In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another. Such functors are ubiquitous in mathematics. Adjoint functors are studied in a branch of mathematics known as category theory. Like much of category theory, the general notion of adjoint functors arises at an abstract level beyond the everyday usage of most mathematicians. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...

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

Motivation

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 Daniel Marinus Kan is a mathematician, more specifically a homotopy theorist. ... Year 1958 (MCMLVIII) was a common year starting on Wednesday (link will display full calendar) of the Gregorian calendar. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ...

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

in the category of abelian groups, where F was the functor –⊗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. In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ... 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 (while being unlikely to introduce errors or cause confusion). ... In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ... Jargon redirects here. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...

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. 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]. The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... In mathematics, the conjugate transpose or adjoint of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ...

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. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In category theory, abstract algebra and other fields of mathematics, frequently constructions are defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...

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 mathematician who cofounded category theory with Samuel Eilenberg. ...

Deep 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. Alexander Grothendieck (born March 28, 1928 in Berlin, Germany) is one of the most important mathematicians of the 20th century. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of 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. In algebraic geometry, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n (and in greater generality) for vector bundles and the more general coherent sheaves. ...

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. The term handwaving is used in mathematics and physics to describe arguments that are not mathematically rigorous. ... In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. ...

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 under certain conditions. ...

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. In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I &#8594; X. The dual notion is that of a terminal object: T is terminal, if to every object X in C... In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ... In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...

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.^{[citation needed]}) 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'. This overt use of impredicativity is honest, in a way that category theory has no intention of being. John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. ... In mathematics, a predicate is a relation. ...

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 associative algebra is unital if it contains a multiplicative identity element, i. ...

The major argument in favour 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 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. In mathematics, especially order theory, a partially ordered set (or poset) is a set equipped with a partial order relation. ... In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets (posets). Galois connections generalize the correspondence between subgroups and subfields investigated in Galois theory. ... In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. ... In mathematics, given a partially ordered set (P, ≤), a closure operator on P is a function C : P → P with the following properties: x ≤ C(x) for all x, i. ...

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 word duality has a variety of different meanings in different contexts: In several spiritual, religious, and philosophical doctrines, duality refers to a two-fold division also called dualism. ... Irving Kaplansky (March 22, 1917) is a Canada 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. 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. In category theory, a monad or triple is a type of functor, together with two associated natural transformations. ... 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 on a set. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... In logic, material implication is a binary operator. ... In logic and mathematics, a propositional calculus (or a sentential calculus) is a formal system in which formulas representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules allows to establish that certain formulas are theorems of the formal system. ... In abstract algebra, if I and J are ideals of a ring A, their ideal quotient (I:J) is the set of all x in A such that xJ is a subset of I. Then (I:J) is itself an ideal in A. Categories: Ring theory ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ...

Formal definitions

A pair of adjoint functors between two categories C and D consists of two functors F : C → D and G : D → C and a natural isomorphism Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...

Φ : Hom_D(F–, –) → Hom_C(–, G–)

consisting of bijections: A bijective function. ...

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

for all objects X in C and Y in D. We then say that F is a left-adjoint of G and G is a right-adjoint of F, and often write F⊣G.

In order to interpret Φ 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). 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: In category theory, a functor is a special type of mapping between categories. ... In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ... In mathematics, specifically in category theory, Hom-sets, i. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...

The horizontal arrows in this diagram are those induced by f and g. Image File history File links No higher resolution available. ...

Unit and co-unit

Every adjoint pair of functors defines a unit η, a natural transformation from the functor 1_C to GF consisting of morphisms In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...

η_X : X → GF(X)

for every X in C. η_X is defined as Φ_X,F(X) (id_F(X)).

Analogously, one may define a co-unit ε, a natural transformation from FG to 1_D consisting of morphisms

ε_Y : FG(Y) → Y.

for every Y in D. ε_Y is defined as Φ_G(Y),Y⁻¹(id_G(Y)).

If the unit and counit are actually isomorphisms, then the adjunction provides an equivalence of categories. For this reason, an adjunction may be considered an even weaker form of equivalence of categories. In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are essentially the same. There are numerous examples of categorical equivalences from many areas of mathematics. ...

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 mathematics, the idea of a free object is one of the basic concepts of abstract algebra. ... A forgetful functor is a type of functor in mathematics. ... In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ... In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. ... The Cayley graph of the free group on two generators a and b In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many...

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 abstract algebra, 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 one and only one way as a finite linear combination of elements of the basis, with integer coefficients. ... In mathematics, a free module is a module having a free basis. ...

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. In category theory, for any object a in any category C where the product a×a exists, there exists the diagonal morphism δa: a → a×a, satisfying Ï€kδa = ida for k=1,2, where Ï€k is the canonical projection morphism to the k-th component. ...

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. In mathematics, the Cartesian product is a direct product of sets. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ... 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 of mathematics, like products and inverse limits. ...

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. In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. ... In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

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. In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... 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 to lie in a field, the scalars may lie in an arbitrary ring. ... 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. 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. 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 fails to be injective. ...

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. In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ...

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. In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ...

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(A, M) 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. In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In mathematics, a group ring is a ring R[G] constructed from a ring R and a multiplicative group G. Sometimes the group ring is written simply as RG. As an R-module, the ring R[G] is the free module over R on the elements G. If R is... This picture illustrates how the hours on a clock form a group under modular addition. ... In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ...

