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Endofunctor on the category V of finite-dimensional vector spaces
polynomial functor is an endofunctor on the category V {\displaystyle {\mathcal {V}}} of finite-dimensional vector spaces that depends polynomially on
Polynomial_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 of
Polynomial functor (type theory)
Polynomial_functor_(type_theory)
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
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
Technique for studying functors
approximating functors are required to be "k-excisive" – such functors are called polynomial functors by analogy with Taylor polynomials – which is a simplifying
Calculus_of_functors
Type of functions in algebra
conjecture, which concerns the sufficiency of a polynomial mapping to be invertible. Polynomial functor Claudio Procesi (2007) Lie Groups: an approach
Polynomial_mapping
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
output type) is also indexed by shape. Container (abstract data type) Polynomial functor (type theory) Michael Abbott; Thorsten Altenkirch; Neil Ghani (2005)
Container_(type_theory)
Algebraic structure
especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally
Polynomial_ring
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
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
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
Free object in the category of associative algebras
analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables. Likewise, the polynomial ring may be regarded
Free_algebra
Mathematical constructs and creation rules
Each W-type is isomorphic to the initial algebra of a so-called polynomial functor. Let 0, 1, 2, etc. be finite types with inhabitants 11 : 1, 12, 22:2
Inductive_type
directed graph. polynomial A functor from the category of finite-dimensional vector spaces to itself is called a polynomial functor if, for each pair
Glossary_of_category_theory
Type of symmetric polynomials in mathematics
k-Schur functions Grothendieck polynomials (K-theoretical analogue of Schur polynomials) LLT polynomials Schur functor Littlewood–Richardson rule, where
Schur_polynomial
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
Type of algorithm in computer science
therefore the powerset functor a -> Bool has no final coalgebra. However, in the case of polynomial functors or quotient polynomial functors, final coalgebras
Corecursion
method Schur complement Schur-convex function Schur decomposition Schur functor Schur index Schur's inequality Schur's lemma (from Riemannian geometry)
List of things named after Issai Schur
List_of_things_named_after_Issai_Schur
Roughly, the number of k-dimensional holes on a topological surface
theorem (based on Tor functors, but in a simple case). The Betti number sequence for a circle is 1, 1, 0, 0, 0, ...; the Poincaré polynomial is 1 + x {\displaystyle
Betti_number
Theory in mathematics
spaces with linear transformations”, then one gets the notion of polynomial functor (after imposing some finiteness condition).[citation needed] Container
Combinatorial_species
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)
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
Software programming optimization technique
construct-memoized-functor(factorial) The above example assumes that the function factorial has already been defined before the call to construct-memoized-functor is
Memoization
"Smallest" commutative algebra that contains a vector space
that the composition of two left adjoint functors is also a left adjoint functor. Here, the forgetful functor from commutative algebras to vector spaces
Symmetric_algebra
American mathematician (born 1936)
dissertation, on Functor Theory, was written under the supervision of Norman Steenrod and David Buchsbaum. Freyd is best known for his adjoint functor theorem
Peter_J._Freyd
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
Particle
Michael H.; Larsen, Michael; Wang, Zhenghan (2002-06-01). "A Modular Functor Which is Universal¶for Quantum Computation". Communications in Mathematical
Fibonacci_anyons
Set of a ring's prime ideals
section functor are contravariant right adjoints between the category of commutative rings and schemes, global Spec and the direct image functor for the
Spectrum_of_a_ring
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
Moduli scheme of subschemes of a scheme, represents the flat-family-of-subschemes functor
is a disjoint union of projective subschemes corresponding to Hilbert polynomials. The basic theory of Hilbert schemes was developed by Alexander Grothendieck (1961)
Hilbert_scheme
Category whose objects are rings and whose morphisms are ring homomorphisms
generators E is the polynomial ring Z[E] whose variables are taken from E. This gives a left adjoint functor to the forgetful functor from CRing to Set
Category_of_rings
Theorem in algebra
duality functor DR gives an anti-equivalence between the categories of Artinian and Noetherian R-modules. In particular the duality functor gives an
Matlis_duality
Operation in algebra
f_{*}N=N_{R}} , formed by restriction of scalars. They are related as adjoint functors: f ∗ : Mod R ⇆ Mod S : f ∗ {\displaystyle f^{*}:{\text{Mod}}_{R}\leftrightarrows
Change_of_rings
Type of mathematical function
the context of a polynomial in one variable x, the constant function is called non-zero constant function because it is a polynomial of degree 0, and
Constant_function
is a polynomial for m >> 0 {\displaystyle m>>0} . This is called the Hilbert polynomial which gives a natural stratification of the quot functor. Again
Quot_scheme
Expression in commutative algebra
homogeneous symmetric polynomials are a specific kind of symmetric polynomials. Every symmetric polynomial can be expressed as a polynomial expression in complete
Complete homogeneous symmetric polynomial
Complete_homogeneous_symmetric_polynomial
constant term 1. In other words the functor Exp from abelian groups to commutative rings is adjoint to the functor from commutative rings to abelian groups
Exp_algebra
Generalization of algebraic variety
X(S) is a 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.
