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{\displaystyle i} -th size functor, with i = 0 , … , n {\displaystyle i=0,\ldots ,n\ } , denoted by F i {\displaystyle F_{i}\ } , is the functor in F u n ( R
Size_functor
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
extension to homology theory (the size functor) was introduced in 2001. The size homotopy group and the size functor are strictly related to the concept
Size_theory
Shape descriptions in a geometrical/topological sense
extension to homology theory (the size functor) was introduced in . The concepts of size homotopy group and size functor are strictly related to the concept
Size_function
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
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
Topics referred to by the same term
property of functors in the mathematical field of category theory; see Full and faithful functors Satiety, the absence of hunger A standard bed size, see Bed
Full
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 limit sketch. Adjoint functors between locally presentable categories have a particularly simple characterization. A functor F : C → D {\displaystyle
Accessible_category
Vertices connected in pairs by edges
permitting loops is the comma category Set ↓ D where D: Set → Set is the functor taking a set s to s × s. The diagram is a schematic representation of the
Graph_(discrete_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
Size function Size functor Size pair Natural pseudodistance Patrizio Frosini, Michele Mulazzani, Size homotopy groups for computation of natural size
Size_homotopy_group
that matching two points of the diagonal has no cost. Size theory Size function Size functor Size homotopy group Natural pseudodistance Michele d'Amico
Matching_distance
dependent polynomial functors) are a generalisation of containers, which can represent a wider class of types, such as vectors (sized lists). The element
Container_(type_theory)
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
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
Subgroup of an abelian group consisting of all elements of finite order
group A / T {\displaystyle A/T} is torsion-free. There is a covariant functor from the category of abelian groups to the category of torsion groups that
Torsion_subgroup
distance Size function Size functor Size homotopy group Patrizio Frosini, Michele Mulazzani, Size homotopy groups for computation of natural size distances
Natural_pseudodistance
Duality for locally compact abelian groups
groups, in order to treat dualization as a functor and prove the identity functor and the dualization functor are not naturally equivalent. Also the duality
Pontryagin_duality
Pointer that points to a function
%g\n", sum); return 0; } Functors, or function objects, are similar to function pointers, and can be used in similar ways. A functor is an object of a class
Function_pointer
Method for computing topological features of a space at different spatial resolutions
{\displaystyle r\leq s\leq t} . Equivalently, we may consider it as a functor from P {\displaystyle P} considered as a category to the category of vector
Persistent_homology
Multi-dimensional generalization of triangle
are simplices themselves. In particular, the convex hull of a subset of size m + 1 (of the n + 1 defining points) is an m-simplex, called an m-face of
Simplex
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
Theory in mathematics
of the category being the bijections between these sets. A species is a functor F : B → B . {\displaystyle F\colon {\mathcal {B}}\to {\mathcal {B}}.} For
Combinatorial_species
Typographical symbol (*)
And more generally the application of any covariant functor, where no doubt exists over which functor is meant. as a unary operator, written as a superscript
Asterisk
a generator. Note that this definition then reduces to saying that the functor Hom ( G , − ) : C → Set {\displaystyle {\text{Hom}}(G,-)\colon C\to {\textbf
Generator_(category_theory)
Array of numbers
particular, let ∅ {\displaystyle \varnothing } be an initial object. The functor ⊗ {\displaystyle \otimes } is distributive over coproducts; i.e., for all
Matrix_(mathematics)
Mathematical object
inclusions. Next choose a total order on the vertex set of K and define a functor F from K {\displaystyle {\mathcal {K}}} to the category of topological
Abstract_simplicial_complex
French mathematician (1928–2014)
cohomology – Weil cohomology theory for schemes X over a base field k Delta-functor – Functor between abelian categories Derivator Derived category – Homological
Alexander_Grothendieck
Programming language
polymorphism, tail recursion, pattern matching, first class lexical closures, functors (parametric modules), exception handling, effect handling, and incremental
OCaml
Arithmetic operation
S^{T})\cong \hom(T\times U,S).} This means the functor "exponentiation to the power T " is a right adjoint to the functor "direct product with T ". This generalizes
