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Relationship between two functors abstracting many common constructions
relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics
Adjoint_functors
Characterizing property of mathematical constructions
concept of adjoint functors was introduced independently by Daniel Kan in 1958. Mathematics portal Free object Natural transformation Adjoint functor Monad
Universal_property
Criteria in Category theory of Mathematics
mathematics, the formal criteria for adjoint functors are criteria for the existence of a left or right adjoint of a given functor. One criterion is the following
Formal criteria for adjoint functors
Formal_criteria_for_adjoint_functors
Mathematical concept
colimits, like the strongly related notions of universal properties and adjoint functors, exist at a high level of abstraction. In order to understand them
Limit_(category_theory)
Concept in category theory
of a forgetful functor with no adjoint. There is no field satisfying a free universal property for a given set. Adjoint functors Functors Projection (set
Forgetful_functor
Mapping between categories
pairs of adjoint functors. Functors sometimes appear in functional programming. For instance, the programming language Haskell has a class Functor where
Functor
General theory of mathematical structures
Adjoint functors: A functor can be left (or right) adjoint to another functor that maps in the opposite direction. Such a pair of adjoint functors typically
Category_theory
Mathematical category
: X → Y {\displaystyle u:X\to Y} is a pair of adjoint functors (u∗,u∗) (where u∗ : Y → X is left adjoint to u∗ : X → Y) such that u∗ preserves finite limits
Topos
Functors which are surjective and injective on hom-sets
category theory, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both properties
Full_and_faithful_functors
Operation in algebra and mathematics
category (an endofunctor is a functor mapping a category to itself). For example, if F , G {\displaystyle F,G} are functors adjoint to each other, then T =
Monad_(category_theory)
Index of articles associated with the same name
Look up adjoint in Wiktionary, the free dictionary. In mathematics, the term adjoint applies in several situations. Several of these share a similar formalism:
Adjoint
Special objects used in (mathematical) category theory
properties and adjoint functors. Let 1 be the discrete category with a single object (denoted by •), and let U : C → 1 be the unique (constant) functor to 1. Then
Initial_and_terminal_objects
Construction in category theory
then just a contravariant functor I → C. Let C I o p {\displaystyle C^{I^{\mathrm {op} }}} be the category of these functors (with natural transformations
Inverse_limit
Conjugate transpose of an operator in infinite dimensions
to the defining properties of pairs of adjoint functors in category theory, and this is where adjoint functors got their name. Mathematical concepts Conjugate
Hermitian_adjoint
Induced map between the dual spaces of the two vector spaces
{\displaystyle u(X)} of u {\displaystyle u} is weakly closed. Adjoint functors – Relationship between two functors abstracting many common constructions Composition
Transpose_of_a_linear_map
Concept in homological algebra
adjoint functors ( j ! , j ∗ , j ∗ ) {\displaystyle (j_{!},j^{*},j_{*})} and ( i ∗ , i ∗ , i ! ) {\displaystyle (i^{*},i_{*},i^{!})} . The functors i
T-structure
Functor type
has a left adjoint. The categorical notions of universal morphisms and adjoint functors can both be expressed using representable functors. Let G : D
Representable_functor
Category admitting tensor products
monoidal structure induced by the cartesian product). Monoidal functors are the functors between monoidal categories that preserve the tensor product and
Monoidal_category
General concept and operation in mathematics
in topology and more generally model categories. Two functors F: C → D and G: D → C are adjoint if for all objects c in C and d in D HomD(F(c), d) ≅ HomC(c
Duality_(mathematics)
Category whose hom sets have algebraic structure
ordinary functors. Additionally, one demands that the diagram commute, which is analogous to the rule F(fg)=F(f)F(g) for ordinary functors. There is
Enriched_category
Collection of maps which give the same result
Cat is naturally a 2-category, with functors as its arrows and natural transformations as the arrows between functors. In this setting, commutative diagrams
Commutative_diagram
Central object of study in category theory
to be a "morphism of functors". Informally, the notion of a natural transformation states that a particular map between functors can be done consistently
Natural_transformation
Type of category in category theory
requirement that the functor –×Y (i.e. the functor from C to C that maps objects X to X×Y and morphisms φ to φ × idY) has a right adjoint, usually denoted
Cartesian_closed_category
Mathematical object that generalizes the standard notions of sets and functions
of all small categories, with functors between them as morphisms. In turn, a functor category has as objects functors between two fixed categories and
Category_(mathematics)
Generalization of a category
category theory. Namely, two functors F : C → D , G : D → C {\displaystyle F:C\to D,\,G:D\to C} are said to be an adjoint pair if there exists a 2-morphism
Quasi-category
Special case of colimit in category theory
{\displaystyle {\mathcal {C}}} admits an alternative description in terms of functors. Any directed set ⟨ I , ≤ ⟩ {\displaystyle \langle I,\leq \rangle } can
Direct_limit
Concept in category theory
theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two
Monoidal_functor
Formalism in homological algebra
tensor product internal Hom The functors f ∗ {\displaystyle f^{*}} and f ∗ {\displaystyle f_{*}} form an adjoint functor pair, as do f ! {\displaystyle
Six_operations
Generalized object in category theory
