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Fiber functors in category theory, topology and algebraic geometry refer to several loosely related functors that generalise the functors taking a covering
Fiber_functor
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
Mathematical concept
Formally, a diagram of shape J {\displaystyle J} in C {\displaystyle C} is a functor from J {\displaystyle J} to C {\displaystyle C} : F : J → C . {\displaystyle
Limit_(category_theory)
family of functions. The notion of topological functors generalizes (and strengthens) that of fibered categories, for which one considers a single morphism
Topological_functor
Concept in category theory
{F}}_{c}} , and a morphism d → c {\displaystyle d\to c} induces a functor from the fibered category structure. Namely, for an object x ∈ Ob ( F c ) {\displaystyle
Fibred_category
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 mathematics, a mapping between categories
f−1(U), one uses the fiber product of U and X over Y. Forming sheaf categories and direct image functors itself defines a functor from the category of
Direct_image_functor
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)
Most general completion of a commutative square given two morphisms with same codomain
and Spec is a contravariant functor, the pullback of two affine schemes Spec(A) and Spec(B) over Spec(R), usually called fiber product, is given by Spec(A ⊗R B)
Pullback_(category_theory)
Exceptional functor
In category theory, a branch of mathematics, certain unusual functors are denoted f ! {\displaystyle f_{!}} and f ! , {\displaystyle f^{!},} with the exclamation
Shriek_map
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
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
Right inverse of a fiber bundle map
In the mathematical field of topology, a section (or cross section) of a fiber bundle E {\displaystyle E} is a continuous right inverse of the projection
Section_(fiber_bundle)
Overview of and topical guide to category theory
Combinatorial species Exact functor Derived functor Dominant functor Enriched functor Kan extension of a functor Hom functor Yoneda lemma Product (category
Outline_of_category_theory
Theory in number theory
mixed-characteristic local fields. Section conjecture Class field theory Fiber functor Neukirch–Uchida theorem Belyi's theorem Inter-universal Teichmüller
Anabelian_geometry
Category in mathematics
category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category
Triangulated_category
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 )
Category_of_elements
involves a functor, the nearby cycle functor, with a definition by means of the higher direct image and pullbacks. The vanishing cycle functor then sits
Vanishing_cycle
Process in mathematics
section of the pullback (fiber-product) bundle f ∗ E {\displaystyle f^{*}E} over M . {\displaystyle M.} Inverse image functor – Construction in algebraic
Pullback
exists that is a final object among all lifts. For example, the forgetful functor QCoh → Sch {\displaystyle {\textrm {QCoh}}\to {\textrm {Sch}}} from the
Cartesian_fibration
Subgroup of the group of invertible n×n matrices
form a tannakian category RepG. In fact, tannakian categories with a "fiber functor" over a field are equivalent to affine group schemes. (Every affine
Linear_algebraic_group
Quotient of a weakly contractible space by a free action
set-valued functor on the homotopy category of topological spaces. The term classifying space can also be used for spaces that represent a set-valued functor on
Classifying_space
Mathematical structure
consequently it was possible to make constructions that imitated the cohomology functor H 1 {\displaystyle H^{1}} . Grothendieck saw that it would be possible
Grothendieck_topology
Abstract approach to algebraic geometry
this is a part of the study of atomic toposes. Tannakian formalism Fiber functor Anabelian geometry Grothendieck, Alexander; et al. (1971). SGA1 Revêtements
Grothendieck's_Galois_theory
Mathematical heuristic
of representable functor can make that point more precise: an object is as good as its representable functor. Representable functors were defined explicitly
Grothendieck's relative point of view
Grothendieck's_relative_point_of_view
Most general completion of a commutative square given two morphisms with same domain
category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the colimit
Pushout_(category_theory)
Concepts in algebraic topology
replaced with a point space, we recover the original functor Δ0. A homotopy pullback (or homotopy fiber-product) is the dual concept of a homotopy pushout
Homotopy_colimit_and_limit
Mathematical concept that extends the intuitive idea of gluing in topology
when this functor is an equivalence of categories. Grothendieck connection Stack (mathematics) Galois descent Grothendieck topology Fibered category Beck's
Descent_(mathematics)