Exponentiation
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
Logical formalism using combinators instead of variables
functor logic. While the expressive power of combinatory logic typically exceeds that of first-order logic, the expressive power of predicate functor
Combinatory_logic
Mathematical function
space, a coordinate vector, the set of matrices of real numbers of a given size, or an R {\displaystyle \mathbb {R} } -algebra, such as the complex numbers
Function_of_a_real_variable
Style of computer programming
parametric polymorphism and generic modules called functors. Both Standard ML and OCaml provide functors, which are similar to class templates and to Ada's
Generic_programming
Software library for the C++ programming language
library itself. It provides four components called algorithms, containers, functors, and iterators. The STL provides a set of common classes for C++, such
Standard_Template_Library
Partially ordered set in which all subsets have both a supremum and infimum
called the lower adjoint and g is called the upper adjoint. By the adjoint functor theorem, a monotone map between any pair of complete lattices preserves
Complete_lattice
Mathematical model for sequential decision making under uncertainty
set A. Let Dist denote the Kleisli category of the Giry monad. Then a functor A → D i s t {\displaystyle {\mathcal {A}}\to \mathbf {Dist} } encodes both
Markov_decision_process
Mathematical concept
group W(V) = F[H(V∗)]. Since passing to group algebras is a contravariant functor, the central extension map H(V) → V becomes an inclusion Sym(V) → W(V)
Symplectic_vector_space
Algebraic structure with addition and multiplication
is the left adjoint functor of the forgetful functor from the category of rings to Set (and it is often called the free ring functor.) Let A, B be algebras
Ring_(mathematics)
Computer programmer and creator of Clojure
nothing." Rich Hickey (February 1995), "Callbacks in C++ using template functors", C++ Report, 7 (2): 43–50. Reprinted in Stanley B. Lippman, ed. (January
Rich_Hickey
Analysis of datasets using techniques from topology
≤ s ≤ t . {\displaystyle r\leq s\leq t.} An equivalent definition is a functor from Z {\displaystyle \mathbb {Z} } considered as a partially ordered set
Topological_data_analysis
In mathematics, invariant of square matrices
category theory, the determinant is a natural transformation between the two functors GL n {\displaystyle \operatorname {GL} _{n}} and ( − ) × {\displaystyle
Determinant
Axiom of set theory
continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left adjoint (the Freyd adjoint functor theorem)
Axiom_of_choice
Group homomorphism into the general linear group over a vector space
an arbitrary category C, a representation of G in C is a functor from G to C. Such a functor selects an object X in C and a group homomorphism from G
Group_representation
Type of logical system
by Alfred Tarski, et al.; Polyadic algebra, by Paul Halmos; Predicate functor logic, primarily by Willard Quine. These algebras are all lattices that
First-order_logic
topological spaces, by post-composing with the geometric realization functor. The size of a filtration refers to the number of simplices in the largest complex
Vietoris–Rips_filtration
Indigenous ethnic group of Borneo
Jason William (31 January 2016). North Borneo Sourcebook: Vocabularies and Functors. University of Hawaii Press. ISBN 978-0-8248-5782-0. Collection of useful
Kadazan_people
a predicate. predicate functor logic A logical system that combines elements of predicate logic with the concept of functors, allowing for a more expressive
Glossary_of_logic
Monster and modular connection
called the monster Lie algebra, is constructed from V using a quantization functor. It is a generalized Kac–Moody Lie algebra with a monster action by automorphisms
Monstrous_moonshine
Object-oriented programming language
Collection is equivalent to the higher-order function filter on an appropriate functor. Control structures do not have special syntax in Smalltalk. They are instead
Smalltalk
Theoretical object in mathematics
{R}},} then defining F1‑schemes to be a particular kind of representable functor on M R . {\displaystyle {\mathfrak {M}}{\mathfrak {R}}.} Using this, they
Field_with_one_element
homological algebra concerned with defining and applying a certain sequence of functors from rings to abelian groups. Algebraic number theory The part of number
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
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
Axiomatic set theories based on the principles of mathematical constructivism
required, for example, to formalize the object map of an internal hom-functor like h o m ( N , − ) . {\displaystyle {\mathrm {hom} }({\mathbb {N} },-)
Constructive_set_theory
