Adjoint functors - Citizendia

"Adjunction" redirects here. For the construction in field theory, see Adjunction (field theory). In Abstract algebra, adjunction is a construction in field theory, where for a given Field extension E / F, subextensions

In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories Such functors are ubiquitous in mathematics. The general notion of adjoint functor is studied in a branch of mathematics known as category theory. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets

Specifically, functors F : C → D and G : D → C between categories C and D form an adjoint pair if there is a family of bijections

$\mathrm{Hom}_{\mathcal{D}}(FX,Y) \cong \mathrm{Hom}_{\mathcal{C}}(X,GY)$

which is natural in the variables X and Y. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal The functor F is called a left adjoint functor, while G is called a right adjoint functor. The relationship “F is left adjoint to G” is sometimes written

$F\dashv G.$

A precise definition and several equivalent formulations are given below.

Motivation

Adjoint functors can be considered from several different points of view. This article starts with a number of introductory sections considering some of these.

Ubiquity of adjoint functors

The idea of an adjoint functor was formulated by Daniel Kan in 1958. Daniel Marinus Kan is a Mathematician working in Homotopy theory. Year 1958 ( MCMLVIII) was a Common year starting on Wednesday (link will display full calendar of the Gregorian calendar. Like many of the concepts in category theory, it was suggested by the needs of homological algebra, which was at the time devoted to computations. Homological algebra is the branch of Mathematics which studies homology in a general algebraic setting Those faced with giving tidy, systematic presentations of the subject would have noticed relations such as

Hom(F(X), Y) = Hom(X, G(Y))

in the category of abelian groups, where F was the functor $- \otimes A$ (i. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the e. take the tensor product with A), and G was the functor Hom(A,–). In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector The use of the equals sign is an abuse of notation; those two groups aren't really identical but there is a way of identifying them that is natural. In Mathematics, abuse of notation occurs when an author uses a Mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition It can be seen to be natural on the basis, firstly, that these are two alternative descriptions of the bilinear mappings from X × A to Y. In Mathematics, a bilinear map is a function of two arguments that is linear in each That's something particular to the case of tensor product, though. What category theory teaches is that 'natural' is a well-defined term of art in mathematics: natural equivalence. Technical terminology is the specialized Vocabulary of a field In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal

The terminology comes from the Hilbert space idea of adjoint operators T, U with <Tx,y> = <x,Uy>, which is formally similar to the above Hom relation. This article assumes some familiarity with Analytic geometry and the concept of a limit. In Mathematics, specifically in Functional analysis, each Linear operator on a Hilbert space has a corresponding adjoint operator. We say that F is left adjoint to G, and G is right adjoint to F. Note that G may have itself a right adjoint that is quite different from F (see below for an example). The analogy to adjoint maps of Hilbert spaces can be made precise in certain contexts [1].

If one starts looking for these adjoint pairs of functors, they turn out to be very common in abstract algebra, and elsewhere as well. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules The example section below provides evidence of this; furthermore, universal constructions, which may be more familiar to some, give rise to numerous adjoint pairs of functors. In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism

In accordance with the thinking of Saunders Mac Lane, any idea such as adjoint functors that occurs widely enough in mathematics should be studied for its own sake. Saunders Mac Lane ( 4 August 1909, Taftville, Connecticut – 14 April 2005, San Francisco) was an American

Problems formulated with adjoint functors

By itself, the generality of the adjoint functor concept isn't a recommendation to most mathematicians. Concepts are judged according to their use in solving problems, at least as much as for their use in building theories. The tension between these two potential motivations for developing a mathematical concept was especially great during the 1950s when category theory was initially developed. Enter Alexander Grothendieck, who used category theory to take compass bearings in foundational, axiomatic work — in functional analysis, homological algebra and finally algebraic geometry. Experimental infobox see Wikipedia talkPersondata before changing --> Alexander Grothendieck (born March 28, 1928 in Berlin, Germany For functional analysis as used in psychology see the Functional analysis (psychology article Homological algebra is the branch of Mathematics which studies homology in a general algebraic setting Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with

It is probably wrong to say that he promoted the adjoint functor concept in isolation: but recognition of the role of adjunction was inherent in Grothendieck's approach. For example, one of his major achievements was the formulation of Serre duality in relative form — one could say loosely, in a continuous family of algebraic varieties. In Algebraic geometry, a branch of Mathematics, Serre duality is a duality present on Non-singular projective algebraic varieties The entire proof turned on the existence of a right adjoint to a certain functor. This is something undeniably abstract, and non-constructive, but also powerful in its own way.

