|
This is a list of category theory topics, by Wikipedia page. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...
Specific categories In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ...
In mathematics, a concrete category is a category in which, roughly speaking, all objects are sets possibly carrying some additional structure, all morphisms are functions between those sets, and the composition of morphisms is the composition of functions. ...
In mathematics, the category K_Vect has all vector spaces over a fixed field K as objects and linear transformations as morphisms. ...
In mathematics, K-Z2Vect is the category with objects are Z2_graded vector spaces over the givenfield K and with morphisms the even and odd linear transformations between two Z2-graded vector spaces. ...
The category Top has topological spaces as objects and continuous maps as morphisms. ...
The category Met has metric spaces as objects and short maps as morphisms. ...
The category Ord has preordered sets as objects and monotonic functions as morphisms. ...
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. ...
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. ...
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ...
In mathematics, the category of magmas (see category, magma for definitions), denoted by Mag, has as objects sets with a binary operation, and morphisms given by homomorphisms of operations (in the universal algebra sense). ...
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 (in the universal algebra sense). ...
Objects In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there...
In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there...
In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there...
In category theory, there is a general definition of subobject extending the idea of subset and subgroup. ...
In mathematics, group objects are certain generalizations of groups which are built on more complicated structures than sets. ...
In mathematics, a magma in a category, or magma object, can be defined in a category with a cartesian product. ...
In category theory, a natural number object (nno) is an object endowed with a recursive structure similar to natural numbers. ...
In mathematics, specifically in category theory, an exponential object is the categorical equivalent of a function space in set theory. ...
In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...
In category theory, a zero morphism is a special kind of trivial morphism. ...
In category theory and its applications to mathematics, a normal monomorphism or normal epimorphism is a particularly well-behaved type of morphism. ...
In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in...
In mathematics, especially in category theory and homotopy theory, a groupoid is a concept (first developed by Heinrich Brandt in 1926) that simultaneously generalises groups, equivalence relations on sets, and actions of groups on sets. ...
Given a category C and a morphism in C, the image of f is a monomorphism satisfying the following universal property: There exists a morphism such that f = hg. ...
In mathematics, particularly in algebra, the coimage of a homomorphism f: A → B is the quotient coim f = A/ker f of the domain and kernel. ...
In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...
In mathematics, in particular category theory, given a functor p:E→C from a category E to a category C, a morphism f : X → Y in E is cartesian (with respect to p) when for each object Z of E and each morphism γ : pZ → pX in C, the function...
A comma category is a construction in category theory, a branch of mathematics. ...
Did somebody just say functor? In category theory, a functor is a special type of mapping between 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. ...
In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...
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. ...
In mathematics, a subcategory S of a category C consists of subsets of the morphisms and of the objects of C, such that the subset X of morphisms is closed under composition in C, and the subset Y of objects contains the source and target of all the f in...
In category theory, a faithful functor is a functor which is injective when restricted to each set of morphisms with a given source and target. ...
In category theory, a full functor is a functor which is surjective when restricted to each set of morphisms with a given source and target. ...
A forgetful functor is a type of functor in mathematics. ...
In mathematics, the Yoneda lemma in category theory is an abstract result on functors of the type morphisms into a fixed object. ...
In category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. ...
In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. ...
The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ...
In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets (posets). Galois connections generalize the correspondence between subgroups and subfields investigated in Galois theory. ...
In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. ...
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ...
In category theory, a monad or triple is a type of functor, together with two associated natural transformations. ...
In category theory, a monad or triple is a type of functor, together with two associated natural transformations. ...
This article is about a concept in combinatorial mathematics. ...
In homological algebra, an exact functor is one which preserves exact sequences. ...
In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. ...
In category theory, an enriched functor is a variant on a special type of mapping between categories. ...
Kan extensions are universal constructs in category theory, a branch of mathematics. ...
Limits In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts of mathematics, like products and inverse limits. ...
In category theory, one defines products to generalize constructions such as the cartesian product of sets, the product of groups, the product of rings and the product of topological spaces. ...
This article is about equalisers in mathematics. ...
In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra. ...
