Back to the index. Or to the chambers

This article has 28 links. View as Cloud or List.

1	Category
1	Functor
1	Commutative Diagram
1	Topological Space
1	Continuous
1	Variable
1	Function
1	Fix
1	Bijection
1	Ordered Pair
1	Commutative Ring
1	Generating Set Of A Group
1	Group
1	Equation
2	Direct Limit
2	Knot Theory
2	Discrete
2	Preservation And Reflection
2	Minimal And Maximal Number
3	Matrix Inverse
3	Free Group
4	Natural Equivalence
4	F O U N D A T I O N S O F M A T H E M A T I C S O V E R V I E W
5	Conjugate Transpose
7	Concrete Category
8	Multifunctor
8	System Definitions
10	Categorical Dynamics

Loading ...

Planetmath Browser (2008—2009)

BSD licence | A django site

All articles taken from PlanetMath.org snapshot under CC-BY-SA licence.

→ The original article on PlanetMath.org

Other Formats: LaTeX

New! You can click on formulas to copy the LaTeX source to your clipboard. | Math Videos

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. is said to be a left adjoint functor to (equivalently, is a right adjoint functor to ) if there is a natural equivalence

$\displaystyle \nu\colon {\mathrm{Hom}}_{\mathcal{D}}(T(-),-) \overset{\cdot}{\longrightarrow} {\mathrm{Hom}}_{\mathcal{C}}(-,S(-)).$

Here the functor ${\mathrm{Hom}}_{\mathcal{D}}(T(-),-)$ 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

to ${\mathrm{Hom}}_{\mathcal{D}}(T(C),D)$ . The functor ${\mathrm{Hom}}_{\mathcal{C}}(-,S(-))$ is defined analogously.

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

$\displaystyle \nu_{C,D} \colon {\mathrm{Hom}}_{\mathcal{D}}(T(C),D) \overset{\sim}{\longrightarrow} {\mathrm{Hom}}_{\mathcal{C}}(C,S(D))$

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

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ {\mathrm{Hom}}_{\mathcal{D}}(T(C... ...ar[rr]^{\nu_{C',D'}} && {\mathrm{Hom}}_{\mathcal{C}}(C',S(D')) \\ } } \end{xy}$

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

$\displaystyle Sg\circ \nu_{C,D}(h)\circ f= \nu_{C',D'}(g\circ h\circ Tf).$

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 is an adjoint pair, and the ordered triple $(T,S,\nu)$ an adjunction from $\mathcal{C}$ to $\mathcal{D}$ , written

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

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:

Let be a commutative ring, and fix an -module . Let
$\displaystyle {-\otimes N}\colon {R\! -\!\mathbf{mod}}\to {R\! -\!\mathbf{mod}}$
be the functor
$\displaystyle M\mapsto N\otimes M,$
and let
$\displaystyle {{\mathrm{Hom}}(N,-)}:{R\! -\!\mathbf{mod}}\to {R\! -\!\mathbf{mod}}$
given by
$\displaystyle L\mapsto\mathrm{Hom}_R(N,L).$
Then one can show that ${-\otimes N}$ is the left adjoint to ${{\mathrm{Hom}}(N,-)}$ . This pair of adjoint functors is the most commonly used and studied, and astonishingly deep facts spring from this adjoint relationship.
Let $U:\mathbf{Top}\to \mathbf{Set}$ be the forgetful functor (i.e. takes topological spaces to their underlying sets, and continuous maps to set functions). Then is right adjoint to the functor $F:\mathbf{Set} \to \mathbf{Top}$ which gives each set the discrete topology.
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 to the free group generated by is left adjoint to .

Remarks on Adjointness:

There are several theorems that link limit and colimit preserving properties of functors to adjointness (e.g., ref. [3]). Thus, a left adjoint functor preserves colimits or acts naturally on the colimit functor (if the latter exists); dually, a right adjoint preserves limits.
According to William F. Lawvere, Adjointness is closely involved with the Foundation of Mathematics.
Adjoint functors define dynamic similarities between general systems in categorical dynamics.

Bibliography

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.