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| Title of object: |
adjoint functor |
| Canonical Name: |
AdjointFunctor |
| Type: |
Definition |
| Created on: |
2002-02-25 12:04:33 |
| Modified on: |
2008-10-19 16:22:00 |
| Classification: |
msc:18A40 |
| Keywords: |
adjoint functor pairs, adjointness, adjunction, adjoint dynamical systems, natural equivalence, natural isomorphism, adjointness theorems, limit and colimit preserving functors |
| Defines: |
adjoint, adjoint pair, adjunction |
| Synonyms: |
adjoint functor=left adjoint
adjoint functor=right adjoint |
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
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%\usepackage{psfrag}
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\DeclareMathOperator{\Hom}{Hom} |
Content:
\PMlinkescapeword{properties}
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 \emph{left adjoint functor} to $S$ (equivalently, $S$ is a \emph{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 $f\colon C'\to C$ is a morphism in $\mathcal{C}$ and $g\colon D\to D'$ is a morphism in $\mathcal{D}$, then the diagram
\[\xymatrix{
\Hom_{\mathcal{D}}(T(C),D)\ar[dd]_{(Tf,g)}\ar[rr]^{\nu_{C,D}} &&
\Hom_{\mathcal{C}}(C,S(D))\ar[dd]^{(f,Sg)} \\ && \\
\Hom_{\mathcal{D}}(T(C'),D')\ar[rr]^{\nu_{C',D'}} &&
\Hom_{\mathcal{C}}(C',S(D')) \\
}\]
is a commutative diagram. If we pick any $h:T(C)\to D$, then we have the equation $$Sg\circ \nu_{C,D}(h)\circ f= \nu_{C',D'}(g\circ h\circ Tf).$$
%
%
% I'm commenting the following out because I moved the mention of the
% natural transformation before the mention of the naturality of the
% bijection. This allows me to give the functors F_1 and F_2 the more
% natural names Hom_D(T(-),-) and Hom_C(-,S(-)).
%
%
%The word ``natural'' in this definition needs some explanation. We can construct a functor
%\begin{align*}
%\mathcal{F}_1\colon\mathcal{C}\times\mathcal{D}&\to\mathbf{Set} \\
% (C,D)&\mapsto \Hom_{\mathcal{D}}(T(C),D)
%\end{align*}
%and a second functor
%\begin{align*}
%\mathcal{F}_2\colon\mathcal{C}\times\mathcal{D}&\to\mathbf{Set} \\
% (C,D)&\mapsto \Hom_{\mathcal{C}}(C,S(D)).
%\end{align*}
%Then the family of bijections $\nu_{C,D}$ should form a natural transformation from $\mathcal{F}_1$ to $\mathcal{F}_2$.
%
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 \emph{adjoint pair}, and the ordered triple $(T,S,\nu)$ an \emph{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 \PMlinkescapetext{injectives} to \PMlinkescapetext{injectives}). An adjoint to any functor is unique up to natural isomorphism.
\textbf{Examples}:
\begin{enumerate}
\item 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.
\item 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.
\item 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 $U$.
\end{enumerate}
\textbf{Remarks on Adjointness Theorems:}
\begin{enumerate}
\item There are several theorems that link limit and colimit preserving properties
of functors to adjointness (e.g., ref. \cite{NP75}).
\item According to William F. Lawvere, Adjointness is closely involved with the Foundation of Mathematics.
\item Adjoint functors define dynamic similarities between general systems in categorical dynamics.
\end{enumerate}
\begin{thebibliography}{9}
\bibitem{K}
Daniel~M.~Kan. Adjoint functors. {\it Transactions of the American Mathematical Society}, Vol. 87, No. 2, (1958), 294--329.
\bibitem{Ma}
S. Mac Lane, \emph{Categories for the Working Mathematician} (2nd edition), Springer-Verlag, 1997.
\bibitem{NP75}
N. Popescu.1975., \emph{Abelian Categories with Applications to Rings and Modules.}
Academic Press: New York and London.
\end{thebibliography} |
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