Z).} Coproduct – the dual of the product Diagonal functor – the left adjoint of the product functor. Limit and colimits – Mathematical concept Equalizer –
Product_(category_theory)
Mathematics construct
identity functor may be replaced with some other functor; this yields a family of categories particularly useful in the study of adjoint functors. For example
Comma_category
Particular correspondence between two partially ordered sets
relative to ≤. The term "adjoint" refers to the fact that monotone Galois connections are special cases of pairs of adjoint functors in category theory as
Galois_connection
Category with direct sums and certain types of kernels and cokernels
it turns out that exact functors, i.e. the functors preserving exact sequences in various senses, are the relevant functors between abelian categories
Abelian_category
Homological construction in category theory
mathematics, specifically category theory, certain functors may be derived to obtain other functors closely related to the original ones. This operation
Derived_functor
See Bousfield localization. calculus of functors The calculus of functors is a technique of studying functors in the manner similar to the way a function
Glossary_of_category_theory
In mathematics, invertible homomorphism
{\displaystyle gf=1_{a}.} Two categories C and D are isomorphic if there exist functors F : C → D {\displaystyle F:C\to D} and G : D → C {\displaystyle G:D\to
Isomorphism
Map (arrow) between two objects of a category
diffeomorphisms. In the category of small categories, the morphisms are functors. In a functor category, the morphisms are natural transformations. For more examples
Morphism
Theorem in algebra
be conceptually explained using the language of adjoint functors and derived categories: the functor between the derived categories of R- and k-modules
Matlis_duality
Mathematical category with weak equivalences, fibrations and cofibrations
and homotopy classes of continuous maps, whence the name. A pair of adjoint functors F : C ⇆ D : G {\displaystyle F:C\leftrightarrows D:G} between two model
Model_category
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
Concept in mathematics
{\displaystyle -\otimes X} and hom-functor Hom ( X , − ) {\displaystyle \operatorname {Hom} (X,-)} form an adjoint pair: Hom ( Y ⊗ X , Z ) ≅ Hom
Tensor–hom_adjunction
Category theory constructs
category theory, a branch of mathematics. They are closely related to adjoints, but are also related to limits and ends. They are named after Daniel M
Kan_extension
Generalization of category
(small) categories, where a 2-morphism is a natural transformation between functors. The concept of a strict 2-category was first introduced by Charles Ehresmann
2-category
Embedding of categories into functor categories
category of functors (contravariant set-valued functors) defined on that category. It also clarifies how the embedded category of representable functors and their
Yoneda_lemma
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
the diagonal functor. Similarly, the colimit functor (which exists if the category is cocomplete) is the left-adjoint of the diagonal functor. For example
Diagonal_functor
Mathematical category whose hom sets form Abelian groups
F:{\text{Hom}}(A,B)\rightarrow {\text{Hom}}(F(A),F(B))} is a group homomorphism. Most functors studied between preadditive categories are additive. For a simple example
Preadditive_category
Generalisations of Serre duality in mathematics
as the existence of a right adjoint functor f ! {\displaystyle f^{!}} , called twisted or exceptional inverse image functor, to a higher direct image with
Coherent_duality
Generalization of monads
for adjoint functors Day & Street 1997, 3. PSEUDOMONOIDS Lack 2000 Marmolejo & Wood 2008 Cheng, Hyland & Power 2003 Note that Remark 2.1. of 2-functor in
Pseudomonad_(category_theory)
isomorphism. The forgetful functors in algebra, such as from Grp to Set, are conservative. More generally, every monadic functor is conservative. In contrast
Conservative_functor
Transforming a function in such a way that it only takes a single argument
describes an adjoint pair of functors: for every fixed set C {\displaystyle C} , the functor B ↦ B × C {\displaystyle B\mapsto B\times C} is left adjoint to the
Currying
Categorical generalization of a function space in set theory
(f\colon X\to Z)\mapsto (f^{Y}\colon X^{Y}\to Z^{Y})} , is a right adjoint to the product functor − × Y {\displaystyle -\times Y} . For this reason, the morphisms
Exponential_object
Functor mapping hom objects to an underlying category
between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category
Hom_functor
Mathematical structures in category theory
a branch of mathematics, a functor category D C {\displaystyle D^{C}} is a category where the objects are the functors F : C → D {\displaystyle F:C\to
Functor_category
American mathematician and philosopher (1937–2023)
universal quantifiers of logic could be characterized as adjoint functors to the substitution functor. This revealed a deep connection between logic and geometry
William_Lawvere
Theorem in category theory
monadicity theorem asserts that a functor U : C → D {\displaystyle U:C\to D} is monadic if and only if U has a left adjoint; U reflects isomorphisms (if U(f)
Beck's_monadicity_theorem
Abstract mathematics relationship
D→D denote the identity functors on C and D, assigning each object and morphism to itself. If F and G are contravariant functors one speaks of a duality
Equivalence_of_categories
Combination of pointed topological spaces
kind of tensor product in an appropriate category of pointed spaces. Adjoint functors make the analogy between the tensor product and the smash product more
Smash_product
Construction in category theory
natural map between constant functors Δ(N), Δ(M) corresponds to a morphism between N and M. In this sense, the diagonal functor acts trivially on arrows.