Adjoint functors as solving optimization problems

One good way to motivate adjoint functors is to explain what problem they solve, and how they solve it.

That can only be done, in some sense, by what mathematicians call 'hand-waving'. The term handwaving is an informal term that describes either the Debate technique of failing to Rigorously address an Argument in an attempt to bypass the It can be said, however, that adjoint functors pin down the concept of the best structure of a type one is interested in constructing. For example, an elementary question in ring theory is how to add a multiplicative identity to a ring that doesn't have one (the definition in this encyclopedia actually assumes one: see ring (mathematics) and glossary of ring theory). In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real Ring theory is the branch of Mathematics in which rings are studied that is structures supporting both an Addition and a Multiplication operation The best way is to add an element '1' to the ring, add nothing extra you don't need (you will need to have r+1 for r in the ring, clearly), and add no relations in the new ring that aren't forced by axioms. This is rather vague, though suggestive.

There are several ways to make precise this concept of best structure. Adjoint functors are one method; the notion of universal properties provides another, essentially equivalent but arguably more concrete approach. In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism

Universal properties are also based on category theory. The idea is to set up the problem in terms of some auxiliary category C; and then identify what we want to do as showing that C has an initial object. This has an advantage that the optimisation — the sense that we are finding the best solution — is singled out and recognisable rather like the attainment of a supremum. To do it is something of a knack: for example, take the given ring R, and make a category C whose objects are ring homomorphisms R → S, with S a ring having a multiplicative identity. The morphisms in C must fill in triangles that are commutative diagrams, and preserve multiplicative identity. In mathematics and especially in Category theory a commutative diagram is a Diagram of objects also known as vertices, and Morphisms also The assertion is that C has an initial object R → R*, and R* is then the sought-after ring.

The adjoint functor method for defining a multiplicative identity for rings is to look at two categories, C₀ and C₁, of rings, respectively without and with assumption of multiplicative identity. There is a functor from C₁ to C₀ that forgets about the 1. We are seeking a left adjoint to it. This is a clear, if dry, formulation.

One way to see what is achieved by using either formulation is to try a direct method. (This is favoured, for example, by John H. Conway. John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups ) One simply adds to R a new element 1, and calculates on the basis that any equation resulting is valid if and only if it holds for all rings that we can create from R and 1. ↔ This is the impredicative method: meaning that the ring we are trying to construct is one of the rings quantified over in 'all rings'. In Mathematics and Logic, impredicativity is the property of a self-referencing Definition.

The answer regarding the way to get a (unital) ring from one that is not unital is simple enough (see examples below); this section has been a discussion of how to formulate the question. In Mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative Identity element (or unit) i

The major argument in favour of the use of adjoint functors is probably this: if one goes through the universal property or impredicative reasoning often enough, it seems like repeating the same kind of steps.

The case of partial orders

Every partially ordered set can be viewed as a category (with a single morphism between x and y if and only if x ≤ y). In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement A pair of adjoint functors between two partially ordered sets is called a Galois connection (or, if it is contravariant, an antitone Galois connection). In Mathematics, especially in Order theory, a Galois connection is a particular correspondence between two Partially ordered sets (posets See that article for a number of examples: the case of Galois theory of course is a leading one. In Mathematics, more specifically in Abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory Any Galois connection gives rise to closure operators and to inverse order-preserving bijections between the corresponding closed elements. A closure operator on a set S is a function cl P ( S) → P ( S) from the Power set of S

As is the case for Galois groups, the real interest lies often in refining a correspondence to a duality (i. e. antitone order isomorphism). A treatment of Galois theory along these lines by Kaplansky was influential in the recognition of the general structure here. Irving Kaplansky ( March 22, 1917 &ndash June 25, 2006) was a Canadian Mathematician.