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). In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... In mathematics, in the field of sheaf theory and especially in algebraic geometry, the direct image functor generalizes the notion of a section of a sheaf to the relative case. ... In mathematics, the inverse image functor is a contravariant construction of sheaves. ... In mathematics, especially in algebraic geometry and the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a sheaf isomorphic with the cokernel of a morphism of OX-modules OXm → OXn. ...

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. In mathematics, K-theory is, firstly, an extraordinary cohomology theory which consists of topological K-theory. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, the Grothendieck group construction in abstract algebra constructs an abelian group from a commutative monoid in the best possible way. ... A negative number is a number that is less than zero, such as &#8722;3. ... In mathematics, an existence theorem is a theorem with a statement beginning there exist(s) .., or more generally for all x, y, ... there exist(s) .... That is, in more formal terms of symbolic logic, it is a theorem with a statement involving the existential quantifier. ... Universal algebra (sometimes called General algebra) is the field of mathematics that studies the ideas common to all algebraic structures. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ...

Frobenius reciprocity in the representation theory of groups: see induced representation. This example foreshadowed the general theory by about half a century. Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mathematics, and in particular group representation theory, the induced representation is one of the major general operations for passing from a representation of a subgroup H to a representation of the (whole) group G itself. ...

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. In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, the Stone–Čech compactification of a Tychonoff topological space is the largest Hausdorff compactification of , in the sense that any Hausdorff compactification of is a quotient of in a way that preserves the embeddings of . ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... 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. 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 mathematics, especially in topology and order theory, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. ... In mathematics, particularly in topology, a topological space X is sober if every irreducible closed subset of X is the closure of exactly one singleton of X. An irreducible closed subset of X is defined to be a nonempty closed subset of X which is not the union of two... In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are essentially the same. There are numerous examples of categorical equivalences from many areas of mathematics. ... Pointless 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). 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. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ... 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.

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. In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...

Relation to universal constructions

All pairs of adjoint functors arise from universal constructions. Let F and G be a pair of adjoint functors with unit η and co-unit ε. Then we have a universal morphism for each object in C and 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 under certain conditions. ...

• For each object X in C, (F(X), η_X) is a universal morphism from X to G. That is, for all f : X → G(Y) there exists a unique g : F(X) → Y for which the following diagrams commute.
• For each object Y in D, (G(Y), ε_Y) is a universal morphism from F to Y. That is, for all g : F(X) → Y there exists a unique f : X → G(Y) for which the following diagrams commute.

Conversely, given any two functors F and G and natural transformations η : 1_C → GF and ε : FG → 1_D which are universal in the above sense, then F and G form an adjoint pair. (Actually it is sufficient to specify only one of η or ε). The isomorphism φ is then determined by the equations Commutative diagram for adjoint functors File links The following pages link to this file: Universal property Adjoint functors Categories: GFDL images ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...

f = φ_X,Y(g) = G(g) O η_X

g = φ_X,Y⁻¹(f) = ε_Y O F(f)

Universal constructions are more general than adjoint functor pairs: as mentioned earlier, 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).

Characterization via unit and co-unit

There exists yet another characterization of adjoint functors via the unit

η : 1_C → GF

and co-unit

ε : FG → 1_D.

These natural transformations have the following properties: the natural transformation

(εF)O(Fη) : F → FGF → F

is equal to 1_F, and the composition

(Gε)o(ηG) : G → GFG → G

is equal to 1_G. These are 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. ...

Conversely, given two natural transformations η and ε with these properties, then the functors F and G form an adjoint pair.

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). 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 of mathematics, like products and inverse limits. ... 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 of mathematics, like products and inverse limits. ...

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 coproduct, or categorical sum, is the dual notion to the categorical product. ... In homological algebra, an exact functor is one which preserves exact sequences. ... In homological algebra, an exact functor is one which preserves exact sequences. ...

Additivity

If the functor F : C → D is left adjoint to G : D → C and both C and D are additive categories, then both F and G are additive functors. In mathematics, specifically in category theory, an additive category is a preadditive category C such that any finitely many objects A1,...,An of C have a biproduct A1 &#8853; ··· &#8853; An in C. (Recall that a category C is preadditive if all its morphism sets are Abelian groups and morphism... A preadditive category is a category that is enriched over the monoidal category of abelian groups. ...

Composition

If the functor F₁ : C → D has G₁ : D → C as right adjoint and the functor F₂ : D → E has G₂ : E → D as right adjoint, then the composition F₂oF₁ : C → E has G₁oG₂ : E → C as right adjoint.

Adjoint pairs extend equivalences

Every adjoint pair extends an equivalence of certain subcategories. Specifically, if F : C → D is left adjoint to G : D → C with unit η and co-unit ε, 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. Then F and G can be restricted to C₁ and D₁ and yield inverse equivalences of these subcategories. In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are essentially the same. There are numerous examples of categorical equivalences from many areas of mathematics. ... A subcategory in Wikipedia is a category that depends on another category. ...

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.

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 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 of mathematics, like products and inverse limits. ... Peter J. Freyd Peter Freyd is a mathematician, perhaps most famous as the author of the sole book on Abelian categories. ... 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 of mathematics, like products and inverse limits. ...

f_i : X → G(Y_i)

where the indices i come from a set I, not a proper class, such that every morphism In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ...

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.

An analogous statement characterizes those functors with a right adjoint.