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Mapping between categories
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic
Functor
Topics referred to by the same term
up functor in Wiktionary, the free dictionary. A functor, in mathematics, is a map between categories. Functor may also refer to: Predicate functor in
Functor_(disambiguation)
Relationship between two functors abstracting many common constructions
relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in
Adjoint_functors
Design pattern in pure functional programming
In functional programming, a functor is a design pattern inspired by the definition from category theory that allows one to apply a function to values
Functor (functional programming)
Functor_(functional_programming)
Mathematical concept
like the strongly related notions of universal properties and adjoint functors, exist at a high level of abstraction. In order to understand them, it
Limit_(category_theory)
In mathematics, a group functor is a group-valued functor on the category of commutative rings. Although it is typically viewed as a generalization of
Group_functor
Embedding of categories into functor categories
category of functors (contravariant set-valued functors) defined on that category. It also clarifies how the embedded category of representable functors and their
Yoneda_lemma
Homological construction in category theory
mathematics, specifically category theory, certain functors may be derived to obtain other functors closely related to the original ones. This operation
Derived_functor
General theory of mathematical structures
contravariant functor acts as a covariant functor from the opposite category Cop to D. A natural transformation is a relation between two functors. Functors often
Category_theory
Functor type
category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an
Representable_functor
Functor mapping hom objects to an underlying category
between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category
Hom_functor
Mathematical structures in category theory
a branch of mathematics, a functor category D C {\displaystyle D^{C}} is a category where the objects are the functors F : C → D {\displaystyle F:C\to
Functor_category
Tool to track locally defined data attached to the open sets of a topological space
direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in
Sheaf_(mathematics)
Central object of study in category theory
mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition
Natural_transformation
Mathematical set of all subsets of a set
contravariant power set functor, P: Set → Set and P: Set op → Set. The covariant functor is defined more simply as the functor which sends a set S to P(S)
Power_set
topological functor is one which has similar properties to the forgetful functor from the category of topological spaces. The domain of a topological functor admits
Topological_functor
Computer programming function
category-theoretic functor axioms for this functor. Functors can also be objects in categories, with "morphisms" called natural transformations. Given two functors F
Map_(higher-order_function)
Functor that preserves short exact sequences
particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations
Exact_functor
Concept in category theory
specifically 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
Forgetful_functor
In category theory, a branch of mathematics, the diagonal functor C → C × C {\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}}
Diagonal_functor
Functors which are surjective and injective on hom-sets
category theory, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both properties
Full_and_faithful_functors
Construction in homological algebra
In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological
Ext_functor
Construction in homological algebra
mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central
Tor_functor
Fiber functors in category theory, topology and algebraic geometry refer to several loosely related functors that generalise the functors taking a covering
Fiber_functor
Mathematical category
the category of contravariant functors from D {\displaystyle D} to the category of sets; such a contravariant functor is frequently called a presheaf
Topos
In category theory, a branch of mathematics, a conservative functor is a functor F : C → D {\displaystyle F:C\to D} such that for any morphism f in C,
Conservative_functor
In mathematics, an effaceable functor is an additive functor F between abelian categories C and D for which, for each object A in C, there exists a monomorphism
Effaceable_functor
Operation in algebra and mathematics
a triple ( T , η , μ ) {\displaystyle (T,\eta ,\mu )} consisting of a functor T from a category to itself and two natural transformations η , μ {\displaystyle
Monad_(category_theory)
Functor between abelian categories
In homological algebra, a δ-functor between two abelian categories A and B is a collection of functors from A to B together with a collection of morphisms
Delta-functor
category theory, the notion of final functor is a generalization of the notion of cofinal set from order theory. A functor F : C → D {\displaystyle F:C\to
