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Functor type
category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations
Representable_functor
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
Functor mapping hom objects to an underlying category
to the tensor product functor – ⊗ {\displaystyle \otimes } R M: Ab → Mod-R. Ext functor Functor category Representable functor Also commonly denoted Cop
Hom_functor
Mapping between categories
Functors like these are called representable functors. An important goal in many settings is to determine whether a given functor is representable. Let
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
Mathematical space
Grassmannian can be constructed as a scheme by expressing it as a representable functor. If E {\displaystyle {\mathcal {E}}} is a quasi-coherent sheaf on
Grassmannian
Central object of study in category theory
covariant functor F X : C → Set {\displaystyle F_{X}:C\to {\textbf {Set}}} . This functor is called representable (more generally, a representable functor is
Natural_transformation
Mathematical structures in category theory
− , X ) {\displaystyle {\text{Hom}}(-,X)} be the contravariant representable functor from C {\displaystyle C} to Set {\displaystyle {\textbf {Set}}}
Functor_category
On representability of a contravariant functor on the category of connected CW complexes
pointed connected CW complexes, to the category of sets Set, to be a representable functor. More specifically, we are given F: Hotcop → Set, and there are
Brown's representability theorem
Brown's_representability_theorem
_{tr}(X)\cong \mathbb {Z} \oplus \mathbb {Z} _{tr}(X,x)} . There is a representable functor associated to the pointed scheme G m = ( A 1 − { 0 } , 1 ) {\displaystyle
Presheaf_with_transfers
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
Mathematical heuristic
concept of representable functor can make that point more precise: an object is as good as its representable functor. Representable functors were defined
Grothendieck's relative point of view
Grothendieck's_relative_point_of_view
Concept in mathematical category theory
{\displaystyle {\mathcal {E}}} fibered over C {\displaystyle {\mathcal {C}}} by a functor π {\displaystyle \pi } whose fibers are the categories { F ( c ) } c ∈
Category_of_elements
Mathematical concept
lift limits uniquely but neither create nor reflect them. Every representable functor C → Set preserves limits (but not necessarily colimits). In particular
Limit_(category_theory)
Mathematical group occurring in algebraic geometry and the theory of complex manifolds
Dolbeault–Grothendieck lemma. The construction of a scheme structure on (the representable functor version of) the Picard group, the Picard scheme, is an important
Picard_group
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
General theory of mathematical structures
most famous basic results of category theory; it describes representable functors in functor categories. Duality: Every statement, theorem, or definition
Category_theory
American mathematician (born 1937)
illustrations. The representable functor point of view is central: Mumford illustrates it by citing contemporary research: Brown's representability theorem in
David_Mumford
Topics referred to by the same term
Representability in mathematics can refer to the existence of a representable functor in category theory Birch's theorem about the representability of
Representability
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
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
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
Suppose that this inclusion morphism G → F is representable by open immersions, i.e., for any representable functor Hom(−, X) and any morphism Hom(−, X) → F
Subfunctor
representable presheaves in a canonical way. For example, by definition, a simplicial set is a presheaf on the simplex category Δ and a representable
Density theorem (category theory)
Density_theorem_(category_theory)
Mathematical functor
the (representable) functor F(X)=[X,B] is half-exact. https://math.stackexchange.com/questions/4615272/showing-a-topological-half-exact-functor-is-topological-exact
Topological half-exact functor
Topological_half-exact_functor
Generalization of algebraic spaces or schemes
{X}}_{\operatorname {Spec} (k)}} is representable as an algebraic space. Another important equivalence of having a representable diagonal is the technical condition
Algebraic_stack
Mathematical concept that extends the intuitive idea of gluing in topology
of existence (see FGA) connecting the descent question with the representable functor question in algebraic geometry in general, and the moduli problem
Descent_(mathematics)
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)
Tool to track locally defined data attached to the open sets of a topological space
to the given presheaf. This construction makes all sheaves into representable functors on certain categories of topological spaces. As above, let F {\displaystyle
Sheaf_(mathematics)
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
Quotient of a weakly contractible space by a free action
introduced) this is a question of whether a certain functor is representable: the contravariant functor from the homotopy category to the category of sets
Classifying_space
1016/0021-8693(78)90160-6. Lurie, Jacob (2009). "5.5.2 Representable Functors and the Adjoint Functor Theorem". Higher Topos Theory. Princeton University
Formal criteria for adjoint functors
Formal_criteria_for_adjoint_functors
space over k. A functor is called representable if it is of the form hX where hX(Y)=hom(X,Y) for some X, and is called pro-representable if it is of the
Schlessinger's_theorem
Adjunction between a category of co/presheaf under the co/Yoneda embedding
X ∈ A {\displaystyle X\in {\mathcal {A}}} to the contravariant representable functor: y ( h ∙ ) : A → [ A o p , V ] {\displaystyle y\;(h^{\bullet }):{\mathcal
Isbell_duality
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
Concept in math