In category theory, a branch of mathematics, the pullback (also called the fiber product) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. ...
In category theory, a branch of mathematics, the pullback (also called the fiber product) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. ...
In mathematics, the inverse limit (also called the projective limit) is a construction which allows one to glue together several related objects, the precise matter of the gluing process being specified by morphisms between the objects. ...
In mathematics, pro-finite groups are groups that are in a certain sense assembled from finite groups; they share many properties with the finite groups. ...
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts of mathematics, like products and inverse limits. ...
In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. ...
In mathematics, a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary category. ...
In abstract algebra, the cokernel of a homomorphism f : X → Y is the quotient of Y by the image of f. ...
In category theory, a branch of mathematics, the pushout is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain. ...
In mathematics, the direct limit (also called the inductive limit) is a general method of taking limits of directed families 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. ...
In category theory and its applications to mathematics, a biproduct is a generalisation of the notion of direct sum that makes sense in any preadditive category. ...
In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...
Additive structure A preadditive category is a category that is enriched over the monoidal category of abelian groups. ...
In mathematics, specifically in category theory, an additive category is a preadditive category C such that any finitely many objects A1,...,An of C have a biproduct A1 ⊕ ··· ⊕ An in C. (Recall that a category C is preadditive if all its morphism sets are Abelian groups and morphism composition is...
In mathematics, specifically in category theory, a pre-Abelian category is an additive category that has all kernels and cokernels. ...
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 nice properties. ...
In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ...
In homological algebra, an exact functor is one which preserves exact sequences. ...
In mathematics, particularly homological algebra, the snake lemma, a statement valid in every Abelian category, is the crucial tool used to construct the long exact sequences that are ubiquitous in homological algebra and its applications, for instance in algebraic topology. ...
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. ...
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. ...
In mathematics, especially homological algebra and other applications of Abelian category theory, the short five lemma is a special case of the five lemma. ...
In mathematics, Mitchells embedding theorem is an important result about abelian categories; it states that these categories, while rather abstractly defined, are all quite concrete categories of modules. ...
In category theory, the concept of an injective cogenerator is drawn from examples such as Pontryagin duality. ...
In mathematics, the derived category D(C) of a 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 (which therefore should already be an abelian category). ...
In mathematics, a triangulated category is a category satisfying some axioms that are based on the properties of a derived category. ...
In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms (arrows): weak equivalences, fibrations and cofibrations. These abstract from a conventional homotopy category, of topological spaces or of chain complexes (derived category theory). ...
In category theory, a 2-category is a category with morphisms between morphisms. It can be formally defined as a category enriched over Cat (the category of catetgories and functors, with the monoidal structure induced by the composition). ...
In mathematics, a bicategory is a concept in category theory used to extend the notion of sameness (i. ...
In mathematics, a monoidal category (or tensor category) is a category equipped with a binary tensor functor and a unit object . ...
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. ...
In mathematics, progress often consists of recognising the same structure in different contexts - so that one method exploiting it has multiple applications. ...
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. ...
A comma category is a construction in category theory, a branch of mathematics. ...
In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms. ...
In category theory and its applications to mathematics, an enriched category is a category whose hom-sets are replaced by objects from some other category, in a well-behaved manner. ...
In mathematics, a bicategory is a concept in category theory used to extend the notion of sameness (i. ...
See also: abstract nonsense, homological algebra Sheaves were introduced into mathematics in the 1940s and, a major theme since then has been to study a space by studying sheaves on that space. ...
In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and...
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 F: Open(X) → C to a category C which initially one takes to be the category of...
In mathematics, the idea of descent has come to stand for a very general idea, extending the intuitive idea of gluing in 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. ...
This page gives some very general background to the mathematical idea of topos. ...
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. Introductory example As an example, the set Ω = {0,1} is a subobject classifier in the category of sets and functions...
Pointless topology is an approach to topology which avoids the mentioning of points. ...
In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebras. ...
Abstract nonsense is a popular term used by mathematicians to describe certain kinds of arguments and concepts in category theory. ...
Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ...
|
Feeds |
Contact