Final_functor
In type theory, a polynomial functor (or container functor) is a kind of endofunctor of a category of types that is intimately related to the concept
Polynomial functor (type theory)
Polynomial_functor_(type_theory)
Concept in algebraic topology
generally, the homology functor is defined axiomatically, as a functor on an abelian category, or, alternately, as a functor on chain complexes, satisfying
Singular_homology
Characterizing property of mathematical constructions
Technically, a universal property is defined in terms of categories and functors by means of a universal morphism (see § Formal definition, below). Universal
Universal_property
Programming construct
In some languages, particularly C++, function objects are often called functors (not related to the functional programming concept). A typical use of a
Function_object
Mathematical construction used in homotopy theory
topological spaces. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were
Simplicial_set
Technique for studying functors
calculus of functors or Goodwillie calculus is a technique for studying functors by approximating them by a sequence of simpler functors; it generalizes
Calculus_of_functors
Mathematical functor in representation theory and algebraic topology
particularly in representation theory and algebraic topology, a Mackey functor is a type of functor that generalizes various constructions in group theory and equivariant
Mackey_functor
signalizer functor is a mapping from a potential finite subgroup to the centralizers of the nontrivial elements of an abelian group. The signalizer functor theorem
Signalizer_functor
Construction in algebraic topology
the inverse image functor is a functor from the category of sheaves on Y to the category of sheaves on X. The direct image functor is the primary operation
Inverse_image_functor
Intermediate structure between functors and monads
an applicative functor, or an applicative for short, is an intermediate structure between functors and monads. Applicative functors allow for functorial
Applicative_functor
Mathematical object that generalizes the standard notions of sets and functions
of all small categories, with functors between them as morphisms. In turn, a functor category has as objects functors between two fixed categories and
Category_(mathematics)
Generalization of category
(small) categories, where a 2-morphism is a natural transformation between functors. The concept of a strict 2-category was first introduced by Charles Ehresmann
2-category
Certain functors from the category of modules over a fixed commutative ring to itself
especially in the field of representation theory, Schur functors (named after Issai Schur) are certain functors from the category of modules over a fixed commutative
Schur_functor
Concept in category theory
theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two
Monoidal_functor
Category equipped with a faithful functor to the category of sets
category that is equipped with a faithful functor to the category of sets (or sometimes to another category). This functor makes it possible to think of the objects
Concrete_category
Map (arrow) between two objects of a category
diffeomorphisms. In the category of small categories, the morphisms are functors. In a functor category, the morphisms are natural transformations. For more examples
Morphism
Construction in category theory
then just a contravariant functor I → C. Let C I o p {\displaystyle C^{I^{\mathrm {op} }}} be the category of these functors (with natural transformations
Inverse_limit
Generalization of a category
general simplicial set there is a functor τ {\displaystyle \tau } from sSet to Cat, the left-adjoint of the nerve functor, and for a quasi-category C, we
Quasi-category
respect to some universe) and the morphisms functors. Fct(C, D), the functor category: the category of functors from a category C to a category D. Set, the
Glossary_of_category_theory
Special case of colimit in category theory
the same as a covariant functor I → C {\displaystyle {\mathcal {I}}\rightarrow {\mathcal {C}}} . The colimit of this functor is the same as the direct
Direct_limit
mathematics, a smooth functor is a type of functor defined on finite-dimensional real vector spaces. Intuitively, a smooth functor is smooth in the sense
Smooth_functor
Category-theoretic construction
Thus the contravariant hom-functor changes coproducts into products. Stated another way, the hom-functor, viewed as a functor from the opposite category
Coproduct
Design pattern in functional programming to build generic types
of any functor with its inverse. Category theory views these collection monads as adjunctions between the free functor and different functors from the
Monad (functional programming)
Monad_(functional_programming)
Mathematical operation with two operands
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally
Binary_operation
Special objects used in (mathematical) category theory
categorical sum. It follows that any functor which preserves limits will take terminal objects to terminal objects, and any functor which preserves colimits will
Initial_and_terminal_objects
In mathematics, invertible homomorphism