Cone_(category_theory)
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
functor U. Mathematics portal Free strict monoidal category Free object Adjoint functors Awodey, Steve (2010). Category theory (2nd ed.). Oxford: Oxford University
Free_category
Category theory
Y_{T})=\mu _{Y}\circ Tf\;} One can show that F and G are indeed functors and that F is left adjoint to G. The counit of the adjunction is given by ε Y T = (
Kleisli_category
Duality between the process of restricting and inducting in representation theory
)&\longmapsto \operatorname {Ind} _{H}^{G}(W,\tau )\end{aligned}}} These functors form an adjoint pair Ind H G ⊣ Res H G {\displaystyle \operatorname {Ind} _{H}^{G}\dashv
Frobenius_reciprocity
Theorem in category theory
Complete Concrete Forgetful functor Pre-abelian Preadditive Commutative diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal
Lawvere's_fixed-point_theorem
Endofunctor on the category V of finite-dimensional vector spaces
are polynomial functors from V {\displaystyle {\mathcal {V}}} to V {\displaystyle {\mathcal {V}}} ; these two are also Schur functors. The notion appears
Polynomial_functor
Most general completion of a commutative square given two morphisms with same codomain
R, is given by the tensor product over R, and Spec is a contravariant functor, the pullback of two affine schemes Spec(A) and Spec(B) over Spec(R), usually
Pullback_(category_theory)
Aspect of category theory
the standard 1-simplex. Coequalizers can be large: There are exactly two functors from the category 1 having one object and one identity arrow, to the category
Coequalizer
Concept in category theory
pullback functor taking bundles on Y to bundles on X. Fibred categories formalise the system consisting of these categories and inverse image functors. Similar
Fibred_category
Category-theoretic construction
\ldots \oplus X_{n}} . Suppose all finite coproducts exist in C, coproduct functors have been chosen as above, and 0 denotes the initial object of C corresponding
Coproduct
Category whose objects and morphisms are inside a bigger category
There is an obvious faithful functor I : S → C {\displaystyle I:{\mathcal {S}}\to {\mathcal {C}}} , called the inclusion functor which takes objects and morphisms
Subcategory
Mathematical category formed by reversing morphisms
object Dual (category theory) Duality (mathematics) Adjoint functor Contravariant functor Opposite functor "Is there an introduction to probability theory
Opposite_category
Most general completion of a commutative square given two morphisms with same domain
we may interpret the theorem as confirming that the fundamental group functor preserves pushouts of inclusions. We might expect this to be simplest when
Pushout_(category_theory)
Mathematical set of all subsets of a set
quantifier can be understood as the right adjoint of a functor between power sets, the inverse image functor of a function between sets; likewise, the
Power_set
Abstract homotopical model for topological spaces
consider globular objects in a category C {\displaystyle {\mathcal {C}}} as functors X ∙ : G o p → C . {\displaystyle X_{\bullet }\colon \mathbb {G} ^{op}\to
∞-groupoid
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
Overview of and topical guide to category theory
categories Subcategory Faithful functor Full functor Forgetful functor Representable functor Functor category Adjoint functors Galois connection Pontryagin
Outline_of_category_theory
Hom functor are adjoint; however, they might not always lift to an exact sequence. This leads to the definition of the Tor functor and the Ext functor. A
Lift_(mathematics)
Left adjoint to a forgetful functor to sets
objects exist in C, the functor F, called the free functor is a left adjoint to the faithful functor U; that is, there is a bijection Hom S e t ( X
Free_object
Functor that preserves short exact sequences
particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations
Exact_functor
Product of two categories, in category theory
I} satisfy: given a family of functors f i : D → C i {\displaystyle f_{i}:D\to C_{i}} , there exists a unique functor f : D → P {\displaystyle f:D\to
Product_category
Generalization of category theory
the category known as Cat, which is the category of small categories and functors is actually a 2-category with natural transformations as its 2-morphisms
Higher_category_theory
Design pattern in functional programming to build generic types