Scheme_(mathematics)
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
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
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
Algebraic structure with addition and multiplication
complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. More formally, a ring
Ring_(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
Type of algebra
such that every element of A {\displaystyle A} can be expressed as a polynomial in a finite number of generators a 1 , … , a n ∈ A {\displaystyle a_{1}
Finitely_generated_algebra
Mathematical object
in positive characteristic. The divided power functor is used in the construction of co-Schur functors. Crystalline cohomology The uniqueness follows
Divided_power_structure
In mathematics, a module that has a basis
{\textbf {Set}}} is the forgetful functor, meaning R ( − ) {\displaystyle R^{(-)}} is a left adjoint of the forgetful functor. Many statements true for free
Free_module
Set of functions between two fixed sets
bifunctor; but as (single) functor, of type [ X , − ] {\displaystyle [X,-]} , it appears as an adjoint functor to a functor of type − × X {\displaystyle
Function_space
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
Association of one output to each input
Higher-order function Homomorphism Morphism Microfunction Distribution Functor Associative array Closed-form expression Elementary function Functional
Function_(mathematics)
Operation on mathematical functions
Generalizations Relation (Binary relation) Set-valued Multivalued Partial Implicit Space Higher-order Morphism Functor List of specific functions v t e
Function_composition
Topics referred to by the same term
change of coordinate system Covariance and contravariance of functors, properties of functors General covariance or simply covariance (inaccurate but common
Covariance_(disambiguation)
Direct summand of a free module (mathematics)
R-module P is projective if and only if the covariant functor Hom(P, -): R-Mod → Ab is an exact functor, where R-Mod is the category of left R-modules and
Projective_module
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
Invariant of mathematical knots
cochain complex. It may be regarded as a categorification of the Jones polynomial. It was developed in the late 1990s by Mikhail Khovanov. To any link diagram
Khovanov_homology
Particular correspondence between two partially ordered sets
that monotone Galois connections are special cases of pairs of adjoint functors in category theory as discussed further below. Other terminology encountered
Galois_connection
Mathematical concept named for Ernst Witt
the functor W {\textstyle W} by appealing to the adjoint functor theorem. One further has that W {\textstyle W} restricts to a fully faithful functor on
Witt_vector
values of integer type due to simplicity of implementation. Integer-valued polynomial Semi-continuity Rank (disambiguation)#Mathematics Grade (disambiguation)#In
Integer-valued_function
Exact sequence used to describe the structure of an object
respectively) functor. The importance of acyclic resolutions lies in the fact that the derived functors RiF (of a left exact functor, and likewise LiF
Resolution_(algebra)
Module over a sheaf of differential operators
expanding on the work of Sato and Joseph Bernstein on the Bernstein–Sato polynomial. Early major results were the Kashiwara constructibility theorem and Kashiwara
D-module
Generalization of vector spaces from fields to rings
a covariant additive functor from C to Ab should be considered a generalized left module over C. These functors form a functor category C-Mod, which
Module_(mathematics)
Mathematical concept
Generalizations Relation (Binary relation) Set-valued Multivalued Partial Implicit Space Higher-order Morphism Functor List of specific functions v t e
Inverse_function
Commutative algebra theorem
(algebraic) vector bundles) is given by the 'globalisation' or 'twiddlification' functor, sending M {\displaystyle M} to M ~ {\displaystyle {\widetilde {M}}} .