The partial order case collapses the adjunction definitions quite noticeably, but can provide several themes:

adjunctions may not be dualities or isomorphisms, but are candidates for upgrading to that status
closure operators may indicate the presence of adjunctions, as corresponding monads (cf. In Category theory, a monad or triple is an (endo- Functor, together with two associated Natural transformations They are important in the theory the Kuratowski closure axioms)
a very general comment of Martin Hyland is that syntax and semantics are adjoint: take C to be the set of all logical theories (axiomatizations), and D the power set of the set of all mathematical structures. In Topology and related branches of Mathematics, the Kuratowski closure axioms are a set of Axioms which can be used to define a Topological structure For a theory T in C, let F(T) be the set of all structures that satisfy the axioms T; for a set of mathematical structures S, let G(S) be the minimal axiomatization of S. We can then say that F(T) is a subset of S if and only if T logically implies G(S): the "semantics functor" F is left adjoint to the "syntax functor" G.
division is (in general) the attempt to invert multiplication, but many examples, such as the introduction of implication in propositional logic, or the ideal quotient for division by ring ideals, can be recognised as the attempt to provide an adjoint. In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" In Abstract algebra, if I and J are ideals of a commutative ring A, their ideal quotient ( I: J) In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring.

Together these observations provide explanatory value all over mathematics.

Formal definitions

There are a variety of equivalent ways of defining adjoint functors. We give three such definitions here. The whole picture, and the relationships between these definitions, will be given in the next section.

Hom set definition

An adjunction between two categories C and D consists of two functors F : C → D and G : D → C and a natural isomorphism

Φ : Hom_D (F–, –) → Hom_C (–, G–)

consisting of bijections:

Φ_X,Y : Hom_D (F(X), Y) → Hom_C (X, G(Y))

for all objects X in C and Y in D. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property

In order to interpret Φ as a natural isomorphism, one must recognize Hom_D(F–, –) and Hom_C(–, G–) as functors. In fact, they are both bifunctors from C^op × D to Set (the category of sets). In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose Morphisms are For details, see the article on Hom functors. In Mathematics, specifically in Category theory, Hom-sets ie sets of Morphisms between objects give rise to important Functors to the Category Explicitly, the naturality of Φ means that for all morphisms f : X ′ → X in C and all morphisms g : Y → Y ′ in D the following diagram commutes:

The horizontal arrows in this diagram are those induced by f and g. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In mathematics and especially in Category theory a commutative diagram is a Diagram of objects also known as vertices, and Morphisms also

Universal morphism definition

An adjunction between two categories C and D is given by a functor G : D → C together with a universal morphism (FX, η_X) from X to G for each X in C. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism The assignment $X\mapsto FX$ defines a functor F : C → D called the left adjoint of G. The morphisms η_X form the components of a natural transformation η : 1_C → GF called the unit of the adjunction. In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal

Dually, an adjunction between two categories C and D is given by a functor F : C → D together with a universal morphism (GY, ε_Y) from F to Y for each Y in D. In Category theory, an abstract branch of Mathematics, the dual category or opposite category C op of a category C is the The assignment $Y\mapsto GY$ defines a functor G : D → C called the right adjoint of F. The morphisms ε_Y form the components of a natural transformation ε : FG → 1_D called the counit of the adjunction.

Unit and counit definition

An adjunction between two categories C and D consists of two functors F : C → D and G : D → C and two natural transformations

$\begin{align} \eta &: 1_{\mathcal C} \to GF \\ \varepsilon &: FG \to 1_{\mathcal D}\end{align}$

called the unit and the counit of the adjunction, respectively. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal These must satisfy

$\begin{align} 1_F &= \varepsilon F\circ F\eta : F \to FGF \to F \\ 1_G &= G\varepsilon \circ \eta G : G \to GFG \to G \end{align}$

where 1_F and 1_G are the identity transformations on F and G respectively (i. e. the transformations whose components are all identity morphisms). These equations are sometimes called the zig-zag equations because of the appearance of the corresponding string diagrams. In Category theory, string diagrams are a way of representing 2-cells in 2-categories. In component form these equations are

$\begin{align} \mathrm{id}_{FX} &= \varepsilon_{FX}\circ F(\eta_X) \\ \mathrm{id}_{GY} &= G(\varepsilon_Y)\circ\eta_{GY} \end{align}$

for each X in C and each Y in D.

