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.

By definition, an adjunction between categories

C

{\displaystyle {\mathcal {C}}}

and

D

{\displaystyle {\mathcal {D}}}

is a pair of functors (assumed to be covariant)

F

:

D

→

C

G

:

C

→

D

{\displaystyle {\begin{aligned}F&:{\mathcal {D}}\rightarrow {\mathcal {C}}\\G&:{\mathcal {C}}\rightarrow {\mathcal {D}}\end{aligned}}}

and, for all objects

c

{\displaystyle c}

in

C

{\displaystyle {\mathcal {C}}}

and

d

{\displaystyle d}

in

D

{\displaystyle {\mathcal {D}}}

, a bijection between the respective morphism sets

h

o

m

C

(

F

d

,

c

)

≅

h

o

m

D

(

d

,

G

c

)

{\displaystyle \mathrm {hom} _{\mathcal {C}}(Fd,c)\cong \mathrm {hom} _{\mathcal {D}}(d,Gc)}

such that this family of bijections is natural in

c

{\displaystyle c}

and

d

{\displaystyle d}

. For locally small categories, naturality here means that there are natural isomorphisms between the pair of functors

C

(

F

−

,

c

)

:

D

→

S

e

t

op

{\displaystyle {\mathcal {C}}(F-,c):{\mathcal {D}}\to \mathrm {Set^{\text{op}}} }

and

D

(

−

,

G

c

)

:

D

→

S

e

t

op

{\displaystyle {\mathcal {D}}(-,Gc):{\mathcal {D}}\to \mathrm {Set^{\text{op}}} }

for a fixed

c

{\displaystyle c}

in

C

{\displaystyle {\mathcal {C}}}

, and also the pair of functors

C

(

F

d

,

−

)

:

C

→

S

e

t

{\displaystyle {\mathcal {C}}(Fd,-):{\mathcal {C}}\to \mathrm {Set} }

and

D

(

d

,

G

−

)

:

C

→

S

e

t

{\displaystyle {\mathcal {D}}(d,G-):{\mathcal {C}}\to \mathrm {Set} }

for a fixed

d

{\displaystyle d}

in

D

{\displaystyle {\mathcal {D}}}

. For other categories, naturality is defined as a generalisation of this.

The functor

F

{\displaystyle F}

is called a left adjoint functor or left adjoint to

G

{\displaystyle G}

, while

G

{\displaystyle G}

is called a right adjoint functor or right adjoint to

F

{\displaystyle F}

. We write

F

⊣

G

{\displaystyle F\dashv G}

.

An adjunction between categories

C

{\displaystyle {\mathcal {C}}}

and

D

{\displaystyle {\mathcal {D}}}

is somewhat akin to a "weak form" of an equivalence between

C

{\displaystyle {\mathcal {C}}}

and

D

{\displaystyle {\mathcal {D}}}

, and indeed every equivalence gives an adjunction, though the equivalence itself is not necessarily an adjunction. In many situations, an adjunction can be "upgraded" to an equivalence, by a suitable natural modification of the involved categories and functors.