Quillen–Suslin_theorem
Ring that is also a vector space or a module
category of the category of commutative rings under R.) The prime spectrum functor Spec then determines an anti-equivalence of this category to the category
Associative_algebra
Concept in mathematics
F2(x,y) − F2(y,x) The natural functor from Lie groups or algebraic groups to Lie algebras can be factorized into a functor from Lie groups to formal group
Formal_group_law
In mathematics, element with a multiplicative inverse
formation of the unit group defines a functor from the category of rings to the category of groups. This functor has a left adjoint which is the integral
Unit_(ring_theory)
Algebraic structure used in topology
derived functors of a left exact functor on an abelian category, while "homology" is used for the left derived functors of a right exact functor. For example
Cohomology
general linear group GL(V). Each Young diagram λ corresponds to a Schur functor Lλ(-) on the category of GL(V)-representations. Given two Young diagrams
Plethysm
Linear operator in mathematics
adjoint to the push-forward; the composition operator is the inverse image functor. Since the domain considered here is that of Borel functions, the above
Composition_operator
1969 result in deformation theory
(A{\text{-algebras}})\to ({\text{sets}}),} be a functor sending filtered colimits to filtered colimits (Artin calls such a functor locally of finite presentation). Then
Artin_approximation_theorem
the language of category theory, the functor sending a set X to the Lie algebra generated by X is the free functor from the category of sets to the category
Free_Lie_algebra
Map raising elements to the pth power, in characteristic p
the Frobenius endomorphism is a natural transformation from the identity functor on the category of characteristic p rings to itself. If the ring R is a
Frobenius_endomorphism
Isomorphism of commutative rings constructed in the theory of Lie algebras
(negative of) the natural derivative operator on the loop algebra. Translation functor Infinitesimal character Humphreys 1978, p. 130. Humphreys 1978, pp. 135–141
Harish-Chandra_isomorphism
Computer hardware technology that uses quantum mechanics
Michael H.; Larsen, Michael; Wang, Zhenghan (1 June 2002). "A Modular Functor Which is Universal for Quantum Computation". Communications in Mathematical
Quantum_computing
Abstract algebra concept
{C} } be the category of integral domains and injective ring maps. The functor from C {\displaystyle \mathbf {C} } to the category of fields that takes
Field_of_fractions
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
Flat_module
Representation of a mathematical function
however, cannot be determined from the graph alone. The graph of the cubic polynomial on the real line f ( x ) = x 3 − 9 x {\displaystyle f(x)=x^{3}-9x} is
Graph_of_a_function
Study of abstract algebraic structures
algebra is not unital, it may be made so in a standard way (see the adjoint functors page); there is no essential difference between modules for the resulting
Algebra_representation
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)
Generalisation of a sheaf; a fibered category that admits effective descent
automorphisms which have been overcounted. A category c {\displaystyle c} with a functor to a category C {\displaystyle C} is called a fibered category over C {\displaystyle
Stack_(mathematics)
Mapping between functions in the quantum phase space
relation Deformation quantization Heisenberg group Moyal bracket Weyl algebra Functor Pseudo-differential operator Wigner quasi-probability distribution Stone–von
Wigner–Weyl_transform
Graphical representation of a morphism
and a monoidal functor to its underlying morphism of signatures, i.e. it forgets the identity, composition and tensor. The free functor C − : M o n S i
String_diagram
Algebraic structure with "nice" duality properties
Frobenius adjunction iff also G ⊣ F {\displaystyle G\dashv F} . A functor F is a Frobenius functor if it is part of a Frobenius adjunction, i.e. if it has isomorphic
Frobenius_algebra
Ring without non-zero nilpotent elements
the nilradical of a commutative ring R {\displaystyle R} . There is a functor R ↦ R / N R {\displaystyle R\mapsto R/{\mathfrak {N}}_{R}} of the category
Reduced_ring
General concept and operation in mathematics