Adjunctions

There are numerous functors and natural transformations associated with every adjunction, and there exists an intricate web of relationships among these objects. These relationships allow one to determine all of the data based on knowledge of only some of the pieces.

An adjunction between categories C and D consists of

A functor F : C → D called the left adjoint
A functor G : D → C called the right adjoint
A natural transformation η : 1_C → GF called the unit
A natural transformation ε : FG → 1_D called the counit
A natural isomorphism Φ : Hom_D(F–,–) → Hom_C(–,G–)

The web of relationships between these functors and transformations can be neatly summarized by a family of commutative diagrams. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal In mathematics and especially in Category theory a commutative diagram is a Diagram of objects also known as vertices, and Morphisms also Let X be an object in C and Y an object in D. Then for every morphism f : X → G(Y) there is a unique morphism g : F(X) → Y such that the following two diagrams commute:

Likewise, for every g : F(X) → Y there is a unique f : X → G(Y) such that the above diagrams commute. Each adjunction, therefore, gives rise to a family of universal morphisms:

For each object X in C, the pair (F(X), η_X) is a universal morphism from X to G. In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism
For each object Y in D, the pair (G(Y), ε_Y) is a universal morphism from F to Y.

The morphisms f and g in the above diagrams are related by the natural isomorphism Φ, which for each X in C and Y in D gives a bijection

$\Phi_{X,Y}:\mathrm{Hom}_{\mathcal{D}}(F(X),Y) \cong \mathrm{Hom}_{\mathcal{C}}(X,G(Y))$

such that

$\begin{align}f &= \Phi_{X,Y}(g) = G(g)\circ \eta_X\\ g &= \Phi_{X,Y}^{-1}(f) = \varepsilon_Y\circ F(f)\end{align}$

If either the unit and counit are given then these equations can be used to define Φ.

We consider two special cases of the above diagrams:

The unit

If Y = F(X) then Hom_D(F(X),Y) contains the identity morphism. The corresponding element of Hom_C(X,G(Y)) is just the unit η_X. That is, if Y = F(X) and g = id_F(X), then f = η_X so that

$\eta_X = \Phi_{X,FX}\left(\mathrm{id}_{FX}\right),\qquad \mathrm{id}_{FX} = \varepsilon_{FX}\circ F(\eta_X).$

The first of these equations gives η in terms of Φ. The second, holding for all X in C, is equivalent to the statement that the composition

$F\xrightarrow{\;F\eta\;}FGF\xrightarrow{\;\varepsilon F\,}F$

is equal to the identity transformation from F to F.

The counit

If X = G(Y) then Hom_C(X,G(Y)) contains the identity morphism. The corresponding element of Hom_D(F(X),Y) is just the counit ε_Y. That is, if X = G(Y) and f = id_G(Y), then g = ε_Y so that

$\mathrm{id}_{GY} = G(\varepsilon_Y)\circ\eta_{GY},\qquad\varepsilon_Y = \Phi_{GY,Y}^{-1}\left(\mathrm{id}_{GY}\right).$

The second of these equations gives ε in terms of Φ, while the first, holding for all Y in D, is equivalent to the statement that the composition

$G\xrightarrow{\;\eta G\;}GFG\xrightarrow{\;G \varepsilon\,}G$

is equal to the identity transformation from G to G.

Examples

Free objects and forgetful functors. In Mathematics, the idea of a free object is one of the basic concepts of Abstract algebra. In Mathematics, in the area of Category theory, a forgetful functor is a type of Functor. If F : Set → Grp is the functor assigning to each set X the free group over X, and if G : Grp → Set is the forgetful functor assigning to each group its underlying set, then the universal property of the free group shows that F is left adjoint to G. In Mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose Morphisms are In Mathematics, the category Grp has the class of all groups for objects and Group homomorphisms for Morphisms As such In Mathematics, a group G is called free if there is a Subset S of G such that any element of G can be The unit of this adjoint pair is the embedding of a set X into the free group over X.

