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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 additional 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$.

Remarks on Adjointness:

  1. 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.
  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.

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.



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See Also: forgetful functor, universal mapping property, similarity and analogous systems: dynamic adjointness and topological equivalence, Galois connection, section functor, natural equivalence, foundations of mathematics: overview

Other names:  left adjoint, right adjoint
Also defines:  adjoint, adjoint pair, adjunction
Keywords:  adjoint functor pairs, adjointness, adjunction, adjoint dynamical systems, natural equivalence, natural isomorphism, adjointness theorems, limit and colimit preserving functors

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unit of adjunction (Definition) by CWoo
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Cross-references: categorical dynamics, general systems, Foundation of Mathematics, preserves, colimit, limit, link, generated by, free group, groups, discrete topology, set functions, continuous maps, topological spaces, forgetful functor, fix, commutative ring, adjoint matrix, inverse, ordered pair, equation, commutative diagram, diagram, morphism, bijection, function, object, variable, bifunctor, natural equivalence, covariant functors, categories
There are 35 references to this entry.

This is version 34 of adjoint functor, born on 2002-02-25, modified 2008-12-19.
Object id is 2691, canonical name is AdjointFunctor.
Accessed 13927 times total.

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