probability distribution function in statistics a Fibonacci number an arbitrary functor a field an event space sigma algebra as part of a probability space, often
Latin letters used in mathematics, science, and engineering
Latin_letters_used_in_mathematics,_science,_and_engineering
Form of data structure
correct operation in an array of callbacks or functors. This requires that the map of types to callbacks or functors be initialized at runtime, but offers more
Scene_graph
Family of formalisms in natural language syntax
structure of a sentence. Functor and argument In a right (left) function application, the node of the type A\B (B/A) is called the functor, and the node of the
Categorial_grammar
argument is used as index. It distinguishes atomic values and the principal functor of compound terms. Nonfirst argument indexing is a variation of first-argument
Term_indexing
One-to-one correspondence
specific spot in a line-up). The set X will be the players on the team (of size nine in the case of baseball) and the set Y will be the positions in the
Bijection
Property of a mathematical operation
composition of morphisms is associative by definition. Associativity of functors and natural transformations follows from associativity of morphisms. Consider
Associative_property
2011 edition of the C++ programming language standard
anything which can be called (function pointers, member function pointers, or functors) whose arguments are compatible with those of the wrapper. An example can
C++11
Form of figurative language
mathematical analogy much further with the concept of functors. Given two categories C and D, a functor f from C to D can be thought of as an analogy between
Analogy
Matrix whose entries are all minors of another matrix
above bases of the exterior powers) is Cr (A). Taking exterior powers is a functor, which means that ∧ r ( A B ) = ( ∧ r A ) ( ∧ r B ) . {\displaystyle \wedge
Compound_matrix
Generalization of associativity properties
^{S_{n}}\to {\mathsf {Oper}}} to this forgetful functor (this is the usual definition of free functor). Given a collection of operations E, Γ ( E ) {\displaystyle
Operad
Subject area in mathematics
hoc descriptions, which remain useful. Throughout, let A be a ring. The functor K0 takes a ring A to the Grothendieck group of the set of isomorphism classes
Algebraic_K-theory
American philosopher and logician (1908–2000)
predicate functor logic, one of several ways that have been proposed for doing logic without quantifiers. For a comprehensive treatment of predicate functor logic
Willard_Van_Orman_Quine
Algebraic structure
following, the morphisms are trivially defined). There is a forgetful functor F : A L G → S E T {\displaystyle \mathrm {F} :\mathrm {ALG} \to \mathrm
Polynomial_ring
Particle
Michael H.; Larsen, Michael; Wang, Zhenghan (2002-06-01). "A Modular Functor Which is Universal¶for Quantum Computation". Communications in Mathematical
Fibonacci_anyons
Technique in mathematical group theory
an analogue of the construction of Deligne and Lusztig, using Zuckerman functors to construct representations. The construction of Deligne-Lusztig characters
Deligne–Lusztig_theory
Correspondence between topics in Lie theory
category of finite-dimensional (real) Lie-algebras. This functor has a left adjoint functor Γ {\displaystyle \Gamma } from (finite dimensional) Lie algebras
Lie group–Lie algebra correspondence
Lie_group–Lie_algebra_correspondence
{\displaystyle {\mathcal {R}}(X)} . The Rips filtration can be expressed as a functor Rips ( X ) : R → S i m p {\displaystyle {\text{Rips}}(X):\mathbb {R} \to
Degree-Rips_bifiltration
Area of mathematics
{\displaystyle e_{j}} . Alternating polynomials Symmetric polynomials Schur functor Robinson–Schensted correspondence Schur–Weyl duality Jucys–Murphy element
Representation theory of the symmetric group
Representation_theory_of_the_symmetric_group
Technique for creating lexically scoped first class functions
programming, and in fact closures are similar to stateful function objects (or functors) with a single call-operator method. In stateful languages, closures can
Closure (computer programming)
Closure_(computer_programming)
Algebra associated to any vector space
exterior algebra ⋀ ( V ) {\displaystyle \textstyle \bigwedge (V)} is a functor from the category of vector spaces to the category of algebras. Rather
Exterior_algebra
Type of group in abstract algebra
the symmetric group plays a fundamental role through the ideas of Schur functors. In the theory of Coxeter groups, the symmetric group is the Coxeter group
Symmetric_group
In number theory, measure of non-unique factorization
algebraic K-theory, with K 0 ( R ) {\displaystyle K_{0}(R)} being the functor assigning to R {\displaystyle R} its ideal class group; more precisely
Ideal_class_group
Swiss mathematician (1888–1977)
sufficiently strong consistent theory cannot contain its own reference functor is known as the Hilbert–Bernays paradox. In seven papers, published between