In general, free constructions in mathematics tend to be left adjoints of forgetful functors. Free rings, free abelian groups, and free modules follow this general pattern. In Mathematics, especially in the area of Abstract algebra known as Ring theory, a free algebra is the noncommutative analogue of a Polynomial ring In Abstract algebra, a free abelian group is an Abelian group that has a "basis" in the sense that every element of the group can be written in In Mathematics, a free module is a Free object in the category of modules Given a set S, a free module on S is a (particular construction

Products. In Category theory, the product of two (or more objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as Let Δ : Grp → Grp² be the diagonal functor which assigns to every group X the pair (X, X) in the product category Grp², and Π : Grp² → Grp the functor which assigns to each pair (Y₁, Y₂) the product group Y₁×Y₂. In Category theory, for any object a in any category C where the product a × a exists there exists the diagonal morphism The universal property of the product group shows that Π is right-adjoint to Δ. The co-unit gives the natural projections from the product to the factors.

The cartesian product of sets, the product of rings, the product of topological spaces etc. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural follow the same pattern; it can also be extended in a straightforward manner to more than just two factors. In fact, any limit or colimit is adjoint to a diagonal functor. In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts

Coproducts. In Category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological If F : Ab² → Ab assigns to every pair (X₁, X₂) of abelian groups their direct sum and if G : Ab → Ab² is the functor which assigns to every abelian group Y the pair (Y, Y), then F is left adjoint to G, again a consequence of the universal property of direct sums. In Mathematics, the category Ab has the Abelian groups as objects and Group homomorphisms as Morphisms This is the prototype The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction The unit of the adjoint pair provides the natural embeddings from the factors into the direct sum.

Analogous examples are given by the direct sum of vector spaces and modules, by the free product of groups and by the disjoint union of sets. The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Abstract algebra, the free product of groups constructs a group from two or more given ones

Kernels. Consider the category D of homomorphisms of abelian groups. If f₁ : A₁ → B₁ and f₂ : A₂ → B₂ are two objects of D, then a morphism from f₁ to f₂ is a pair (g_A, g_B) of morphisms such that g_Bf₁ = f₂g_A. Let G : D → Ab be the functor which assigns to each homomorphism its kernel and let F : Ab → D be the morphism which maps the group A to the homomorphism A → 0. In the various branches of Mathematics that fall under the heading of Abstract algebra, the kernel of a Homomorphism measures the degree to which the homomorphism Then G is right adjoint to F, which expresses the universal property of kernels, and the co-unit of this adjunction yields the natural embedding of a homomorphism's kernel into the homomorphism's domain.

A suitable variation of this example also shows that the kernel functors for vector spaces and for modules are right adjoints. Analogously, one can show that the cokernel functors for abelian groups, vector spaces and modules are left adjoints.

Making a ring unital This example was discussed in section 1. In Mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative Identity element (or unit) i 3 above. Given a non-unital ring R, a multiplicative identity element can be added by taking RxZ and defining a Z-bilinear product with (r,0)(0,1) = (0,1)(r,0) = (r,0), (r,0)(s,0) = (rs,0), (0,1)(0,1) = (0,1). This constructs a left adjoint to the functor taking a ring to the underlying non-unital ring.

Ring extensions. Suppose R and S are rings, and ρ : R → S is a ring homomorphism. In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication Then S can be seen as a (left) R-module, and the tensor product with S yields a functor F : R-Mod → S-Mod. In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector Then F is left adjoint to the forgetful functor G : S-Mod → R-Mod.

Tensor products. If R is a ring and M is a right R module, then the tensor product with M yields a functor F : R-Mod → Ab. The functor G : Ab → R-Mod, defined by G(A) = Hom_Z(M,A) for every abelian group A, is a right adjoint to F.