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
Operation in algebra
category theory, the extension of scalars functor is left adjoint to the restriction of scalars functor. Fiber product of schemes § Base change and descent
Change_of_rings
such that π {\displaystyle \pi } is injective as a function. fiber functor fiber functor. Frobenius reciprocity The Frobenius reciprocity states that
Glossary of representation theory
Glossary_of_representation_theory
Construct in mathematics
of infinitesimal thickenings Twisted forms of projective varieties Fiber functors for motives H 3 ( X , Z ) {\displaystyle H^{3}(X,\mathbb {Z} )} and
Gerbe
Mathematical object that generalizes the standard notions of sets and functions
table. Fiber bundles with bundle maps between them form a concrete category. The category Cat consists of all small categories, with functors between
Category_(mathematics)
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
a covariant functor from the category of spaces to the category of abelian groups, while a cohomology theory is a contravariant functor from the category
Bivariant_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
Category whose objects are topological spaces and whose morphisms are continuous maps
with T o p {\displaystyle \mathbf {Top} } (such as fiber-completeness, discrete and indiscrete functors, and unique lifting of limits). The category T o
Category of topological spaces
Category_of_topological_spaces
stabilization of an ∞-category C having finite limits and base point is a functor from the stable ∞-category S to C. It preserves limits. The objects in
Stable_∞-category
Generalization of a fiber bundle
categories. The functor taking each manifold to its tangent bundle is an example of a section of this bundle object. Fiber bundle Fibration Fibered manifold
Bundle_(mathematics)
Geometric space whose points represent algebro-geometric objects of some fixed kind
{\displaystyle \phi (s_{i})=s_{i}'} . This means the associated moduli functor P Z n : Sch → Sets {\displaystyle \mathbf {P} _{\mathbb {Z} }^{n}:{\text{Sch}}\to
Moduli_space
Intuitively, the Puppe sequence allows us to think of homology theory as a functor that takes spaces to long-exact sequences of groups. It is also useful
Puppe_sequence
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
Branch of mathematics
These functors are used to construct fiber sequences and cofiber sequences. Namely, if f : X → Y {\displaystyle f:X\to Y} is a map, the fiber sequence
Homotopy_theory
Algebraic geometry category satisfying lifting conditions
topology is a category together with a functor p: F → C satisfying a certain lifting condition and such that (when the fibers are groupoids) locally isomorphic
Prestack
Type of mathematical object
and inverse axioms) a functor from schemes over S to the category of groups, such that composition with the forgetful functor to sets is equivalent to
Group_scheme
History of maths
analogy of ring theory with geometric cases. 1960 Alexander Grothendieck Fiber functors 1960 Daniel Kan Kan extensions 1960 Alexander Grothendieck Formal algebraic
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
Category where every morphism is invertible; generalization of a group
{G}}_{0}} with functors s , t : G 1 → G 0 {\displaystyle s,t:{\mathcal {G}}_{1}\to {\mathcal {G}}_{0}} and an embedding given by an identity functor i : G 0
Groupoid
Theorem in category theory
Beck's monadicity theorem gives a criterion that characterises monadic functors, introduced by Jonathan Mock Beck (1968). It is often stated in dual form
Beck's_monadicity_theorem
Generalisation of a sheaf; a fibered category that admits effective descent
overcounted. A category c {\displaystyle c} with a functor to a category C {\displaystyle C} is called a fibered category over C {\displaystyle C} if for any
Stack_(mathematics)
Concept in category theory
F be a functor from the category of k-algebras to the category of sets. Then, for any k-point p ∈ F ( k ) {\displaystyle p\in F(k)} , the fiber of π :
Tangent_space_to_a_functor
Branch of mathematics
we could consider the functor F : Sch → Sets {\displaystyle F:{\text{Sch}}\to {\text{Sets}}} where F ( S ) = { X ↓ S : each fiber is a degree d hypersurface
Deformation_(mathematics)
Relates the geometric vector bundles to algebraic projective modules
above theorem is that for any connected smooth manifold M, the section functor Γ from the category of smooth vector bundles over M to the category of
Serre–Swan_theorem
Continuous deformation between two continuous functions
homotopy equivalent. Then a functor on the category of topological spaces is homotopy invariant if it can be expressed as a functor on the homotopy category
Homotopy
In algebraic geometry, a Fourier–Mukai transform ΦK is a functor between derived categories of coherent sheaves D(X) → D(Y) for schemes X and Y, which
Fourier–Mukai_transform