groups. Singular cohomology has an even better property: it is a representable functor on the homotopy category. That is, for each abelian group A and
Homotopy_category
Moduli space in the Grothendieck category of schemes
problems is to set them up as a representable functor question, then apply a criterion that singles out the representable functors for schemes. When this programmatic
Moduli_scheme
Theoretical object in mathematics
{M}}{\mathfrak {R}},} then defining F1‑schemes to be a particular kind of representable functor on M R . {\displaystyle {\mathfrak {M}}{\mathfrak {R}}.} Using this
Field_with_one_element
Category where every morphism is invertible; generalization of a group
{\displaystyle G} -action on the set G {\displaystyle G} . The (unique) representable functor F : G r → S e t {\displaystyle F:\mathrm {Gr} \to \mathrm {Set}
Groupoid
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
that this functor is representable by the scheme X. The affine Grassmannian is a functor from k-algebras to sets which is not itself representable, but which
Affine_Grassmannian
Type of Grothendieck topology on the category of schemes
F on X is called finite locally constant if it is a representable functor which can be represented by an étale cover of X. It is called constructible if
Étale_topology
mapping induced by the pullback with f, is representable. The universal element representing this functor is the pair (Av, P). This association is a duality
Dual_abelian_variety
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
In category theory, a branch of mathematics, the diagonal functor C → C × C {\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}}
Diagonal_functor
Contravariant functor to Set
the contravariant hom-functor Hom(–, A) for some object A of C is called a representable presheaf. Some authors refer to a functor F : C o p → V {\displaystyle
Presheaf_(category_theory)
Yoneda's lemma, as a way of identifying S {\displaystyle S} with the representable functor h S {\displaystyle h_{S}} it sets up. Historically there was a process
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Concept in mathematics
statement that the tensor product − ⊗ X {\displaystyle -\otimes X} and hom-functor Hom ( X , − ) {\displaystyle \operatorname {Hom} (X,-)} form an adjoint
Tensor–hom_adjunction
Type of topological space
basic result is that the representable functors on the homotopy category have a simple characterisation (the Brown representability theorem). Abstract cell
CW_complex
In mathematics, process for extending a category
shape theory. Pro-objects also arise via their connection to pro-representable functors, for example in Grothendieck's Galois theory, and also in Schlessinger's
Ind-completion
Mathematical category whose hom sets form Abelian groups
homomorphism from R {\displaystyle R} to S {\displaystyle S} is represented by an additive functor from C R {\displaystyle C_{R}} to C S {\displaystyle C_{S}}
Preadditive_category
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
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
Category whose hom sets have algebraic structure
usual cartesian product, the definitions of enriched category, enriched functor, etc... reduce to the original definitions from ordinary category theory
Enriched_category
Category
pre-abelian category, exact functors can be described in particularly simple terms. First, recall that an additive functor is a functor F: C → D between preadditive
Pre-abelian_category
Structure in algebraic geometry
{\displaystyle \mathbf {Z} _{tr}(Y)(Z):={\text{Hom}}_{cor}(Z,Y)} is the representable functor over the category of presheaves with transfers. For the Nisnevich
Nisnevich_topology
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
Duality for locally compact abelian groups
{\displaystyle {\widehat {G}}} is a contravariant functor LCA → LCA, represented (in the sense of representable functors) by the circle group T {\displaystyle \mathbb
Pontryagin_duality
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
1960–67 foundational treatise on algebraic geometry by Alexander Grothendieck
"scheme" by "separated scheme", and heavily emphasizes the use of representable functors. The new preface of the second edition also includes a slightly
Éléments de géométrie algébrique
Éléments_de_géométrie_algébrique
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
presentation is not used much as it is not subcanonical; in other words, representable functors need not be sheaves. Unfortunately the terminology for flat topologies
Flat_topology
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
American mathematician (born 1934)
compact manifolds. His work on the problem of characterising the representable functors in the category of schemes has led to the Artin approximation theorem
Michael_Artin
Formalization in mathematical topos theory
nature. The third insight is that over a certain category, these are representable functors. Furthermore, their representatives are related to the algebras
Synthetic differential geometry
Synthetic_differential_geometry
Topological space
In mathematics, more specifically in topology, the Volodin space X {\displaystyle X} of a ring R is a subspace of the classifying space B G L ( R ) {\displaystyle
Volodin_space
Type of category in category theory
must be additive functors (see here). Most of the interesting functors studied in category theory are adjoints. When considering functors between R-linear
Additive_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
Category theory
{\displaystyle {\hat {\mathbf {C} }}} (the presheaves over C) of retracts of representable functors. The category of presheaves on C is equivalent to the category of
Karoubi_envelope
faithfully-flat descent, the discovery of Grothendieck that the representable functors are sheaves for it (i.e. a very general gluing axiom holds). Function
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
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
Mathematical object that generalizes the standard notions of sets and functions