{\displaystyle FG=1_{D}} (the identity functor on D) and G F = 1 C {\displaystyle GF=1_{C}} (the identity functor on C). In a concrete category (roughly
Isomorphism
Category with direct sums and certain types of kernels and cokernels
category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These
Abelian_category
Algebraization of first-order logic
In mathematical logic, predicate functor logic (PFL) is one of several ways to express first-order logic (also known as predicate logic) by purely algebraic
Predicate_functor_logic
Topic in abstract algebra
It turns out that there are applications of our functors which make use of the analogous transformations which we like to think of as a change of basis
Tilting_theory
Endofunctor on the category V of finite-dimensional vector spaces
In algebra, a polynomial functor is an endofunctor on the category V {\displaystyle {\mathcal {V}}} of finite-dimensional vector spaces that depends polynomially
Polynomial_functor
the formal criteria for adjoint functors are criteria for the existence of a left or right adjoint of a given functor. One criterion is the following
Formal criteria for adjoint functors
Formal_criteria_for_adjoint_functors
Construction in category theory
a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category
Cone_(category_theory)
Mathematical group occurring in algebraic geometry and the theory of complex manifolds
lemma. The construction of a scheme structure on (the representable functor version of) the Picard group, the Picard scheme, is an important step in
Picard_group
Branch of mathematics
Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development
Homological_algebra
Mathematical category whose hom sets form Abelian groups
{\displaystyle C} and D {\displaystyle D} are preadditive categories, then a functor F : C → D {\displaystyle F:C\rightarrow D} is additive if it too is enriched
Preadditive_category
Category whose objects and morphisms are inside a bigger category
There is an obvious faithful functor I : S → C {\displaystyle I:{\mathcal {S}}\to {\mathcal {C}}} , called the inclusion functor which takes objects and morphisms
Subcategory
Mathematical group formed from the automorphisms of an object
{\displaystyle C_{2}} , and if F : C 1 → C 2 {\displaystyle F:C_{1}\to C_{2}} is a functor mapping X 1 {\displaystyle X_{1}} to X 2 {\displaystyle X_{2}} , then F
Automorphism_group
Type of category in category theory
The third condition is equivalent to the requirement that the functor –×Y (i.e. the functor from C to C that maps objects X to X×Y and morphisms φ to φ × idY)
Cartesian_closed_category
In mathematics, a mapping between categories
In mathematics, the direct image functor describes how structured data assigned to one space can be systematically transferred to another space using
Direct_image_functor
Abstract mathematics relationship
equivalence of categories consists of a functor between the involved categories, which is required to have an "inverse" functor. However, in contrast to the situation
Equivalence_of_categories
Contravariant functor to Set
branch of mathematics, a presheaf on a category C {\displaystyle C} is a functor F : C o p → S e t {\displaystyle F\colon C^{\mathrm {op} }\to \mathbf {Set}
Presheaf_(category_theory)
Generalized object in category theory
the components and projections. If we regard this diagram as a functor, it is a functor from the index set I {\displaystyle I} considered as a discrete
Product_(category_theory)
Collection of maps which give the same result
Cat is naturally a 2-category, with functors as its arrows and natural transformations as the arrows between functors. In this setting, commutative diagrams
Commutative_diagram
translation functor is a functor taking representations of a Lie algebra to representations with a possibly different central character. Translation functors were
Translation_functor
Category whose hom sets have algebraic structure
properties. An enriched functor is the appropriate generalization of the notion of a functor to enriched categories. Enriched functors are then maps between
Enriched_category
Left adjoint to a forgetful functor to sets
that is equipped with a faithful functor to Set, the category of sets. Let C be a concrete category with a faithful functor U : C → Set. Let X be a set (that
Free_object
In mathematics, specifically in category theory, a functor F : C → D {\displaystyle F:C\to D} is essentially surjective if each object d {\displaystyle
Essentially surjective functor
Essentially_surjective_functor
Formalism in homological algebra
operations are six functors. Usually these are functors between derived categories and so are actually left and right derived functors. the direct image
Six_operations
In mathematics, a Zuckerman functor is used to construct representations of real reductive Lie groups from representations of Levi subgroups. They were
Zuckerman_functor
Concept in category theory
pullback functor taking bundles on Y to bundles on X. Fibred categories formalise the system consisting of these categories and inverse image functors. Similar
Fibred_category
Indexed collection of objects and morphisms in a category