From monoids and groups to rings The integral monoid ring construction gives a functor from monoids to rings. In Abstract algebra, a monoid ring is a new ring constructed from some other ring and a Monoid. In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation This functor is left adjoint to the functor that associates to a given ring its underlying multiplicative monoid. Similarly, the integral group ring construction yields a functor from groups to rings, left adjoint to the functor that assigns to a given ring its group of units. In Mathematics, a group ring is a ring R constructed from a ring R and a group G (written multiplicatively In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i One can also start with a field K and consider the category of K-algebras instead of the category of rings, to get the monoid and group rings over K. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive

Suspensions and loop spaces Given topological spaces X and Y, the space [SX, Y] of homotopy classes of maps from the suspension SX of X to Y is naturally isomorphic to the space [X, ΩY] of homotopy classes of maps from X to the loop space ΩY of Y. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical In Topology, the suspension SX of a Topological space X is the Quotient space: SX = (X \times I/\{(x_10\sim(x_20\mbox{ In Mathematics, the space of loops or (free loop space of a Topological space X is the space of loops from the Unit circle This is an important fact in homotopy theory. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical

Direct and inverse images of sheaves Every continuous map f : X → Y between topological spaces induces a functor f _∗ from the category of sheaves (of sets, or abelian groups, or rings. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. . . ) on X to the corresponding category of sheaves on Y, the direct image functor. In Mathematics, in the field of Sheaf theory and especially in Algebraic geometry, the direct image functor generalizes the notion of a Section of It also induces a functor f ⁻¹ from the category of sheaves of abelian groups on Y to the category of sheaves of abelian groups on X, the inverse image functor. In Mathematics, the inverse image functor is a Contravariant construction of sheaves f ⁻¹ is left adjoint to f _∗. Here a more subtle point is that the left adjoint for coherent sheaves will differ from that for sheaves (of sets). In Mathematics, especially in Algebraic geometry and the theory of Complex manifolds coherent sheaves are specific class of sheaves having

The Grothendieck group. In K-theory, the point of departure is to observe that the category of vector bundles on a topological space has a commutative monoid structure under direct sum. In Mathematics, K-theory is a tool used in several disciplines In Mathematics, a vector bundle is a topological construction which makes precise the idea of a family of Vector spaces parameterized by another space Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction One may make an abelian group out of this monoid, the Grothendieck group, by formally adding an additive inverse for each bundle (or equivalence class). An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, the Grothendieck group construction in Abstract algebra constructs an Abelian group from a Commutative Monoid in the Alternatively one can observe that the functor that for each group takes the underlying monoid (ignoring inverses) has a left adjoint. This is a once-for-all construction, in line with the third section discussion above. That is, one can imitate the construction of negative numbers; but there is the other option of an existence theorem. A negative number is a Number that is less than zero, such as −2 In Mathematics, an existence theorem is a theorem with a statement beginning 'there exist(s. For the case of finitary algebraic structures, the existence by itself can be referred to universal algebra, or model theory; naturally there is also a proof adapted to category theory, too. Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models" In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models

Frobenius reciprocity in the representation theory of groups: see induced representation. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In Mathematics, and in particular Group representation theory the induced representation is one of the major general operations for passing from a representation This example foreshadowed the general theory by about half a century.

Stone-Čech compactification. Let D be the category of compact Hausdorff spaces and G : D → Top be the forgetful functor which treats every compact Hausdorff space as a topological space. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. Then G has a left adjoint F : Top → D, the Stone–Čech compactification. In the mathematical discipline of General topology, Stone–Čech compactification is a technique for constructing a universal map from a topological space X to a The unit of this adjoint pair yields a continuous map from every topological space X into its Stone-Čech compactification. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function This map is an embedding (i. In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group e. injective, continuous and open) if and only if X is a Tychonoff space. In Topology and related branches of Mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of Topological spaces

Soberification. The article on Stone duality describes an adjunction between the category of topological spaces and the category of sober spaces that is known as soberification. In Mathematics, there is an ample supply of categorical dualities between certain categories of Topological spaces and categories of Partially ordered In Mathematics, particularly in Topology, a sober space is a particular kind of Topological space. Notably, the article also contains a detailed description of another adjunction that prepares the way for the famous duality of sober spaces and spatial locales, exploited in pointless topology. In Category theory, an abstract branch of Mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are In Mathematics, pointless topology (also called point-free or pointfree topology is an approach to Topology which avoids the mentioning of points

A functor with a left and a right adjoint. Let G be the functor from topological spaces to sets that associates to every topological space its underlying set (forgetting the topology, that is). Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. G has a left adjoint F, creating the discrete space on a set Y, and a right adjoint H creating the trivial topology on Y. In Topology, a discrete space is a particularly simple example of a Topological space or similar structure one in which the points are " isolated " In Topology, a Topological space with the trivial topology is one where the only Open sets are the Empty set and the entire space

The functor π₀ which assigns to a category its sets of connected components is left-adjoint to the functor D which assigns to a set the discrete category on that set. Moreover, D is left-adjoint to the object functor U which assigns to each category its set of objects, and finally U is left-adjoint to A which assigns to each set the antidiscrete category on that set.

