Adjoint functors

In mathematics, specifically in category theory, the adjunction It is a relationship between two funtors that frequently appears through the different branches of mathematics and that captures an intuitive notion of solution to a problem of optimization. Two funtors ${displaystyle Fcolon {cal {D}{cal {C}}}}$ and ${displaystyle Gcolon {cal {C}}{cal {D}}}}$ they say attached to each other, if there is a family of bijections

${displaystyle mathrm {hom} _{mathcal {C}}(FY,X)cong mathrm {hom} _{mathcal {D}(Y,GX)}}}$

which is natural for any ${displaystyle X}$ e ${displaystyle Y}$. The relationship that ${displaystyle F}$ Whatever. attached to the left of ${displaystyle G}$or, equivalently, that ${displaystyle G}$ Whatever. attached to right of ${displaystyle F}$, it is noted as ${displaystyle Fdashv G}$.

Motivation

The ubiquity of adjoint functors

The idea of an adjoint functor was formulated by Daniel Kan in 1958. As with many of the concepts in category theory, it was suggested by the needs of homological algebra. Those mathematicians concerned with giving ordered or systematic presentations of the subject observed relationships such as

Hom.FB, C) = Hom (B, GC)

in the category of abelian groups, where the functor F was 'take the tensor product with A', and G was the functor Hom(A,.). Here

Hom.X, And)

means 'all homomorphisms of abelian groups'. The use of the equals sign is an abuse of notation; the two groups are not really identical but there is a way to identify them that is natural. It can be seen as natural on the ground, first of all, that these are two alternative descriptions of the bilinear functions from BxA to C. This, however, is somewhat peculiar to the tensor product. What category theory teaches is that 'natural' is a well-defined technical term in mathematics: natural equivalence.

The terminology comes from Hilbert space and the idea of the adjoint operator of T, U with <Tx, y > = <x, Uy>, which is formally similar to the previous Hom relation. We say that F is left adjoint of G, and G is right adjoint of F. Since G can itself be a right adjoint, quite different from F (see below for an example), the analogy collapses at that point. If one starts looking for these pairs of functor adjoints, they turn out to be very common in abstract algebra and elsewhere as well. The examples section below provides the evidence; moreover, universal constructions, which may be more familiar, give rise to numerous pairs of adjoint functors.

According to the thinking of Saunders MacLane, any ideas, such as adjoint functors, that occur to a sufficient extent in mathematics should be studied on their own.

Deep problems formulated with adjoint functors

Alexander Grothendieck used category theory to orient himself in certain foundational, axiomatic works of functional analysis, homological algebra, and finally algebraic geometry.

Recognition of the role of attachment was inherent in Grothendieck's approach. For example, one of his important achievements was the formulation of Serre's duality in a relative form - one can say: in a continuous family of algebraic varieties . The whole proof revolved around the existence of a right adjoint for a certain functor.

Adjoint functors as a solution to optimization problems

A good way to motivate adjoint functors is to explain what problem they solve, and how they solve it. That can only be done, in a certain sense, by gesturing. It can be said, however, that with adjoint functors the concept of the best structure is created, one of the type that one is interested in building. For example, an elementary question in ring theory is how to add a multiplicative identity to a ring that has no such thing (Wikipedia's definition actually assumes one: see ring (mathematics) and glossary of ring theory). The best way is to add an 1 element to the ring, and not add anything extra that isn't needed (you'll need to have r+1 for each r in the ring, of course), and not add any relation in the new ring that is not forced by the axioms. This is somewhat vague, though suggestive.

There are several ways to make this concept of best structure exact. Adjoint functors are a method; the notion of universal properties provides others, essentially equivalent but probably with a more concrete focus.

Universal properties are also based on category theory. The idea is to present the problem in terms of some auxiliary category C; and then identify what we want to do like prove that C has initial object. This has an advantage that optimization - the feeling that we are finding the best solution - is selected and recognizable as a supreme achievement. Doing so is a matter of skill: for example, take a given ring R, and make a category C whose objects are ring homomorphisms R → < i>S, with S a ring having a multiplicative identity. The morphisms on C must complete 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 ring sought.

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

One way to see what is achieved using either formulation is to try a direct method. (Some are more fond of these methods, for example John Conway.) Simply add a new element, 1, to R, and compute on the basis that any equation The resultant is valid if and only if it holds for all rings that we can create from R and 1. This is the impredicative method: which means that the ring we are trying to construct is one of the quantized rings in < i>all rings. This open use of impredicativity is honest, in a different way than it is in category theory.

The answer regarding how to get the (unital) ring from a non-unital one is fairly simple (see examples below); this section has been a discussion of how to formulate the question.

The main argument in favor of adjoint functors is probably this: if one proceeds with universal properties or impredicative reasoning, quite often, they seem like a repetition of the same steps.