theory viewpoint, duality can also be seen as a functor, at least in the realm of vector spaces. This functor assigns to each space its dual space, and the
Duality_(mathematics)
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
In algebra, completion w.r.t. powers of an ideal
In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion
Completion_of_a_ring
Mathematical operation on vector spaces
not injective. Higher Tor functors measure the defect of the tensor product being not left exact. All higher Tor functors are assembled in the derived
Tensor_product
Theorem classifying finite simple groups
The signalizer functor method for odd primes. The main problem is to prove a signalizer functor theorem for nonsolvable signalizer functors. This was solved
Classification of finite simple groups
Classification_of_finite_simple_groups
Analogue of a complex analytic space over a nonarchimedean field
complete nonarchimedean field k. The Tate algebra is the completion of the polynomial ring in n variables under the Gauss norm (taking the supremum of coefficients)
Rigid_analytic_space
Algebraic structure in mathematics
quadratic polynomial with coefficients in the ring. There are free and graded quadratic algebras. Given a commutative ring R, and the ring of polynomials R[X]
Quadratic_algebra
Generalized concept of a set element
this way – is due to Grothendieck, and is often called the method of the functor of points. Suppose C is any category and A, T are two objects of C. A T-valued
Element_(category_theory)
Characteristic classes of vector bundles
types of algebraic geometry), the Chern classes can be expressed as polynomials in the coefficients of the curvature form. There are various ways of
Chern_class
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
Branch of algebra that studies commutative rings
commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers Z
Commutative_algebra
by J, consisting of all elements x such that xJ⊆I. [] R[x,y,...] is a polynomial ring over R. [[]] R[[x,y,...]] is a formal power series ring over R. {}
Glossary of commutative algebra
Glossary_of_commutative_algebra
In algebraic geometry, a point with rational coordinates
determined up to isomorphism by the functor S ↦ X(S); this is the philosophy of identifying a scheme with its functor of points. Another formulation is
Rational_point
exp1m – exponential minus 1 function. (Also written as expm1.) Ext – Ext functor. ext – exterior. extr – a set of extreme points of a set. FFT – fast Fourier
List of mathematical abbreviations
List_of_mathematical_abbreviations
submodule of M intersects non-trivially. exact exact sequence Ext functor Ext functor extension Extension of scalars uses a ring homomorphism from R to
Glossary_of_module_theory
Algebraic term
square matrix M is a unipotent matrix if and only if its characteristic polynomial P(t) is a power of t − 1. Thus all the eigenvalues of a unipotent matrix
Unipotent
POLYNOMIAL FUNCTOR
POLYNOMIAL FUNCTOR
POLYNOMIAL FUNCTOR
POLYNOMIAL FUNCTOR
Boy/Male
Hindu
One who brings good luck
Boy/Male
Indian
Answer of All Prayers
Boy/Male
Arabic, Muslim, Sindhi
Commendable
Girl/Female
Danish, Indian, Sanskrit
Satisfying
Female
Italian
Italian diminutive form of Latin Maria, MARIETTA means "little rebel."
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi
The Eye
Surname or Lastname
Spanish (LucÃa) and southern Italian
Spanish (LucÃa) and southern Italian : from the female personal name Lucia, feminine derivative of Latin lux ‘light’.English : from a Latinized form of Luce.Respelling of French Lussier.
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Telugu
Courage
Boy/Male
Indian, Punjabi, Sikh
Glow of Heart
Boy/Male
Muslim
Courageous. Brave.
POLYNOMIAL FUNCTOR
POLYNOMIAL FUNCTOR
POLYNOMIAL FUNCTOR
POLYNOMIAL FUNCTOR
POLYNOMIAL FUNCTOR
a.
Consisting of two or more words; having names consisting of two or more words; as, a polynomial name; polynomial nomenclature.
a.
Containing many names or terms; multinominal; as, the polynomial theorem.
n.
A polynomial of four terms connected by the signs plus or minus.
a.
Possessing the same number of factors of a given kind; as, a homogeneous polynomial.
n. & a.
Same as Polynomial.
n.
A polynomial name or term.
n.
An expression composed of two or more terms, connected by the signs plus or minus; as, a2 - 2ab + b2.