# Dictionary Definition

adjunction n : an act of joining or adjoining
things [syn: junction]

# Extensive Definition

In mathematics, adjoint
functors are pairs of functors which stand in a
particular relationship with one another. Such functors are
ubiquitous in mathematics. The general notion of adjoint functor is
studied in a branch of mathematics known as category
theory.

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_(FX,Y) \cong \mathrm_(X,GY)

- F\dashv G.

## 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. 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. 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.e. take the
tensor
product with A), and G was the functor
Hom(A,–).

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. 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. 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.

The terminology comes from the Hilbert
space idea of adjoint
operators T, U with = , which is formally similar to the above
Hom relation. 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 http://www.arxiv.org/abs/q-alg/9609018.

If one starts looking for these adjoint pairs of
functors, they turn out to be very common in abstract
algebra, and elsewhere as well. 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 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.

### 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.

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. 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'.
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). 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.

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. 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.) 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'.

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.

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). A pair of adjoint functors
between two partially ordered sets is called a Galois
connection (or, if it is contravariant, an antitone Galois
connection). See that article for a number of examples: the case of
Galois
theory of course is a leading one. Any Galois connection gives
rise to closure
operators and to inverse order-preserving bijections between
the corresponding closed elements.

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.

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. 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. 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.

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 : HomD (F–, –) → HomC (–, G–)

- ΦX,Y : HomD (F(X), Y) → HomC (X, G(Y))

In order to interpret Φ as a natural isomorphism,
one must recognize HomD(F–, –) and
HomC(–, G–) as functors. In fact, they are both
bifunctors from Cop
× D to Set (the category
of sets). For details, see the article on Hom functors.
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.

### 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, ηX) from X to G for each X in C. The assignment
X\mapsto FX defines a functor F : C → D called the left
adjoint of G. The morphisms ηX form the components of a natural
transformation η : 1C → GF called the unit of the
adjunction.

Dually,
an adjunction between two categories C and D is given by a functor
F : C → D together with a universal morphism (GY, εY) from
F to Y for each Y in D. The assignment Y\mapsto GY defines a
functor G : D → C called the right adjoint of F. The
morphisms εY form the components of a natural transformation ε : FG
→ 1D 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

- \begin

- \begin

## 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 : 1C → GF called the unit
- A natural transformation ε : FG → 1D called the counit
- A natural isomorphism Φ : HomD(F–,–) → HomC(–,G–)

The web of relationships between these functors
and transformations can be neatly summarized by a family of
commutative
diagrams. 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), ηX) is a universal morphism from X to G.
- For each object Y in D, the pair (G(Y), εY) 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_:\mathrm_(F(X),Y) \cong \mathrm_(X,G(Y))

- \beginf &= \Phi_(g) = G(g)\circ \eta_X\\

We consider two special cases of the above
diagrams:

### The unit

If Y = F(X) then HomD(F(X),Y) contains the identity morphism. The corresponding element of HomC(X,G(Y)) is just the unit ηX. That is, if Y = F(X) and g = idF(X), then f = ηX so that- \eta_X = \Phi_\left(\mathrm_\right),\qquad

- F\xrightarrowFGF\xrightarrowF

### The counit

If X = G(Y) then HomC(X,G(Y)) contains the identity morphism. The corresponding element of HomD(F(X),Y) is just the counit εY. That is, if X = G(Y) and f = idG(Y), then g = εY so that- \mathrm_ = G(\varepsilon_Y)\circ\eta_,\qquad\varepsilon_Y = \Phi_^\left(\mathrm_\right).

- G\xrightarrowGFG\xrightarrowG

## Examples

- Free objects and forgetful functors. 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. 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.

- Products. 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. 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. 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.

- Coproducts. 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. 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.

- 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 (gA, gB) of morphisms such that gBf1 = f2gA. 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. 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 1.3 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. Then S can be seen as a (left) R-module, and the tensor product with S yields a functor F : R-Mod → S-Mod. 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) = HomZ(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. 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. 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.

- 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. This is an important fact in homotopy theory.

- 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...) on X to the corresponding category of sheaves on Y, the direct image functor. 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. 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).

- 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. 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). 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. 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.

- Frobenius reciprocity in the representation theory of groups: see induced 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. Then G has a left adjoint F : Top → D, the Stone–Čech compactification. The unit of this adjoint pair yields a continuous map from every topological space X into its Stone-Čech compactification. This map is an embedding (i.e. injective, continuous and open) if and only if X is a Tychonoff space.

- 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. 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.

- 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). 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.

- 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.

## Properties

### Uniqueness of adjoints

If the functor F : C → D has two right adjoints
G1 and G2, then G1 and G2 are naturally
isomorphic. 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′

- \begin

### 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 \to \mathcal

- G_1 \circ G_2 : \mathcal \to \mathcal.

- \begin

One can then form a category whose objects are
all small
categories and whose morphisms are adjunctions.

### 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.e.
commutes with colimits).

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.

### 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_ : \mathrm_(FX,Y) \cong \mathrm_(X,GY)

Moreover, if both C and D are additive
categories (i.e. preadditive categories with all finite
biproducts), then any
pair of adjoint functors between them are automatically
additive.

### 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

- fi : X → G(Yi)

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 fi

for some i in I and some morphism

- t : Yi → Y in D.

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. 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. If F : C → D and
G : D → C are functors with natural isomorphisms η : 1C
→ GF and ε : FG → 1D 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
ηX is an isomorphism, and define D1 as the full subcategory of D
consisting of those objects Y of D for which εY is an isomorphism.
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 1D) 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. The functor

- T : \mathcal \to \mathcal

- \eta : 1_ \to T

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

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.

## References

- Abstract and Concrete Categories
- Categories for the Working Mathematician

## External links

- Adjunctions Seven short lectures on adjunctions.

adjunction in German: Adjunktion
(Kategorientheorie)

adjunction in Spanish: Funtores adjuntos

adjunction in French: Adjoint (foncteur)

adjunction in Korean: 수반 펑터

adjunction in Russian: Сопряжённые
функторы

adjunction in Chinese: 伴隨函子