Heinrich (1965). "Every standard construction is induced by a pair of adjoint functors" (PDF). Proceedings of the American Mathematical Society. 16 (3): 544–546
Monad (functional programming)
Monad_(functional_programming)
Mathematical structure
Continuous functors induce functors between the corresponding topoi by sending a sheaf F {\displaystyle F} to F u {\displaystyle Fu} . These functors are called
Grothendieck_topology
Category
pre-abelian category, exact functors can be described in particularly simple terms. First, recall that an additive functor is a functor F: C → D between preadditive
Pre-abelian_category
an amnestic functor F : A → B is a functor for which an A-isomorphism ƒ is an identity whenever Fƒ is an identity. An example of a functor which is not
Amnestic_functor
Mathematical construction used in homotopy theory
contravariant functor X : Δ → Set where Set is the category of sets. (Alternatively and equivalently, one may define simplicial sets as covariant functors from
Simplicial_set
Graphical representation of a morphism
C_{-}:\mathbf {MonSig} \to \mathbf {MonCat} } , i.e. the left adjoint to the forgetful functor, sends a monoidal signature Σ {\displaystyle \Sigma } to the
String_diagram
Applications of category theory
Complete Concrete Forgetful functor Pre-abelian Preadditive Commutative diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal
Applied_category_theory
Indexed collection of objects and morphisms in a category
natural transformation between functors. One can then interpret the category of diagrams of type J in C as the functor category CJ, and a diagram is then
Diagram_(category_theory)
Relation of categories in category theory
functor category C1, with objects functors c: 1 → C, selecting an object c∈Ob(C), and arrows natural transformations f: c → d between these functors,
Isomorphism_of_categories
Symbol in mathematical logic
{\displaystyle F\dashv G} , is used to indicate that the functor F is left adjoint to the functor G. More rarely, a turnstile ( ⊢ {\displaystyle \vdash }
Turnstile_(symbol)
On representability of a contravariant functor on the category of connected CW complexes
a (covariant) functor F: C → D between triangulated categories satisfying certain technical conditions to have a right adjoint functor. Namely, if C and
Brown's representability theorem
Brown's_representability_theorem
"Smallest" commutative algebra that contains a vector space
asserts 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
Concept in mathematical theory of categories
is said to be reflective in B when the inclusion functor from A to B has a left adjoint. This adjoint is sometimes called a reflector, or localization
Reflective_subcategory
1017/S0960129597002375. ISSN 0960-1295. Kock, Anders (1972-12-01). "Strong functors and monoidal monads". Archiv der Mathematik. 23 (1): 113–120. doi:10.1007/BF01304852
Strong_monad
Construction of a ring of fractions
properties, by using the fact that the composition of two left adjoint functors is a left adjoint functor. If R = Z {\displaystyle R=\mathbb {Z} } is the ring of
Localization (commutative algebra)
Localization_(commutative_algebra)
Injective homomorphism
Complete Concrete Forgetful functor Pre-abelian Preadditive Commutative diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal
Monomorphism
Connects set theory with category theory
replaces sets with categories, functions with functors, and equations with natural isomorphisms of functors satisfying additional properties. The term was
Categorification
Tool to track locally defined data attached to the open sets of a topological space
same as a contravariant functor from O ( X ) {\displaystyle O(X)} to C {\displaystyle C} . Morphisms in this category of functors, also known as natural
Sheaf_(mathematics)
Category whose objects are rings and whose morphisms are ring homomorphisms
forgetful functors A : Ring → Ab M : Ring → Mon which "forget" multiplication and addition, respectively. Both of these functors have left adjoints. The left
Category_of_rings
Quotient space of a codomain of a linear map by the map's image
Complete Concrete Forgetful functor Pre-abelian Preadditive Commutative diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal
Cokernel
ADJOINT FUNCTORS
ADJOINT FUNCTORS
Biblical
to call; to anoint
Girl/Female
Arabic, Muslim
Connection; Joint
Girl/Female
British, English, Latin
Dyer; Skillful; Dexterous; Adroit; Right-handed
Boy/Male
Muslim
Skillful, Adroit (1)
Male
English
Anglicized form of Irish Gaelic Comhghall, COWAL means "joint pledge."
