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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
defined by means of diagrams in C. Formally, a diagram of shape J in C is a functor from J to C: F:J→C. The category J is thought of as an index category,
Limit_(category_theory)
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
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
family of functions. The notion of topological functors generalizes (and strengthens) that of fibered categories, for which one considers a single morphism
Topological_functor
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)
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
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)
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
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
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
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
Type of category in mathematics
the category of contravariant functors from D {\displaystyle D} to the category of sets; such a contravariant functor is frequently called a presheaf
Topos
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)
Process in mathematics
f∗s=s∘f of s with f is a section of the pullback (fiber-product) bundle f∗E over M. Inverse image functor – Construction in algebraic topology Pullback (category
Pullback
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)
Topics referred to by the same term
brand of cat food Whiskering, an operation in category theory between a functor and a natural transformation This disambiguation page lists articles associated
Whisker_(disambiguation)
Concept in mathematical category theory
category ℰ fibered over 𝒞 by a functor π whose fibers are the categories {F(c)}c∈𝒞. Let C be a category and let F:Cop→𝐒𝐞𝐭𝐬 be a set-valued functor. The
Category_of_elements
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
geometry. They are those homology cycles of a smooth fiber in a family which vanish in the singular fiber. For example, in a map from a connected complex surface
Vanishing_cycle
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
Construct in mathematics
of infinitesimal thickenings Twisted forms of projective varieties Fiber functors for motives H3(X,ℤ) and 𝒪X∗-gerbes: Jean-Luc Brylinski's approach Gerbes
Gerbe
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
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
Mathematical structure
continuous functor sends covering sieves to covering sieves. If J is the topology defined by a pretopology, and if u commutes with fibered products, then
Grothendieck_topology
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
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
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
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 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
indiscrete spaces is continuous, both of these functors give full embeddings of 𝐒𝐞𝐭 into 𝐓𝐨𝐩. 𝐓𝐨𝐩 is also fiber-complete meaning that the category of
Category of topological spaces
Category_of_topological_spaces
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
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
Collection of objects and morphisms
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)
Mathematical concept that extends the intuitive idea of gluing in topology
says: The functor F↦(F′,α) is fully faithful. Descent theory tells conditions for which there is a fully faithful descent, and when this functor is an equivalence
Descent_(mathematics)
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
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
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
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)
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
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
Concept in algebraic topology
For a fibration p:E→B with fiber F and a fixed commutative ring R with a unit, there exists a contravariant functor from the fundamental groupoid of
Fibration
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
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
Category where every morphism is invertible; generalization of a group
is the functor category [Gr,Set], where Gr is the groupoid (category) with one element and isomorphic to the group G. Indeed, every functor F of this
Groupoid
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
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
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
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
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
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 and a continuous map f:B′→B
Pullback_bundle
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
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)
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
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−1(Spec(K))
Essential_dimension
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
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
Mathematical sequence
particular gives a functor f∗ from sheaves of abelian groups on X to sheaves of abelian groups on Y. Composing this with the functor Γ of taking sections
Leray_spectral_sequence
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
Operation in algebra
right modules mutatis mutandis. Each of these constructions extends to a functor, so that we have an adjoint triple where R M o d {\displaystyle {}_{R}{\mathsf
Change_of_rings
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
Moduli scheme of subschemes of a scheme, represents the flat-family-of-subschemes functor
determinants. Its fundamental property is that for a scheme T, it represents the functor whose T-valued points are the closed subschemes of ℙn×T that are flat over
Hilbert_scheme
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)
Abstract homotopical model for topological spaces
categories of functors ℒ∙:Π∞X→D(Ab) from their truncations ΠnX and the maps τ≤n−1:ΠnX→Πn−1X whose fibers should be the categories of n-functors Πn(K(πnX,n))→D(Ab)
∞-groupoid
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
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
Mathematical theory
p-adic fields. This suggested relation became known as the mysterious functor. To improve the Hodge–Tate conjecture to one involving the de Rham cohomology
P-adic_Hodge_theory
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
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
Mathematical space
the fiber 𝐆𝐫(k,ℰ)s to the usual Grassmannian 𝐆𝐫(k,ℰ⊗OSK(s)) over the residue field K(s). Since the Grassmannian scheme represents a functor, it comes
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)
Concept in algebraic topology
restriction functor i∗:ΔopSets→Δ≤nopSets has a left adjoint, denoted i∗. (The notations i∗,i∗ are comparable with the one of image functors for sheaves
N-skeleton
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
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
Relate the direct image and the pull-back of sheaves
A'-modules): (M⊗BB′)A′≅(MA)⊗AA′. Here the subscript indicates the forgetful functor, i.e., MA is M, but regarded as an A-module. Indeed, such an isomorphism
Base_change_theorems
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
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 to 𝐃, denoted F:𝐂→𝐃
Kernel_(algebra)
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
Cohomology theory for Lie algebras
(see Ext functor for the definition of Ext). Equivalently, these are the right derived functors of the left exact invariant submodule functor M↦M𝔤:={m∈M∣xm=0 for
Lie_algebra_cohomology
Concept in algebraic geometry
f:U→V in C and a bundle isomorphism f∗F→∼E. Let p:Vectn→C be the forgetful functor. Via p, Vectn is a prestack over C. That it is a stack over C is precisely
Moduli stack of vector bundles
Moduli_stack_of_vector_bundles
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
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
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)
Construction for categories
[p+q+1])=∏i+j+1=mHom([i],[p])×Hom([j],[q]), which can be extended by colimits to a functor −∗−:𝐬𝐒𝐞𝐭×𝐬𝐒𝐞𝐭→𝐬𝐒𝐞𝐭, which together with the empty simplicial
Join_(simplicial_sets)
Category theory concept
relative to a fixed object, such as the category of schemes over S, Sch/S. Fiber products in these categories can be considered intersections (e.g. the scheme-theoretic
Overcategory
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:𝑪𝒐𝒗(𝑿)⟶𝑮−𝑺𝒆𝒕:p↦p−1(x) is an equivalence of categories
Covering_space
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
Japanese mathematician
Jørgen Ellegaard; Ueno, Kenji (2007). "Geometric construction of modular functors from conformal field theory". Journal of Knot Theory and Its Ramifications
Kenji_Ueno
Universal construction in multilinear algebra
free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing
Tensor_algebra
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
Mathematical parametrization of vector spaces by another space
X,π,ℝk) defines a fiber bundle; the additional data of the gUV specifies a GL(k) structure group in which the action on the fiber is the standard action
Vector_bundle
Connection on a vector bundle
. The fibers of the vector bundle are the de Rham cohomology groups H d R k ( V s ) {\displaystyle H_{\mathrm {dR} }^{k}(V_{s})} of the fibers V s {\displaystyle
Gauss–Manin_connection
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
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
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
necessary and sufficient conditions on deformation functors which prove the representability of these functors as either algebraic spaces or as algebraic stacks
Artin's_criterion
Algebraic structure in ring theory
Equivalently, an R-module M is flat if the tensor product with M is an exact functor; that is if, for every short exact sequence of R-modules 0→K→L→J→0, the
Flat_module
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