Two theorems needed for Quillen's Q-construction in algebraic K-theory
follows. Quillen's Theorem A—If f : C → D {\displaystyle f:C\to D} is a functor such that the classifying space B ( d ↓ f ) {\displaystyle B(d\downarrow
Quillen's_theorems_A_and_B
Transforming a function in such a way that it only takes a single argument
adjoint functor that maps suspensions to loop spaces, and uncurrying is the dual. The duality between the mapping cone and the mapping fiber (cofibration
Currying
Generalization of algebraic variety
functor from commutative R-algebras to sets. It is an important observation that a scheme X over R is determined by this functor of points. The fiber
Scheme_(mathematics)
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
Mathematical sequence
be a continuous map of topological spaces, which in particular gives a functor f ∗ {\displaystyle f_{*}} from sheaves of abelian groups on X {\displaystyle
Leray_spectral_sequence
Process of extending a representation of a subgroup to the parent group
respectively. With the addition of the normalizing factors this induction functor takes unitary representations to unitary representations. One other variation
Induced_representation
Type of vector bundle
the trivial object O X {\displaystyle {\mathcal {O}}_{X}} and the fiber functor x ∗ {\displaystyle x^{*}} is a Tannakian category. The k {\displaystyle
Essentially finite vector bundle
Essentially_finite_vector_bundle
or a colimit of presheaves on a category C is a limit or colimit in the functor category C ^ = F c t ( C op , S e t ) {\displaystyle {\widehat {C}}=\mathbf
Limit and colimit of presheaves
Limit_and_colimit_of_presheaves
Topological concept in algebraic geometry
geometrically this is the fiber of Y → X {\displaystyle Y\to X} over x , {\displaystyle x,} and abstractly it is the Yoneda functor represented by x {\displaystyle
Étale_fundamental_group
Generalization of algebraic spaces or schemes
surjective morphisms of fibered categories. Here U {\displaystyle {\mathcal {U}}} is the algebraic stack from the representable functor h U {\displaystyle
Algebraic_stack
Concept in algebraic topology
E\to B} with fiber F {\displaystyle F} and a fixed commutative ring R {\displaystyle R} with a unit, there exists a contravariant functor from the fundamental
Fibration
Abstract homotopical model for topological spaces
_{\leq n-1}:\Pi _{n}X\to \Pi _{n-1}X} whose fibers should be the categories of n {\displaystyle n} -functors Π n ( K ( π n X , n ) ) → D ( Ab ) {\displaystyle
∞-groupoid
Long exact sequence
exact sequence which relates the cohomology classes of the base space, the fiber and the total space of a sphere bundle. The Gysin sequence is a useful tool
Gysin_homomorphism
Mathematics glossary
homology is the singular homology of X. 2. The singular simplices functor is the functor T o p → s S e t {\displaystyle \mathbf {Top} \to s\mathbf {Set}
Glossary of algebraic topology
Glossary_of_algebraic_topology
Theory for associative algebras over rings
over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by Gerhard Hochschild (1945) for
Hochschild_homology
Fiber bundle induced by a map of its base space
pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle π : E → B {\displaystyle \pi :E\rightarrow
Pullback_bundle
Mathematical theory
representations based on how nice they are, and also provides faithful functors to categories of linear algebraic objects that are easier to study. The
P-adic_Hodge_theory
Branch of mathematics
the formula involves the Tor functor and thus, unless higher Tor vanish, the scheme-theoretic intersection (i.e., fiber product of immersions) does not
Derived_algebraic_geometry
the fiber p−1(A) form a set. Then we get a functor Fp : Fields/k → Set taking a field extension K/k to the set of isomorphism classes in the fiber p −
Essential_dimension
Transformations induced by a mathematical group
is then nothing but a (covariant) functor from G to the category of sets, and a group representation is a functor from G to the category of vector spaces
Group_action
Concept in homological algebra
localization functors L whose essential image is closed under extension, meaning that if X → Y → Z {\displaystyle X\to Y\to Z} is a fiber sequence with
T-structure
Moduli scheme of subschemes of a scheme, represents the flat-family-of-subschemes functor
property is that for a scheme T {\displaystyle T} , it represents the functor whose T {\displaystyle T} -valued points are the closed subschemes of P
Hilbert_scheme
Set of a ring's prime ideals
contravariant functor from the category of commutative rings to the category of locally ringed spaces. In fact it is the universal such functor, and hence
Spectrum_of_a_ring
Elements taken to zero by a homomorphism
visualized with the commutative diagram: Functors between categories can also have a kernel. A (covariant) functor from a category C {\displaystyle {\mathbf
Kernel_(algebra)