The case of partial order

Each partially ordered set can be viewed as a category (with a single morphism between x and y if and only if x ≤ < i>y). A pair of adjoint functors between two partially ordered sets is called a Galois connection (or, if contravariant, an antitone Galois connection). See the article for a number of examples: the case for Galois theory is of course paramount. Any Galois connection gives rise to closure operators and inverse bijections that preserve order between the corresponding closed elements.

As is the case for Galois groups, often the real interest lies in refining a correspondence to a duality (ie isomorphism of antitone order). A treatment of Galois theory along these lines by Kaplansky was influential in recognizing the general structure.

The partial order reduces the attachment definitions quite noticeably, but can provide several issues:

• attachments may not be dualities or isomorphisms, but they are candidates to reach that stage
• the closing operators may indicate the presence of adjunctions, as corresponding monads (cf. Kuratowski closing axioms)
• a very general comment by Martin Hyland is that syntax and semantics are attached: take C as the whole of all logical theories (axiomatizations), and D the set of parts of all mathematical structures. For a theory T in CI mean, F(T) the set of all structures that satisfy axioms Tfor a set of mathematical structures SI mean, G(S) the minimum axiomatization S. We can then say that F(T) is a subset of S Yes and only if T logically G(S): the "semantic fan" F is attached left to the syntax filter G.
• The division is (generally) the attempt to investment multiplication, but many examples, such as the introduction of involvement in propositional logic, or division by ring ideals, can be recognized as an attempt to provide an attachment.

These observations taken together provide explanatory value about the set of all mathematics.

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 consisting of functions bijectives

φ_{X, Y}: Mor_D(F(X), And) → Mor_C(X, G(And) for all objects X in C and And in D. Then we say that F That's it. Deputy left of G and that G That's it. Deputy of F.

Each pair of adjoint functors defines a unit η, a natural transformation from functor Id_C to GF consisting of morphisms

MIL_X: X - 2005 GF(X) for each X in C. MIL_X is defined as φ_{X, F(X)} (sighs)_F(X)). Similarly, you can define a co-unity ε, a natural transformation consisting of morphisms

ε_And: FG(And) → And. for each And in D.

Examples

Free objects. If F: Set → Grp is the functor that assigns to each set X the free group on X, and if G: Grp → Set is the functor of forgetting that assigns to each group its underlying set, then the universal property of the free group shows that F is the left adjoint of G. The unit of this adjoint pair is the inclusion of a set X in the free group over X.

Free rings, free abelian groups, and free modules follow the same pattern.

Products. Let F: Grp → Grp² be the functor that assigns to each group X the pair (X, X) in the product category Group², and G: Group² → Grp the functor that assigns to each pair (Y₁, < i>Y₂) the product group Y₁xY_{2. The universal property of the product group shows that G is the right adjoint of F. The co-unit gives the natural projections of 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 direct way to more than just two factors.

Coproducts. If F: Ab² → Ab assigns to each pair (X₁, X₂) of abelian groups their direct sum and if G: Ab → Ab² is the functor that assigns to each abelian group Y the pair (Y, Y), then F is the left adjoint of G, again a consequence of the universal property of direct sums. The unit of the adjoint pair provides the natural immersions of the factors in the direct sum. Analogous examples are given by direct addition of vector spaces and modules, by 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 → B_{2 is two D objects, so a map from f1 to f2 is a pair (gA, gB< /sub>) of morphisms such that gBf1 = f< sub>2gA. Let G: D → Ab be the functor that assigns to each homomorphism its kernel and let F: Ab → D the morphism that maps the group A to the homomorphism A → 0. Then G is the right adjoint of F, which expresses the universal property of kernels, and the co-unity of this adjunct gives the natural fit of the kernel of a homomorphism in the domain of the homomorphism.}

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

Making a unital ring. This example was discussed in section 1.3 above. Given a non-unitary ring R, an identity multiplicative element can be added by taking RxZ and defining a product Z-bilinear by (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 ring-carrying functor to the underlying non-unit ring.

Ring extensions. Suppose R and S are rings, and ρ: R → S is a ring homomorphism. Then S can be thought of as an R-module (left), and the tensor product with S gives a functor F: R-Mod → S-Mod. Then F is the left adjoint of the forgetting functor G: S-Mod → R-Mod.

Tensor products. If R is a ring and M is a R right modulus, then the product tensor with M gives a functor F: R-Mod → Ab. the functor G: Ab → R-Mod, defined by G(< i>A) = Hom_Z(A, M) for each abelian group A, is a right adjoint of F.

From monoids and groups to rings. The monoid ring construction gives a functor from monoids to rings. This functor is the left adjoint to the functor that maps to a ring given its underlying multiplicative monoid. Similarly, the ring-group construction gives a functor from groups to rings, left adjoint to the functor that assigns 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, and get monoids and group rings on K .