Paul_Bernays
Features in Haskell programming language
allows for mutable variables to be modified in transactions. Applicative Functors Arrows As Haskell is a pure functional language, functions cannot have
Haskell_features
faithfully-flat descent, the discovery of Grothendieck that the representable functors are sheaves for it (i.e. a very general gluing axiom holds). Function field
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
a functor π 1 {\displaystyle \pi _{1}} from the category of directed spaces to the category of small categories. The fundamental category functor satisfies
Directed_algebraic_topology
Second homology group of a group
Zbl 0047.25703 Dennis, R.K. (1976), In search of new "Homology" functors having a close relationship to K-theory, Cornell University Brown, R.;
Schur_multiplier
Type of symmetric polynomials in mathematics
polynomials (K-theoretical analogue of Schur polynomials) LLT polynomials Schur functor Littlewood–Richardson rule, where one finds some identities involving Schur
Schur_polynomial
embedded in the category of graphs. Every subfunctor of an accessible functor is accessible. (In a definable classes setting) For every natural number
Vopěnka's_principle
Branch of mathematics
between two elements is at most singleton. Functions between orders become functors between categories. Many ideas of order theory are just concepts of category
Order_theory
Each species of structures leads to a functor from Set* to itself. Example. For the species of groups, the functor F maps a set X to the set F(X) of all
Equivalent definitions of mathematical structures
Equivalent_definitions_of_mathematical_structures
Possible axiom for set theory in mathematics
free abelian group, where E x t {\displaystyle \mathrm {Ext} } is the Ext functor. The existence of a definable well-order of all sets (the formula for which
Axiom_of_constructibility
Indigenous ethnic group of Borneo
Jason William (31 January 2016). North Borneo Sourcebook: Vocabularies and Functors. University of Hawaii Press. ISBN 978-0-8248-5782-0. Sareb Putra, R. Masri
Bisaya_(Borneo)
Topological group that is in a certain sense assembled from a system of finite groups
groups with continuous transition maps is profinite and the inverse limit functor is exact on the category of profinite groups. Further, being profinite
Profinite_group
Technique in topological data analysis
filtration shrink as k {\displaystyle k} increases, we can regard it as a functor S ~ ( − ) : N op → S i m p {\displaystyle {\tilde {\mathcal {S}}}(-):\mathbb
Subdivision_bifiltration
Indigenous ethnic group of Borneo
Jason William (31 January 2016). North Borneo Sourcebook: Vocabularies and Functors. University of Hawaii Press. ISBN 978-0-8248-5782-0. Media related to Murut
Murut_people
automatically for all types. To use a modified hash function, use the functor interface Hashtbl.Make to create a module, such as with Map. Finally, functional
Comparison of programming languages (associative array)
Comparison_of_programming_languages_(associative_array)
of the spectrum form a closed subset. Ext The Ext functors, the derived functors of the Hom functor. extension 1. An extension of an ideal is the ideal
Glossary of commutative algebra
Glossary_of_commutative_algebra
{\displaystyle V_{\lambda _{1},\dots ,\lambda _{n}}} in terms of a Schur functor. The dimension of V λ μ {\displaystyle V_{\lambda \mu }} where λ = ( λ
Representations of classical Lie groups
Representations_of_classical_Lie_groups
Writing Lie algebra sets as matrices
{\displaystyle \operatorname {Ind} _{\mathfrak {h}}^{\mathfrak {g}}} is an exact functor from the category of h {\displaystyle {\mathfrak {h}}} -modules to the
Lie_algebra_representation
Z/n where n is the characteristic of R. change A change of rings is a functor (between appropriate categories) induced by a ring homomorphism. Clifford
Glossary_of_ring_theory
Type of monoidal category in category theory
{\displaystyle P} in a monoidal category C {\displaystyle C} is a strict monoidal functor from P {\displaystyle P} to C {\displaystyle C} . Every PRO P {\displaystyle
PROP_(category_theory)
Algebra for manipulating geographic data
modelling. Prentice Hall. Frank, Andrew U. (2005). "Map algebra extended with functors for temporal data". In Akoka, Jacky (ed.). Perspectives in Conceptual Modeling:
Map_algebra
Bisayan language spoken in the Philippines
CV(C) pattern. There are monosyllabic words; however, most of them are functors that have no lexical meaning. Most of the disyllabic words contain an affix
Masbateño_language
Technical treatment of Boolean algebras
complemented distributive lattice, as a Boolean ring, and as a product-preserving functor from a certain category (Lawvere). Two more definitions worth mentioning
Boolean algebras canonically defined
Boolean_algebras_canonically_defined
SIZE FUNCTOR
SIZE FUNCTOR
Boy/Male
Native American
He sits at home.