Analogue of homotopy type for algebraic varieties
homotopical object in simplicial profinite sets. This has a forgetful functor to the homotopy category of profinite simplicial sets. Artin, Michael;
Étale_homotopy_type
for any representable functor Hom(−, X) and any morphism Hom(−, X) → F, the fibered product G×FHom(−, X) is a representable functor Hom(−, Y) and the morphism
Subfunctor
Type of Grothendieck topology on the category of schemes
their fiber product over X. Ét(X) is a large category, meaning that its objects do not form a set. An étale presheaf on X is a contravariant functor from
Étale_topology
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 a scheme
Grassmannian
Mapping equal to its square under mapping composition
self-adjoint idempotent linear operator. In differential topology, any fiber bundle includes a projection map as part of its definition. Locally at least
Projection_(mathematics)
Algebraic structure
homological methods, such as the Ext functor. This functor is the derived functor of the functor HomR(M, −). The latter functor is exact if M is projective, but
Commutative_ring
Concept in mathematics
correspondence (for regular singular connections): there is a functor Sol called the local solutions functor, that is an equivalence from the category of flat connections
Riemann–Hilbert correspondence
Riemann–Hilbert_correspondence
Concept in algebraic topology
in degrees i > n {\displaystyle i>n} . More precisely, the restriction functor i ∗ : Δ o p S e t s → Δ ≤ n o p S e t s {\displaystyle i_{*}:\Delta ^{op}Sets\rightarrow
N-skeleton
Concept in algebraic geometry
For example, working with a category-valued (pseudo-)functor instead of a set-valued functor leads to the notion of a stack, which allows one to keep
Morphism_of_schemes
Mathematical operation on vector bundles
∗ : E ∗ → X {\displaystyle \pi ^{*}:E^{*}\to X} whose fibers are the dual spaces to the fibers of E {\displaystyle E} . Equivalently, E ∗ {\displaystyle
Dual_bundle
to keep track of certain information that is only latent in the moduli functor or moduli stack. — Kollár, János, Chapter 1, "Book on Moduli of Surfaces"
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Construction for simplicial sets
Y {\displaystyle Y} . Let Y {\displaystyle Y} be a simplicial set. The functor Y ⋄ − : s S e t → Y ∖ s S e t , X ↦ ( Y ↦ X ⋄ Y ) {\displaystyle Y\diamond
Diamond_operation
Relate the direct image and the pull-back of sheaves
{\mathcal {F}}} under f, i.e., the derived functor of the direct image (also known as pushforward) functor f ∗ {\displaystyle f_{*}} . This map exists
Base_change_theorems
Mathematical parametrization of vector spaces by another space
in a functorial manner. This is made precise in the language of smooth functors. An operation of a different nature is the pullback bundle construction
Vector_bundle
Mathematical operation
turns several constructions in differential geometry into contravariant functors. Let ϕ : M → N {\displaystyle \phi :M\to N} be a smooth map between (smooth)
Pullback (differential geometry)
Pullback_(differential_geometry)
Connection on a vector bundle
MR 2641188. Reiter, Stefan (2002). "On applications of Katz' middle convolution functor (Deformation of differential equations and asymptotic analysis)" (PDF)
Gauss–Manin_connection
Topological construction on a map between spaces
notated C f {\displaystyle Cf} . Its dual, a fibration, is called the mapping fiber. The mapping cone can be understood to be a mapping cylinder M f {\displaystyle
Mapping_cone_(topology)
Fiber bundle
In mathematics, the theory of fiber bundles with a structure group G {\displaystyle G} (a topological group) allows an operation of creating an associated
Associated_bundle
Special kind of model structure
morphism f : X → Y {\displaystyle f\colon X\rightarrow Y} in it, there is a functor f ∗ : Y ∖ M → X ∖ M {\displaystyle f^{*}\colon Y\backslash {\mathcal {M}}\rightarrow
Proper_model_structure
Mathematical group of the homotopy classes of loops in a topological space
associating to a topological space its fundamental group is therefore a functor π 1 : T o p ∗ → G r p ( X , x 0 ) ↦ π 1 ( X , x 0 ) {\displaystyle {\begin{aligned}\pi
Fundamental_group
Topological group structure arising in Fourier analysis
follows that the adeles are self-dual. Pontryagin duality asserts that the functor G ↦ G ^ {\displaystyle G\mapsto {\hat {G}}} induces an equivalence of categories
Locally_compact_abelian_group
Type of continuous map in topology
evaluating at the endpoint of the lift, a group action on the fiber of a covering, the functor F : C o v ( X ) ⟶ G − S e t : p ↦ p − 1 ( x ) {\displaystyle
Covering_space
FIBER FUNCTOR
FIBER FUNCTOR
Surname or Lastname
English
English : occupational name for a refiner of gold and other metals, from Middle English fine(n) ‘to refine or purify’ (a derivative of fine ‘fine’, ‘pure’).Probably a translated form of German Feiner.