Direct and inverse images of beams. Each continuous function f: X → Y between topological spaces induces a functor f_* of the category of bundles (of sets, or of abelian groups, or of rings...) in X to the corresponding category of beams in Y, the direct image functor. It also induces a functor f^* from the category of bundles in Y to the category of bundles in X, the inverse image functor. f^* is the left adjoint to f_*.

The Grothendieck construction. In K-theory, the starting point is to observe that the category of vector bundles in a topological space has a monoid commutative structure using direct addition. To make this monoid an abelian group, one can follow the method of extending a group, formally adding the inverse of the addition for each bundle (or the equivalence class). Alternatively one can observe that the functor that for each group takes the underlying monoid (ignoring the inverse) has a left adjoint. This is a once and for all construct, in line with the third argument in the previous section. That is, one can imitate the construction of negative numbers; but there is the other option of an existence theorem. For the case of finite algebraic structures, existence by itself can refer to a universal algebra, or to model theory; naturally there is also a proof adapted to category theory.

Frobenius reciprocity in group representation theory: see induced representation. This example advanced the general theory by about half a century.

Stone-Čech compactification. Let D be the category of Hausdorff compacts and G: D → Top be the forgetting functor that treats every compact Hausdorff space as a topological space. So G has a left adjoint F: Top → D, the Stone-Čech compaction. The unity of this adjoint pair gives the continuous function of each topological space X in its Stone-Čech compaction. This function is an immersion (ie injective, continuous, and open) if and only if X is a Tychonoff space.

Soberification. Stone's article on duality describes an adjunct between the category of topological spaces and the category of sober spaces known as soberification. Notably, the article also contains a detailed description of another adjunction that paves the way for the famous duality of sober spaces and spatial locals, 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 each topological space with its underlying set (that is,, which forgets about topology). the G has a left adjoint F, creating the discrete space in a set Y, and a right adjoint H creating the trivial topology on Y (cf. trivial structure).

Properties

Relation to universal constructions

All pairs of adjoint functors arise from universal constructions. The constructions in the above examples can all be explained with a universal property, and in fact some of the relevant articles do. Universal constructions are more general than pairs of adjoint functors: as mentioned above, a universal construction is like an optimization problem; results in an adjoint pair if and only if this problem has a solution for every D object.

Uniqueness of attachments

If functor F: C → D had two right adjoints G_{1 and G2, then G1 and G2 are naturally isomorphic. The same is true for left adjoints.}

Attachments preserve certain limits

The most important property of adjoints is their continuity: every functor that has a left adjoint (and is therefore a right adjoint) is continuous (i.e. it commutes with limits in the theoretical sense of the category); every functor that has a right adjoint (and is therefore a left adjoint) is cocontinuous (ie it commutes with colimits).

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

• the application of an attachment right to an object product gives the product of the images;
• the application of a left attachment to a co-product of objects gives the co-product of the images;
• each right attachment is right left; * each left attachment is right.

Additivity

If the functor F: C → D is the left adjoint of G: D → C and C and D are additive categories, so F and G are additive functors.

Composition

If functor F₁: C → D has G< sub>1: D → C as right adjoint and functor F₂: D → E has G₂: E → D as right adjunct, then the composition F₂or F₁: C → E has G₁orG₂: E → C as right adjunct.

Characterization via unity and co-unity

The unit η: 1_C → GF and the co-unit ε: FG → 1< sub>D have the following properties: composition (εF)o(Fη), a natural transformation F→FGF→F, is equal to 1_F, and the composition (< i>Gε)o(ηG): G→GFG→G is equal to 1_G. Conversely, given two natural transformations η 1_C → GF and ε: FG → 1_D with these properties, then the functors F and G form an adjoint pair.

Adjoint pairs extend equivalences

Each adjoint pair extends the equivalence of certain subcategories. Specifically, if F: C → D is the left adjoint of G: D > → C with unit η and co-unit ε, define C₁ as complete subcategory of C consisting of those objects X of C for which η_X is an isomorphism, and define D ₁ as the complete 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 D ₁ and give inverse equivalences of these subcategories. In a sense, then, adjuncts are "generalized" inverses. Note however that the right inverse of F (ie a functor G such that FG is naturally isomorphic to 1_D) need not be a right (or left) adjunct to F.

Adjoints generalizes bilateral inverses.

General Existence Theorem

Not every functor G: D → C admits a left adjoint. If D is complete, then functors with left adjoints can be characterized by Freyd's Adjoint Functor Theorem: G has a left adjoint if and only if it is continuous and a certain condition of smallness is satisfied: for each object X of C there exists a family of morphisms f_{< i>i}: X → G(Y_i) (where the indices i come from an array I, not a proper class -- that's the whole point), such that each morphism h: X → G(Y) can be written as h = G( t) or f_i for some i in I and some morphism t: Y_i → Y on D.

An analogous statement characterizes functors with a right adjoint.