Field of fractions. In Mathematics, every Integral domain can be embedded in a field; the smallest field which can be used is the field of fractions or field of quotients

Properties

Uniqueness of adjoints

If the functor F : C → D has two right adjoints G₁ and G₂, then G₁ and G₂ are naturally isomorphic. In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal The same is true for left adjoints.

Conversely, if F is left adjoint to G₁, and G₁ is naturally isomorphic to G₂ then F is also left adjoint to G₂. More generally, if 〈F, G, η, ε〉 is an adjunction and

σ : F → F′

τ : G → G′

are natural isomorphisms then 〈F′, G′, η′, ε′〉 is an adjunction where

$\begin{align} \eta' &= (\tau\ast\sigma)\circ\eta \\ \varepsilon' &= \varepsilon\circ(\sigma^{-1}\ast\tau^{-1}). \end{align}$

Here $\circ$ denotes vertical composition of natural transformations, and $\ast$ denotes horizontal composition.

Composition

Adjunctions can be composed in a natural fashion. Specifically, if 〈F₁, G₁, η₁, ε₁〉 is an adjunction between C and D and 〈F₂, G₂, η₂, ε₂〉 is an adjunction between D and E then the functor

$F_2 \circ F_1 : \mathcal{C} \to \mathcal{E}$

is left adjoint to

$G_1 \circ G_2 : \mathcal{E} \to \mathcal{C}.$

The unit and the counit of this adjunction are given by the compositions:

$\begin{align} &1_{\mathcal C}\xrightarrow{\eta_1}G_1F_1\xrightarrow{G_1\eta_2F_1}G_1G_2F_2F_1\\ &F_2F_1G_1G_2\xrightarrow{F_2\varepsilon_1G_2}F_2G_2\xrightarrow{\varepsilon_2}1_{\mathcal E}. \end{align}$

One can then form a category whose objects are all small categories and whose morphisms are adjunctions. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships

Adjoints preserve certain limits

The most important property of adjoints is their continuity: every functor that has a left adjoint (and therefore is a right adjoint) is continuous (i. e. commutes with limits in the category theoretical sense); every functor that has a right adjoint (and therefore is a left adjoint) is cocontinuous (i. In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts e. commutes with colimits). In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts

Since many common constructions in mathematics are limits or colimits, this provides a wealth of information. For example:

applying a right adjoint functor to a product of objects yields the product of the images;
applying a left adjoint functor to a coproduct of objects yields the coproduct of the images;
every right adjoint functor is left exact;
every left adjoint functor is right exact. In Category theory, the product of two (or more objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as In Category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological In Homological algebra, an exact functor is a Functor, from some category to another which preserves Exact sequences Exact functors are very In Homological algebra, an exact functor is a Functor, from some category to another which preserves Exact sequences Exact functors are very

Additivity

If C and D are preadditive categories and F : C → D is an additive functor with a right adjoint G : D → C, then G is also an additive functor and the Hom-set bijections

$\Phi_{X,Y} : \mathrm{Hom}_{\mathcal D}(FX,Y) \cong \mathrm{Hom}_{\mathcal C}(X,GY)$

are, in fact, isomorphisms of abelian groups. In Mathematics, specifically in Category theory, a preadditive category is a category that is enriched over the Monoidal category In Mathematics, specifically in Category theory, a preadditive category is a category that is enriched over the Monoidal category Dually, if G is additive with a left adjoint F, then F is also additive.

