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# adjoint functor

Let $\mathcal{C}$ and $\mathcal{D}$ be (small) categories, and let $T:\mathcal{C}\to\mathcal{D}$ and $S:\mathcal{D}\to\mathcal{C}$ be covariant functors. $T$ is said to be a *left adjoint functor* to $S$ (equivalently, $S$ is a *right adjoint functor* to $T$) if there is a natural equivalence

$\nu\colon\Hom_{{\mathcal{D}}}(T(-),-)\overset{\cdot}{\longrightarrow}\Hom_{{% \mathcal{C}}}(-,S(-)).$ |

Here the functor $\Hom_{{\mathcal{D}}}(T(-),-)$ is a bifunctor $\mathcal{C}\times\mathcal{D}\to\mathbf{Set}$ which is contravariant in the first variable, is covariant in the second variable, and sends an object $(C,D)$ to $\Hom_{{\mathcal{D}}}(T(C),D)$. The functor $\Hom_{{\mathcal{C}}}(-,S(-))$ is defined analogously.

This definition needs additional explanation. Essentially, it says that for every object $C$ in $\cal{C}$ and every object $D$ in $\cal{D}$ there is a function

$\nu_{{C,D}}\colon\Hom_{{\mathcal{D}}}(T(C),D)\overset{\sim}{\longrightarrow}% \Hom_{{\mathcal{C}}}(C,S(D))$ |

which is a natural bijection of hom-sets. Naturality means that if $f\colon C^{{\prime}}\to C$ is a morphism in $\mathcal{C}$ and $g\colon D\to D^{{\prime}}$ is a morphism in $\mathcal{D}$, then the diagram

$\xymatrix{\Hom_{{\mathcal{D}}}(T(C),D)\ar[dd]_{{(Tf,g)}}\ar[rr]^{{\nu_{{C,D}}}% }&&\Hom_{{\mathcal{C}}}(C,S(D))\ar[dd]^{{(f,Sg)}}\\ &&\\ \Hom_{{\mathcal{D}}}(T(C^{{\prime}}),D^{{\prime}})\ar[rr]^{{\nu_{{C^{{\prime}}% ,D^{{\prime}}}}}}&&\Hom_{{\mathcal{C}}}(C^{{\prime}},S(D^{{\prime}}))\\ }$ |

is a commutative diagram. If we pick any $h:T(C)\to D$, then we have the equation

$Sg\circ\nu_{{C,D}}(h)\circ f=\nu_{{C^{{\prime}},D^{{\prime}}}}(g\circ h\circ Tf).$ |

If $T:\mathcal{C}\to\mathcal{D}$ is a left adjoint of $S:\mathcal{D}\to\mathcal{C}$, then we say that the ordered pair $(T,S)$ is an *adjoint pair*, and the ordered triple $(T,S,\nu)$ an *adjunction* from $\mathcal{C}$ to $\mathcal{D}$, written

$(T,S,\nu):\mathcal{C}\to\mathcal{D},$ |

where $\nu$ is the natural equivalence defined above.

An adjoint to a functor is in some ways like an inverse (as in the case of an adjoint matrix); often formal properties about a functor lead to formal properties of its adjoint (for example the right adjoint to a left-exact functor takes injectives to injectives). An adjoint to any functor is unique up to natural isomorphism.

Examples:

1. Let $R$ be a commutative ring, and fix an $R$-module $N$. Let

${-\otimes N}\colon{R\!-\!\mathbf{mod}}\to{R\!-\!\mathbf{mod}}$ be the functor

$M\mapsto N\otimes M,$ and let

${\Hom(N,-)}:{R\!-\!\mathbf{mod}}\to{R\!-\!\mathbf{mod}}$ given by

$L\mapsto\mathrm{Hom}_{R}(N,L).$ Then one can show that ${-\otimes N}$ is the left adjoint to ${\Hom(N,-)}$. This pair of adjoint functors is the most commonly used and studied, and astonishingly deep facts spring from this adjoint relationship.

2. Let $U:\mathbf{Top}\to\mathbf{Set}$ be the forgetful functor (i.e. $U$ takes topological spaces to their underlying sets, and continuous maps to set functions). Then $U$ is right adjoint to the functor $F:\mathbf{Set}\to\mathbf{Top}$ which gives each set the discrete topology.

3. If $U:\mathbf{Grp}\to\mathbf{Set}$ is again the forgetful functor, this time on the category of groups, the functor $F:\mathbf{Set}\to\mathbf{Grp}$ which takes a set $A$ to the free group generated by $A$ is left adjoint to $U$.

Remarks on Adjointness:

1. 2. According to William F. Lawvere, Adjointness is closely involved with the Foundation of Mathematics.

3. Adjoint functors define dynamic similarities between general systems in categorical dynamics.

# References

- 1 Daniel M. Kan. Adjoint functors. Transactions of the American Mathematical Society, Vol. 87, No. 2, (1958), 294–329.
- 2
S. Mac Lane,
*Categories for the Working Mathematician*(2nd edition), Springer-Verlag, 1997. - 3
N. Popescu.1975.,
*Abelian Categories with Applications to Rings and Modules.*Academic Press: New York and London.

## Mathematics Subject Classification

18A40*no label found*

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