Adjoint functors

In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certain universal property), such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology.

For the construction in field theory, see Adjunction (field theory). For the construction in topology, see Adjunction space.

By definition, an adjunction between categories ${\mathcal {C}}$ and ${\mathcal {D}}$ is a pair of functors (assumed to be covariant)

and, for all objects $X$ in ${\mathcal {C}}$ and $Y$ in ${\mathcal {D}}$ , a bijection between the respective morphism sets

such that this family of bijections is natural in $X$ and $Y$ . Naturality here means that there are natural isomorphisms between the pair of functors ${\mathcal {C}}(F-,X):{\mathcal {D}}\to \mathrm {Set^{\text{op}}}$ and ${\mathcal {D}}(-,GX):{\mathcal {D}}\to \mathrm {Set^{\text{op}}}$ for a fixed $X$ in ${\mathcal {C}}$ , and also the pair of functors ${\mathcal {C}}(FY,-):{\mathcal {C}}\to \mathrm {Set}$ and ${\mathcal {D}}(Y,G-):{\mathcal {C}}\to \mathrm {Set}$ for a fixed $Y$ in ${\mathcal {D}}$ .

The functor $F$ is called a left adjoint functor or left adjoint to $G$ , while $G$ is called a right adjoint functor or right adjoint to $F$ . We write $F\dashv G$ .

An adjunction between categories ${\mathcal {C}}$ and ${\mathcal {D}}$ is somewhat akin to a "weak form" of an equivalence between ${\mathcal {C}}$ and ${\mathcal {D}}$ , and indeed every equivalence is an adjunction. In many situations, an adjunction can be "upgraded" to an equivalence, by a suitable natural modification of the involved categories and functors.

The definitions via universal morphisms are easy to state, and require minimal verifications when constructing an adjoint functor or proving two functors are adjoint. They are also the most analogous to our intuition involving optimizations.

The definition via hom-sets makes symmetry the most apparent, and is the reason for using the word adjoint.

The definition via counit–unit adjunction is convenient for proofs about functors which are known to be adjoint, because they provide formulas that can be directly manipulated.

Examples[edit]

Free groups[edit]

The construction of free groups is a common and illuminating example.

Let F : Set → Grp be the functor assigning to each set Y the free group generated by the elements of Y, and let G : Grp → Set be the forgetful functor, which assigns to each group X its underlying set. Then F is left adjoint to G:

Initial morphisms. For each set Y, the set GFY is just the underlying set of the free group FY generated by Y. Let $\eta _{Y}:Y\to GFY$ be the set map given by "inclusion of generators". This is an initial morphism from Y to G, because any set map from Y to the underlying set GW of some group W will factor through $\eta _{Y}:Y\to GFY$ via a unique group homomorphism from FY to W. This is precisely the universal property of the free group on Y.

Terminal morphisms. For each group X, the group FGX is the free group generated freely by GX, the elements of X. Let $\varepsilon _{X}:FGX\to X$ be the group homomorphism which sends the generators of FGX to the elements of X they correspond to, which exists by the universal property of free groups. Then each $(GX,\varepsilon _{X})$ is a terminal morphism from F to X, because any group homomorphism from a free group FZ to X will factor through $\varepsilon _{X}:FGX\to X$ via a unique set map from Z to GX. This means that (F,G) is an adjoint pair.

Hom-set adjunction. Group homomorphisms from the free group FY to a group X correspond precisely to maps from the set Y to the set GX: each homomorphism from FY to X is fully determined by its action on generators, another restatement of the universal property of free groups. One can verify directly that this correspondence is a natural transformation, which means it is a hom-set adjunction for the pair (F,G).

counit–unit adjunction. One can also verify directly that ε and η are natural. Then, a direct verification that they form a counit–unit adjunction $(\varepsilon ,\eta ):F\dashv G$ is as follows:

The first counit–unit equation $1_{F}=\varepsilon F\circ F\eta$ says that for each set Y the composition

A F : D → C called the left adjoint

functor

A functor G : C → D called the right adjoint

A Φ : hom_C(F–,–) → hom_D(–,G–)

natural isomorphism

A ε : FG → 1_C called the counit

natural transformation

A natural transformation η : 1_D → GF called the unit

F has a right adjoint if and only if F preserves small colimits

F has a left adjoint if and only if F preserves small limits and is an

accessible functor

Relationships[edit]

Universal constructions[edit]

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 : C → D from every object of D, 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 D (equivalently, every object of C).

Equivalences of categories[edit]

If a functor F : D → C is one half of an equivalence of categories then it is the left adjoint in an adjoint equivalence of categories, i.e. an adjunction whose unit and counit are isomorphisms.

Every adjunction 〈F, G, ε, η〉 extends an equivalence of certain subcategories. Define C₁ as the full subcategory of C consisting of those objects X of C for which ε_X is an isomorphism, and define D₁ as the full subcategory of D consisting of those objects Y of D for which η_Y is an isomorphism. Then F and G can be restricted to D₁ and C₁ 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[edit]

Every adjunction 〈F, G, ε, η〉 gives rise to an associated monad 〈T, η, μ〉 in the category D. The functor

is given by T = GF. The unit of the monad

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

is given by μ = GεF. Dually, the triple 〈FG, ε, FηG〉 defines a comonad in C.

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.

Adámek, Jiří; Herrlich, Horst; Strecker, George E. (1990). (PDF). John Wiley & Sons. ISBN 0-471-60922-6. Zbl 0695.18001.

Abstract and Concrete Categories. The joy of cats

(1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5 (2nd ed.). Springer-Verlag. ISBN 0-387-98403-8. Zbl 0906.18001.

Mac Lane, Saunders

playlist on YouTube – seven short lectures on adjunctions by Eugenia Cheng of The Catsters

Adjunctions

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