda lemma
  Yoneda lemma
  In mathematics, specifically in category theory, the Yoneda lemma is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory...
  
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• Representable functor
  Representable functor
  In mathematics, particularly category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures In mathematics, particularly category theory, a...
  
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• Functor category
  Functor category
  In category theory, a branch of mathematics, the functors between two given categories form a category, where the objects are the functors and the morphisms are natural transformations between the functors...
  
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• Adjoint functors
  Adjoint functors
  In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another, called an adjunction. The relationship of adjunction is ubiquitous in mathematics, as it rigorously reflects the intuitive notions of optimization and efficiency...
  
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• Galois connection
  Galois connection
  In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets . The same notion can also be defined on preordered sets or classes; this article presents the common case of posets. Galois connections generalize the correspondence...
  
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• Pontryagin duality
  Pontryagin duality
  In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact groups, such as R, the circle or finite cyclic groups.-Introduction:...
  
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• Affine scheme –
• Monad (category theory)
  Monad (category theory)
  In category theory, a branch of mathematics, a monad, Kleisli triple, or triple is an functor, together with two natural transformations...
  
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• Comonad –
• Combinatorial species
  Combinatorial species
  In combinatorial mathematics, the theory of combinatorial species is an abstract, systematic method for analysing discrete structures in terms of generating functions. Examples of discrete structures are graphs, permutations, trees, and so on; each of these has an associated generating function...
  
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• Exact functor
  Exact functor
  In homological algebra, an exact functor is a functor, from some category to another, which preserves exact sequences. Exact functors are very convenient in algebraic calculations, roughly speaking because they can be applied to presentations of objects easily...
  
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• Derived functor
  Derived functor
  In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.- Motivation :...
  
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• Enriched functor –
• Kan extension of a functor
  Kan extension
  Kan extensions are universal constructs in category theory, a branch of mathematics. They are closely related to adjoints, but are also related to limits and ends. They are named after Daniel M...
  
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• Hom functor
  Hom functor
  In mathematics, specifically in category theory, hom-sets, i.e. sets of morphisms between objects, give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics.-Formal...
  
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Limits

• Product (category theory)
  Product (category theory)
  In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces...
  
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• Equaliser (mathematics) –
• Kernel (category theory)
  Kernel (category theory)
  In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernels from algebra...
  
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• Pullback (category theory)
  Pullback (category theory)
  In category theory, a branch of mathematics, a pullback is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain; it is the limit of the cospan X \rightarrow Z \leftarrow Y...
  
  /fiber product –
• Inverse limit
  Inverse limit
  In mathematics, the inverse limit is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects...
  
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• Pro-finite group
  Pro-finite group
  In mathematics, profinite groups are topological groups that are in a certain sense assembled from finite groups; they share many properties with their finite quotients.- Definition :...
  
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  • Colimit –
  • Coproduct
    Coproduct
    In category theory, the coproduct, or categorical sum, is the category-theoretic construction which includes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...
    
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  • Coequalizer
    Coequalizer
    In category theory, a coequalizer is a generalization of a quotient by an equivalence relation to objects in an arbitrary category...
    
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  • Cokernel
    Cokernel
    In mathematics, the cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y/im of the codomain of f by the image of f....
    
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  • Pushout (category theory)
    Pushout (category theory)
    In category theory, a branch of mathematics, a pushout is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain: it is the colimit of the span X \leftarrow Z \rightarrow Y.The pushout is the...
    
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  • Direct limit
    Direct limit
    In mathematics, a direct limit is a colimit of a "directed family of objects". We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category.- Algebraic objects :In this section objects are understood to be...
    
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  • Biproduct
    Biproduct
    In category theory and its applications to mathematics, a biproduct of a finite collection of objects in a category with zero object is both a product and a coproduct. In a preadditive category the notions of product and coproduct coincide for finite collections of objects...
    
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  • Direct sum
    Direct sum
    In mathematics, one can often define a direct sum of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets , together with a suitably defined structure. More abstractly, the direct sum is often, but not always, the coproduct in the category in question...
    
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Additive structure

• Preadditive category
  Preadditive category
  In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups...
  
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• Additive category
  Additive category
  In mathematics, specifically in category theory, an additive category is a preadditive category C such that all finite collections of objects A1,...,An of C have a biproduct A1 ⊕ ⋯ ⊕ An in C....
  
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• Pre-Abelian category
  Pre-Abelian category
  In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels.Spelled out in more detail, this means that a category C is pre-abelian if:...
  
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• Abelian category
  Abelian category
  In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
  
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• Exact sequence
  Exact sequence
  An exact sequence is a concept in mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory...
  
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• Exact functor
  Exact functor
  In homological algebra, an exact functor is a functor, from some category to another, which preserves exact sequences. Exact functors are very convenient in algebraic calculations, roughly speaking because they can be applied to presentations of objects easily...
  
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• Snake lemma
  Snake lemma
  The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance in algebraic topology...
  
