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adjoint functor (Definition)

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

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

This definition needs a bit of explanation. Essentially, it says that for every object $ C$ in $ \cal{C}$ and every object $ D$ 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 $ (T,S)$ 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:

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

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.



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"adjoint functor" is owned by mps. [ full author list (6) | owner history (5) ]
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See Also: forgetful functor, universal mapping property

Other names:  left adjoint, right adjoint
Also defines:  adjoint, adjoint pair, adjunction

Attachments:
unit of adjunction (Definition) by CWoo
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Cross-references: generated by, free group, groups, discrete topology, set functions, continuous maps, topological spaces, forgetful functor, fix, commutative ring, natural isomorphism, adjoint matrix, inverse, ordered pair, equation, commutative diagram, morphism, bijection, function, object, variable, bifunctor, natural equivalence, covariant functors, categories
There are 25 references to this entry.

This is version 20 of adjoint functor, born on 2002-02-25, modified 2007-07-29.
Object id is 2691, canonical name is AdjointFunctor.
Accessed 12495 times total.

Classification:
AMS MSC: 18A40 (Category theory; homological algebra :: General theory of categories and functors :: Adjoint functors )
Pending Errata and Addenda
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