Boy/Male
Australian, Biblical, Christian, French, German, Greek
The Son of Tiber; Of the Tiber (River)
Boy/Male
Italian
From the Tiber.
Boy/Male
British, English, Greek
Gujarati Words for String which Made by Coconut's Fibers
Boy/Male
Australian, Czechoslovakian, Danish, German, Hungarian, Slavic
Sacred Place; Of the River Tiber
Girl/Female
Afghan, American, Arabic, Hindu, Indian, Marathi, Telugu
Superior; Finer; Rising; Ascending; High-born; The High; Exalted One
Boy/Male
Latin
Dionysus.
Girl/Female
American, Australian, British, English, Portuguese
Bright Guardian; Of High Value; Of the Tiber
Male
Romanian
Romanian form of Roman Tiberius, TIBERIU means "of the Tiber (river)."
Boy/Male
English Latin
Derived from the Roman clan name Fabius; a name given several Roman emperors and 16 saints.
Boy/Male
American, Anglo, Australian, British, English, Portuguese
Bright Guardian; Of the Tiber; River
Boy/Male
Australian, Irish, Jamaican, Latin
Another Name for Dionysus; Free
Surname or Lastname
English
English : occupational name for a maker or user of files, from an agent derivative of Middle English file ‘file’.English : occupational name for a spinner, from an agent derivative of Middle English, Old French fil ‘thread’ (Latin filum).English : Americanized spelling of German Feiler, cognate of 1.
Male
Yiddish
 Variant spelling of Yiddish Lieber, LIBER means "beloved." Compare with another form of Liber.
Biblical
the son of Tiber
Girl/Female
Latin
From the Tiber.
Male
Czechoslovakian
, of the Tiber (river).
Boy/Male
American, British, English, French, Latin
Bean Grower; Derived from the Roman Clan Name Fabius; A Name Given Several Roman Emperors and 16 Saints; One who Grows Beans
Boy/Male
Biblical
The son of Tiber.
Girl/Female
Italian Latin
From the Tiber.
FIBER FUNCTOR
FIBER FUNCTOR
Boy/Male
Muslim/Islamic
Worship
Boy/Male
Tamil
Yajnadhar | யஜநாதர
Lord Vishnu
Girl/Female
Hindu
Fishlike beautiful eyes
Girl/Female
Indian, Kannada, Sanskrit
Ruler
Male
Hebrew
(ש×ְמוּ×ֵל) Hebrew name SHEMUWEL means "heard of God," "his name is El," or "name of God." In the bible, this is the name of several characters, including a son of Elkanah by Hannah.
Male
English
Great Ambition
Girl/Female
Latin
Happy. Feminine of Felix.
Boy/Male
Shakespearean
King Henry IV, Part 2' The play's presenter.
Female
English
Latin form of French Louise, LOUISA means "famous warrior."Â
Boy/Male
Indian, Sanskrit
Self Restrained
FIBER FUNCTOR
FIBER FUNCTOR
FIBER FUNCTOR
FIBER FUNCTOR
FIBER FUNCTOR
n.
A small cord, ligature, or fiber.
n.
A small fiber; the branch of a fiber; a very slender thread; a fibrilla.
a.
Having fibers; made up of fibers.
n.
The inner bark of plants, lying next to the wood. It usually contains a large proportion of woody, fibrous cells, and is, therefore, the part from which the fiber of the plant is obtained, as that of hemp, etc.
a.
Alt. of Fibre-faced
n.
Alt. of Fibre
n.
The plant which yields the fiber.
n.
Sinew; strength; toughness; as, a man of real fiber.
n.
One of the delicate, threadlike portions of which the tissues of plants and animals are in part constituted; as, the fiber of flax or of muscle.
n.
Gomuti fiber. See Gomuti.
n.
The longer and finer fiber of flax.
n.
An enlargement or swelling in a vessel, fiber, or the like; a varix; as, the varicosities of nerve fibers.
a.
Having no fibers; destitute of fibers or fiber.
a.
Having a visible fiber embodied in the surface of; -- applied esp. to a kind of paper for checks, drafts, etc.
a.
Having the form of a fiber or fibers; resembling a fiber.
a.
Composed of, or resembling, muscular fiber.
n.
Any fine, slender thread, or threadlike substance; as, a fiber of spun glass; especially, one of the slender rootlets of a plant.