Girl/Female
Gaelic, German, Irish, Latin
Blind One; Form of Sheila
Boy/Male
Hindu
Of extra ordinary size
Boy/Male
British, English
An American Girl Doll
Boy/Male
Hindu, Indian
Placing Side by Side
Surname or Lastname
English
English : variant spelling of Vise.
Boy/Male
Tamil
Of extra ordinary size
Girl/Female
Irish
Good.
Girl/Female
Gaelic Irish Scottish
Surname or Lastname
English
English : unexplained.
Male
Native American
Native American Navajo name SIKE means "he sits at home."
Girl/Female
Latin
Wife of Orion.
Female
English
Anglicized form of Irish Gaelic Sadhbh, SIVE means "sweet."
Male
African
country, nation.
Boy/Male
Indian
Side
Girl/Female
Australian, Hebrew
Lily
Girl/Female
Australian, Hebrew
Pledged to God
Girl/Female
Hindu, Indian, Marathi
Size of Moon
Girl/Female
Gaelic Irish
Surname or Lastname
English
English : status name or occupational name from Middle English sysour ‘assizer’, i.e. a member of the court of assize.
SIZE FUNCTOR
SIZE FUNCTOR
Female
Spanish
Diminutive form of Spanish Concha, CONCHITA means "conception."
Girl/Female
Indian, Punjabi, Sikh
Excellent; Abundant
Boy/Male
Tamil
Boy/Male
Indian
Righteousness
Boy/Male
Muslim
Knowledgeable
Surname or Lastname
English
English : habitational name from any of various places so called, especially the city at the mouth of the river Wear. This, like other places so called in Cumbria, Lancashire, and southern Scotland, derives its name from Old English sundor ‘separate’ + land ‘land’; a further example in Northumbria has the same origin as Sutherland.
Girl/Female
Gujarati, Hindu, Indian, Kannada, Marathi, Sindhi, Tamil, Telugu
Shines Like a Pearl
Girl/Female
Hindu, Indian, Marathi, Sanskrit
Sensitive; Compassionate; Loving
Boy/Male
Muslim/Islamic
Servant of the Loving
Surname or Lastname
English
English : variant of Rathbone.
SIZE FUNCTOR
SIZE FUNCTOR
SIZE FUNCTOR
SIZE FUNCTOR
SIZE FUNCTOR
n.
An instrument consisting of a number of perforated gauges fastened together at one end by a rivet, -- used for ascertaining the size of pearls.
a.
Adjusted according to size.
n.
The perpendicular itself. See Sine of angle, below.
n.
An instrument or tool for bringing anything to an exact size.
v. i.
To take greater size; to increase in size.
a.
Sizelike; viscous; glutinous; as, sizy blood.
n.
Extent of superficies or volume; bulk; bigness; magnitude; as, the size of a tree or of a mast; the size of a ship or of a rock.
a.
Having a medium size; as, a medium-sized man.
a.
Hence, indirect; oblique; collateral; incidental; as, a side issue; a side view or remark.
n.
Bulk; largeness. [Obs.] See Size.
a.
Having a particular size or magnitude; -- chiefly used in compounds; as, large-sized; common-sized.
n.
Figurative bulk; condition as to rank, ability, character, etc.; as, the office demands a man of larger size.
a.
Of or pertaining to a side, or the sides; being on the side, or toward the side; lateral.
imp. & p. p.
of Size
n.
An instrument or contrivance to size articles, or to determine their size by a standard, or to separate and distribute them according to size.
a.
Of full size; of the natural size.
v. i.
To lean on one side.
v. t.
To cover with size; to prepare with size.
v. t.
To be or stand at the side of; to be on the side toward.
v. t.
To adjust or arrange according to size or bulk.