Boy/Male
Muslim/Islamic
Skillful Adroit
Female
Irish
Variant spelling of Irish Éadan, ÉADAOIN means "face" or perhaps "against" or "opposite."
Male
Irish
Contracted form of Irish Gaelic Comhghall, COMGAL means "joint pledge."
Boy/Male
Arabic, Australian, Muslim, Sindhi
Skillful; Adroit
Surname or Lastname
English
English : presumably from Old French joint ‘united’, ‘joined’. The application as a surname is unclear.
Male
Japanese
(1-å·§, 2-åŒ , 3-å·¥) Japanese name TAKUMI means 1) "adroit," 2) "artisan," or 3) "skilful."
Girl/Female
Biblical
To call, to anoint.
Girl/Female
Latin
Adroit; skillful.
Girl/Female
Indian, Telugu
Love; To Joint
Surname or Lastname
English
English : from a pet form of Fulcher.German (also Füge) : nickname for a skillful, adroit person, from Middle High German vüege ‘skillful’, ‘fitting’ (see Fiegel).
ADJOINT FUNCTORS
ADJOINT FUNCTORS
Boy/Male
Tamil
Longing
Girl/Female
Assamese, Christian, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Mythological, Oriya, Sanskrit, Sindhi, Telugu, Traditional
Basil; Tulsi; Goddess Radha
Boy/Male
Muslim
Inspired
Boy/Male
Tamil
Dilshith | திலà¯à®·à®¿à®¤
Boy/Male
American, Australian, British, Christian, English, French, German, Greek
Lover of Horses; Form of Phillip
Male
Greek
(Λουκανός) Greek form of Latin Lucanus, LOUKANOS means "from Lucania," a region of southern Italy. Lucania probably comes from the word lux, meaning "light."
Surname or Lastname
English
English : metonymic occupational name for an official responsible for obtaining the supplies required by a monastery or manor house, from Anglo-Norman French purchacer ‘to acquire or buy’ (Old French pourchacier, from chacier ‘to chase or catch’ + the intensive prefix p(o)ur, Latin pro).
Girl/Female
Irish
muirgheal “bright as the sea.†The Irish form of the name Muriel.
Girl/Female
Arabic, Muslim
Princess; Success; Beautiful
Boy/Male
Arthurian Legend
Name of a king.
ADJOINT FUNCTORS
ADJOINT FUNCTORS
ADJOINT FUNCTORS
ADJOINT FUNCTORS
ADJOINT FUNCTORS
v. t.
To join or unite to; to lie contiguous to; to be in contact with; to attach; to append.
v. t.
To separate the joints; of; to divide at the joint or joints; to disjoint; to cut up into joints, as meat.
v. t.
To unite by a joint or joints; to fit together; to prepare so as to fit together; as, to joint boards.
p. pr. & vb. n.
of Adjoin
n.
The space between the adjacent surfaces of two bodies joined and held together, as by means of cement, mortar, etc.; as, a thin joint.
a.
United, joined, or sharing with another or with others; not solitary in interest or action; holding in common with an associate, or with associates; acting together; as, joint heir; joint creditor; joint debtor, etc.
imp. & p. p.
of Adjoin
a.
Joined; united; combined; concerted; as joint action.
v. t.
To provide with a joint or joints; to articulate.
v. t.
To reunite the joints of; to joint anew.
n.
An adjunct; a helper.
v. i.
To lie or be next, or in contact; to be contiguous; as, the houses adjoin.
a.
Shared by, or affecting two or more; held in common; as, joint property; a joint bond.
n.
The place or part where two things or parts are joined or united; the union of two or more smooth or even surfaces admitting of a close-fitting or junction; junction as, a joint between two pieces of timber; a joint in a pipe.
n.
The part or space included between two joints, knots, nodes, or articulations; as, a joint of cane or of a grass stem; a joint of the leg.
v. i.
To join one's self.
n.
A joining of two things or parts so as to admit of motion; an articulation, whether movable or not; a hinge; as, the knee joint; a node or joint of a stem; a ball and socket joint. See Articulation.
a.
Dexterous in the use of the hands or in the exercise of the mental faculties; exhibiting skill and readiness in avoiding danger or escaping difficulty; ready in invention or execution; -- applied to persons and to acts; as, an adroit mechanic, an adroit reply.
v. i.
To fit as if by joints; to coalesce as joints do; as, the stones joint, neatly.