Moreover, if both C and D are additive categories (i. In Mathematics, specifically in Category theory, an additive category is a Preadditive category C such that any finitely many objects A e. preadditive categories with all finite biproducts), then any pair of adjoint functors between them are automatically additive. In Category theory and its applications to Mathematics, a biproduct is a generalisation of the notion of Direct sum that makes sense in any Preadditive

General existence theorem

Not every functor G : D → C admits a left adjoint. If D is complete, then the functors with left adjoints can be characterized by the adjoint functor theorem of Peter J. Freyd: G has a left adjoint if and only if it is continuous and a certain smallness condition is satisfied: for every object X of C there exists a family of morphisms

f_i : X → G(Y_i)

where the indices i come from a set I, not a proper class, such that every morphism

h : X → G(Y)

can be written as

h = G(t) o f_i

for some i in I and some morphism

t : Y_i → Y in D. In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts Peter J Freyd is an American Mathematician, a professor at the University of Pennsylvania, known for work in Category theory. In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously

An analogous statement characterizes those functors with a right adjoint.

Relationship to other categorical concepts

Universal constructions

As stated earlier, an adjunction between categories C and D gives rise to a family of universal morphisms, one for each object in C and one for each object in D. In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism Conversely, if there exists a universal morphism to a functor G : D → C from every object of C, then G has a left adjoint.

However, universal constructions are more general than adjoint functors: a universal construction is like an optimization problem; it gives rise to an adjoint pair if and only if this problem has a solution for every object of C (equivalently, every object of D).

Equivalences of categories

Every equivalence of categories defines an adjunction. In Category theory, an abstract branch of Mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are If F : C → D and G : D → C are functors with natural isomorphisms η : 1_C → GF and ε : FG → 1_D then (F, G) form an adjoint pair with unit η and counit ε. Conversely, an adjunction 〈F, G, η, ε〉 defines an equivalence of categories if and only if, the unit and counit are natural isomorphisms (and not just natural transformations).

If (F, G) define an equivalence of categories, then F is not only a left adjoint of G but a right adjoint as well. Explicitly, if 〈F, G, η, ε〉 is an adjoint equivalence then so is 〈G, F, ε⁻¹, η⁻¹〉.

Every adjunction 〈F, G, η, ε〉 extends an equivalence of certain subcategories. Define C₁ as the full subcategory of C consisting of those objects X of C for which η_X is an isomorphism, and define D₁ as the full subcategory of D consisting of those objects Y of D for which ε_Y is an isomorphism. In Mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in Then F and G can be restricted to C₁ and D₁ and yield inverse equivalences of these subcategories.

In a sense, then, adjoints are "generalized" inverses. Note however that a right inverse of F (i. e. a functor G such that FG is naturally isomorphic to 1_D) need not be a right (or left) adjoint of F. Adjoints generalize two-sided inverses.

Monads

Every adjunction 〈F, G, η, ε〉 gives rise to an associated monad 〈T, η, μ〉 in the category C. In Category theory, a monad or triple is an (endo- Functor, together with two associated Natural transformations They are important in the theory The functor

$T : \mathcal{C} \to \mathcal{C}$

is given by T = GF. The unit

$\eta : 1_{\mathcal{C}} \to T$

is just the unit η of the adjunction and the multiplication transformation

$\mu : T^2 \to T\,$

is given by μ = GεF. Dually, the triple 〈FG, ε, FηG〉 defines a comonad in D. In Category theory, a monad or triple is an (endo- Functor, together with two associated Natural transformations They are important in the theory

Every monad arises from some adjunction—in fact, typically from many adjunctions—in the above fashion. Two constructions, called the category of Eilenberg-Moore algebras and the Kleisli category are two extremal solutions to the problem of constructing an adjunction that gives rise to a given monad. In Category theory, a monad or triple is an (endo- Functor, together with two associated Natural transformations They are important in the theory In Category theory, a monad or triple is an (endo- Functor, together with two associated Natural transformations They are important in the theory

References

Adámek, Jiří; Horst Herrlich, and George E. Strecker (1990). Abstract and Concrete Categories. John Wiley & Sons. ISBN 0-471-60922-6.
Mac Lane, Saunders (1998). Saunders Mac Lane ( 4 August 1909, Taftville, Connecticut – 14 April 2005, San Francisco) was an American Categories for the Working Mathematician, (2nd ed. Categories for the Working Mathematician is a textbook in Category theory written by American Mathematician Saunders Mac Lane, who ), Graduate Texts in Mathematics 5, Springer-Verlag. ISBN 0-387-98403-8.

External links

Adjunctions Seven short lectures on adjunctions.

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