two categories compatible with their respective structures is called a functor. Well-known categories are denoted by a short capitalized word or abbreviation
Category_(mathematics)
Topological concept in algebraic geometry
functor F {\displaystyle F} is typically not representable in C {\displaystyle C} ; however, it is pro-representable in C {\displaystyle C} , in fact by Galois
Étale_fundamental_group
Concept in algebraic geometry
local finiteness may hold) failure of components to be representable as schemes, although representable topologically. It is the third point that motivated
Geometric_invariant_theory
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)
Collection of maps which give the same result
diagram in a category C can be interpreted as a functor from an index category J to C; one calls the functor a diagram. More formally, a commutative diagram
Commutative_diagram
category. 2. A regular morphism. representable A set-valued contravariant functor F on a category C is said to be representable if it belongs to the essential
Glossary_of_category_theory
Topological group structure arising in Fourier analysis
{T} )} . More abstractly, these are both examples of representable functors, being represented respectively by K {\displaystyle K} and T {\displaystyle
Locally_compact_abelian_group
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
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
Category in mathematics
D^{\text{op}}\to {\text{Ab}}} a cohomological functor which takes coproducts to products. Then H is representable. (That is, there is an object W of D such
Triangulated_category
Restriction of scalars
mathematics, restriction of scalars (also known as "Weil restriction") is a functor which, for any finite extension of fields L/k and any algebraic variety
Weil_restriction
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
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
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
Concept in geometric topology
an assembly map is a universal approximation of a homotopy invariant functor by a homology theory from the left. From the geometric viewpoint, assembly
Assembly_map
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
Relation of categories in category theory
isomorphic if there exist functors F : C → D and G : D → C that are mutually inverse to each other, i.e. FG = 1D (the identity functor on D) and GF = 1C. This
Isomorphism_of_categories
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
Type of mathematical object
scheme G, the functor that takes an S-scheme T to G(T)/H(T) is in general not a sheaf, and even its sheafification is in general not representable as a scheme
Group_scheme
Product of two categories, in category theory
I} satisfy: given a family of functors f i : D → C i {\displaystyle f_{i}:D\to C_{i}} , there exists a unique functor f : D → P {\displaystyle f:D\to
Product_category
History of maths
presheaves. Yoneda lemma plays a critical role in the study of representable functors in algebraic geometry. For example, even though it is never mentioned
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
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
Category theory constructs
Kan extension from 1956 was in homological algebra to compute derived functors. In Categories for the Working Mathematician, Saunders Mac Lane titled
Kan_extension
Mathematical concept
In category theory, an end of a functor S : C o p × C → X {\displaystyle S\colon \mathbf {C} ^{\mathrm {op} }\times \mathbf {C} \to \mathbf {X} } is a
End_(category_theory)
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
coaugmented functor. A coaugmented functor is a pair (L,l) where L:C → C is an endofunctor and l:Id → L is a natural transformation from the identity functor to
Localization_of_a_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)
Mathematical category formed by reversing morphisms
Dual (category theory) Duality (mathematics) Adjoint functor Contravariant functor Opposite functor "Is there an introduction to probability theory from
Opposite_category
Algebraic geometry category satisfying lifting conditions
\Delta } is strongly representable, then every morphism U → X {\displaystyle U\to X} from a scheme U is strongly representable since U × X T ≃ ( U ×
Prestack
REPRESENTABLE FUNCTOR
REPRESENTABLE FUNCTOR
REPRESENTABLE FUNCTOR
REPRESENTABLE FUNCTOR
Boy/Male
Irish
Surname.
Female
Greek
(Αλκιππη) Greek name ALKIPPE means "mighty mare." In mythology, this was the name of a daughter of Ares.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Telugu
Lord Krishna
Boy/Male
Hindu, Indian
Dispersing Darkness; Fire; Lamp; The Moon
Boy/Male
Hindu
Lord of the sacrificial fire
Boy/Male
Indian, Punjabi, Sikh
Invincible Brave
Girl/Female
Hindu
The Moon
Girl/Female
Hindu, Indian, Marathi, Sanskrit
Goddess of Bliss
Surname or Lastname
English (Devon)
English (Devon) : habitational name from any of three places so named. Hingston, Cornwall and Hingston Down in Moretonhampstead, Devon are both named from the Old English byname Hengest (or from Old English hengest ‘stallion’) + Old English dÅ«n ‘hill’, while Hingston in Bigbury, Devon is named from Old English hind ‘hind’ + stÄn ‘stone’.
Girl/Female
Tamil
Nibodhitha | நீபோதீதா
Having been enlightened
REPRESENTABLE FUNCTOR
REPRESENTABLE FUNCTOR
REPRESENTABLE FUNCTOR
REPRESENTABLE FUNCTOR
REPRESENTABLE FUNCTOR
a.
Capable of being represented.
a.
Not capable of being represented or portrayed.
a.
Capable or admitting of being presented; suitable to be exhibited, represented, or offered; fit to be brought forward or set forth; hence, fitted to be introduced to another, or to go into society; as, ideas that are presentable in simple language; she is not presentable in such a gown.
a.
Admitting of the presentation of a clergiman; as, a church presentable.
a.
Having a well-formed body, or person; graceful; comely; of good appearance; presentable; as, a personable man or woman.
n.
Representation; likeness.