equivalently, a functor from a fixed index category to some category. Formally, a diagram of type J in a category C is a (covariant) functor D : J → C. The
Diagram_(category_theory)
Surjective homomorphism
-)&\rightarrow &\operatorname {Hom} (X,-)\end{matrix}}} being a monomorphism in the functor category SetC. Every coequalizer is an epimorphism, a consequence of the
Epimorphism
Monoidal category
gist of the theory is that the fiber functor Φ of the Galois theory is replaced by an exact and faithful tensor functor F from C to the category of finite-dimensional
Tannakian_formalism
Most general completion of a commutative square given two morphisms with same codomain
R, is given by the tensor product over R, and Spec is a contravariant functor, the pullback of two affine schemes Spec(A) and Spec(B) over Spec(R), usually
Pullback_(category_theory)
Hom functor are adjoint; however, they might not always lift to an exact sequence. This leads to the definition of the Tor functor and the Ext functor. A
Lift_(mathematics)
Overview of and topical guide to category theory
categories Subcategory Faithful functor Full functor Forgetful functor Representable functor Functor category Adjoint functors Galois connection Pontryagin
Outline_of_category_theory
Connects set theory with category theory
replaces sets with categories, functions with functors, and equations with natural isomorphisms of functors satisfying additional properties. The term was
Categorification
Endofunctor on the category of simplicial sets
(extension functor or Ex functor) is an endofunctor on the category of simplicial sets. Due to many remarkable properties, the extension functor has plenty
Extension_(simplicial_set)
Abstract homotopical model for topological spaces
consider globular objects in a category C {\displaystyle {\mathcal {C}}} as functors X ∙ : G o p → C . {\displaystyle X_{\bullet }\colon \mathbb {G} ^{op}\to
∞-groupoid
geometry, a functor represented by a scheme X is a set-valued contravariant functor on the category of schemes such that the value of the functor at each
Functor represented by a scheme
Functor_represented_by_a_scheme
Applications of category theory
Complete Concrete Forgetful functor Pre-abelian Preadditive Commutative diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal
Applied_category_theory
Theorem relating to algebraic topology
In mathematics, the Landweber exact functor theorem, named after Peter Landweber, is a theorem in algebraic topology. It is known that a complex orientation
Landweber exact functor theorem
Landweber_exact_functor_theorem
Category mapping
category to the category Cat of (small) categories that is just like a functor except that F ( f ∘ g ) = F ( f ) ∘ F ( g ) {\displaystyle F(f\circ g)=F(f)\circ
Pseudo-functor
Category admitting tensor products
a strict monoidal category with the composition of functors as the product and the identity functor as the unit. Just like for any category E, the full
Monoidal_category
Theorem in category theory
Complete Concrete Forgetful functor Pre-abelian Preadditive Commutative diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal
Lawvere's_fixed-point_theorem
Exact sequence used to describe the structure of an object
functors RiF(En) vanish for all i > 0 and n ≥ 0. Dually, a left resolution is acyclic with respect to a right exact functor if its derived functors vanish
Resolution_(algebra)
Multi-dimensional generalization of triangle
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex
Simplex
Mathematical functor
In mathematics, a topological half-exact functor F is a functor from a fixed topological category (for example CW complexes or pointed spaces) to an abelian
Topological half-exact functor
Topological_half-exact_functor
Every Boolean algebra is isomorphic to a certain field of sets
other words, there is a contravariant functor that gives an equivalence between the categories. The inverse functor's action on morphisms maps a continuous
Stone's representation theorem for Boolean algebras
Stone's_representation_theorem_for_Boolean_algebras
Injective homomorphism
Complete Concrete Forgetful functor Pre-abelian Preadditive Commutative diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal
Monomorphism
FUNCTOR
FUNCTOR
FUNCTOR
FUNCTOR
Girl/Female
Hindu, Indian
Creeper of Love
Boy/Male
German
Graceful
Boy/Male
Hindu
Living
Girl/Female
Hindu
Test, Exam
Boy/Male
African, American, Australian, British, Chinese, Christian, Danish, English, French, Gaelic, German, Greek, Indian, Irish, Jamaican, Scottish, Tamil
Victory of the People; Young Boy; Abbreviation of Nicholas People's Victory; Young Creature; Victor; People's Victory; Cub; Pup; Dove; Oak Meadow
Boy/Male
Bengali, Hindu, Indian
Best; King Among Beautiful Persons
Girl/Female
Muslim/Islamic
Name of a tree
Surname or Lastname
English (mainly northern)
English (mainly northern) : habitational name from any of various minor places, in Lancashire and elsewhere, named from Middle English sc(h)ole ‘hut’ (see Scales) + feld ‘pasture’, ‘open country’.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Extreme Delight
Boy/Male
Tamil
King of water
FUNCTOR
FUNCTOR
FUNCTOR
FUNCTOR
FUNCTOR