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• Nine lemma
  Nine lemma
  In mathematics, the nine lemma is a statement about commutative diagrams and exact sequences valid in any abelian category, as well as in the category of groups. It states: ifis a commutative diagram and all columns as well as the two bottom rows are exact, then the top row is exact as well...
  
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  • Five lemma
    Five lemma
    In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams....
    
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• Short five lemma
  Short five lemma
  In mathematics, especially homological algebra and other applications of abelian category theory, the short five lemma is a special case of the five lemma....
  
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  • Mitchell's embedding theorem
    Mitchell's embedding theorem
    Mitchell's embedding theorem, also known as the Freyd–Mitchell theorem, is a result stating that every abelian category admits a full and exact embedding into the category of R-modules...
    
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  • Injective cogenerator
    Injective cogenerator
    In category theory, the concept of an injective cogenerator is drawn from examples such as Pontryagin duality. Generators are objects which cover other objects as an approximation, and cogenerators are objects which envelope other objects as an approximation...
    
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  • Derived category
    Derived category
    In mathematics, the derived category D of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C...
    
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  • Triangulated category
    Triangulated category
    A triangulated category is a mathematical category satisfying some axioms that are based on the properties of the homotopy category of spectra, and the derived category of an abelian category. A t-category is a triangulated category with a t-structure.- History :The notion of a derived category...
    
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  • Model category
    Model category
    In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes...
    
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  • 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...
    
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  • Bicategory
    Bicategory
    In mathematics, a bicategory is a concept in category theory used to extend the notion of category to handle the cases where the composition of morphisms is not associative, but only associative up to an isomorphism. The notion was introduced in 1967 by Jean Bénabou.Formally, a bicategory B...
    
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Dagger categories

• Dagger symmetric monoidal category –
• Dagger compact category
  Dagger compact category
  In mathematics, dagger compact categories first appeared in 1989 in the work of Doplicher and Roberts on the reconstruction of compact topological group from their category of finite-dimensional continuous unitary representations...
  
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• Strongly ribbon category –

Monoidal categories

• Closed monoidal category
  Closed monoidal category
  In mathematics, especially in category theory, aclosed monoidal category is a context where we can take tensor products of objects and also form 'mapping objects'. A classic example is the category of sets, Set, where the tensor product of sets A and B is the usual cartesian product A \times B, and...
  
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• Braided monoidal category –

Structure

• Semigroupoid
  Semigroupoid
  In mathematics, a semigroupoid is a partial algebra which satisfies the axioms for a small category, except possibly for the requirement that there be an identity at each object...
  
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• Comma category
  Comma category
  In mathematics, a comma category is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become objects in their own right. This notion was introduced in 1963 by F. W...
  
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• Localization of a category
  Localization of a category
  In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms. This is formally similar to the process of localization of a ring; it in general makes objects isomorphic that were not so before...
  
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• Enriched category
  Enriched category
  In category theory and its applications to mathematics, enriched category is a generalization of category that abstracts the set of morphisms associated with every pair of objects to an opaque object in some fixed monoidal category of "hom-objects" and then defines composition and identity solely...
  
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• Bicategory
  Bicategory
  In mathematics, a bicategory is a concept in category theory used to extend the notion of category to handle the cases where the composition of morphisms is not associative, but only associative up to an isomorphism. The notion was introduced in 1967 by Jean Bénabou.Formally, a bicategory B...
  
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Topoi, toposes

• Sheaf
  Sheaf (mathematics)
  In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
  
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• Gluing axiom
  Gluing axiom
  In mathematics, the gluing axiom is introduced to define what a sheaf F on a topological space X must satisfy, given that it is a presheaf, which is by definition a contravariant functor...
  
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• Descent (category theory)
  Descent (category theory)
  In mathematics, the idea of descent has come to stand for a very general idea, extending the intuitive idea of 'gluing' in topology. Since the topologists' glue is actually the use of equivalence relations on topological spaces, the theory starts with some ideas on identification.A sophisticated...
  
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• Grothendieck topology
  Grothendieck topology
  In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.Grothendieck topologies axiomatize the notion...
  
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• Introduction to topos theory –
• Subobject classifier
  Subobject classifier
  In category theory, a subobject classifier is a special object Ω of a category; intuitively, the subobjects of an object X correspond to the morphisms from X to Ω. As the name suggests, what a subobject classifier does is to identify/classify subobjects of a given object according to which elements...
  
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• Pointless topology
  Pointless topology
  In mathematics, pointless topology is an approach to topology that avoids mentioning points. The name 'pointless topology' is due to John von Neumann...
  
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• Heyting algebra
  Heyting algebra
  In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b...
  
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History of category theory

Main article: History of category theory

See also

• Abstract nonsense
  Abstract nonsense
  In mathematics, abstract nonsense, general abstract nonsense, and general nonsense are terms used facetiously by some mathematicians to describe certain kinds of arguments and methods related to category theory. roughly speaking, category theory is the study of the general form of mathematical...
  
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• Homological algebra
  Homological algebra
  Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
  
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• Glossary of category theory
  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....
  
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