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Version 18 |
| \PMlinkescapeword{properties} |
\PMlinkescapeword{properties} |
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| 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 |
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 |
| \[ |
\[ |
| \nu\colon \Hom_{\mathcal{D}}(T(-),-) \overset{\cdot}{\longrightarrow} \Hom_{\mathcal{C}}(-,S(-)). |
\nu\colon \Hom_{\mathcal{D}}(T(-),-) \overset{\cdot}{\longrightarrow} \Hom_{\mathcal{C}}(-,S(-)). |
| \] |
\] |
| 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. |
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. |
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| 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 |
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 |
| \[ |
\[ |
| \nu_{C,D} \colon \Hom_{\mathcal{D}}(T(C),D) \overset{\sim}{\longrightarrow} \Hom_{\mathcal{C}}(C,S(D)) |
\nu_{C,D} \colon \Hom_{\mathcal{D}}(T(C),D) \overset{\sim}{\longrightarrow} \Hom_{\mathcal{C}}(C,S(D)) |
| \] |
\] |
| 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 |
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{ |
\[\xymatrix{ |
| \Hom_{\mathcal{D}}(T(C),D)\ar[dd]_{(Tf,g)}\ar[rr]^{\nu_{C,D}} && |
\Hom_{\mathcal{D}}(T(C),D)\ar[dd]_{(Tf,g)}\ar[rr]^{\nu_{C,D}} && |
| \Hom_{\mathcal{C}}(C,S(D))\ar[dd]^{(f,Sg)} \\ && \\ |
\Hom_{\mathcal{C}}(C,S(D))\ar[dd]^{(f,Sg)} \\ && \\ |
| \Hom_{\mathcal{D}}(T(C'),D')\ar[rr]^{\nu_{C',D'}} && |
\Hom_{\mathcal{D}}(T(C'),D')\ar[rr]^{\nu_{C',D'}} && |
| \Hom_{\mathcal{C}}(C',S(D')) \\ |
\Hom_{\mathcal{C}}(C',S(D')) \\ |
| }\] |
}\] |
| is a commutative diagram. If we pick any $h:T(C)\to D$, then we have the equation $$Sg\circ \nu_{C,D}(h)\circ f= \nu_{C',D'}(g\circ h\circ Tf).$$ |
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 |
% I'm commenting the following out because I moved the mention of the |
| % natural transformation before the mention of the naturality 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 |
% bijection. This allows me to give the functors F_1 and F_2 the more |
| % natural names Hom_D(T(-),-) and Hom_C(-,S(-)). |
% natural names Hom_D(T(-),-) and Hom_C(-,S(-)). |
| % |
% |
| % |
% |
| %The word ``natural'' in this definition needs some explanation. We can construct a functor |
%The word ``natural'' in this definition needs some explanation. We can construct a functor |
| %\begin{align*} |
%\begin{align*} |
| %\mathcal{F}_1\colon\mathcal{C}\times\mathcal{D}&\to\mathbf{Set} \\ |
%\mathcal{F}_1\colon\mathcal{C}\times\mathcal{D}&\to\mathbf{Set} \\ |
| % (C,D)&\mapsto \Hom_{\mathcal{D}}(T(C),D) |
% (C,D)&\mapsto \Hom_{\mathcal{D}}(T(C),D) |
| %\end{align*} |
%\end{align*} |
| %and a second functor |
%and a second functor |
| %\begin{align*} |
%\begin{align*} |
| %\mathcal{F}_2\colon\mathcal{C}\times\mathcal{D}&\to\mathbf{Set} \\ |
%\mathcal{F}_2\colon\mathcal{C}\times\mathcal{D}&\to\mathbf{Set} \\ |
| % (C,D)&\mapsto \Hom_{\mathcal{C}}(C,S(D)). |
% (C,D)&\mapsto \Hom_{\mathcal{C}}(C,S(D)). |
| %\end{align*} |
%\end{align*} |
| %Then the family of bijections $\nu_{C,D}$ should form a natural transformation from $\mathcal{F}_1$ to $\mathcal{F}_2$. |
%Then the family of bijections $\nu_{C,D}$ should form a natural transformation from $\mathcal{F}_1$ to $\mathcal{F}_2$. |
| % |
% |
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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.
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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 $\mathcal{D}$, where $\nu$ is the natural equivalence defined above.
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| 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. |
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. |
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| \textbf{Examples}: |
\textbf{Examples}: |
| \begin{enumerate} |
\begin{enumerate} |
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| \item Let $R$ be a commutative ring, and fix an $R$-module $N$. Let |
\item Let $R$ be a commutative ring, and fix an $R$-module $N$. Let |
| \[ |
\[ |
| {-\otimes N}\colon {R\! -\!\mathbf{mod}}\to {R\! -\!\mathbf{mod}} |
{-\otimes N}\colon {R\! -\!\mathbf{mod}}\to {R\! -\!\mathbf{mod}} |
| \] |
\] |
| be the functor |
be the functor |
| \[ |
\[ |
| M\mapsto N\otimes M, |
M\mapsto N\otimes M, |
| \] |
\] |
| and let |
and let |
| \[ |
\[ |
| {\Hom(N,-)}:{R\! -\!\mathbf{mod}}\to {R\! -\!\mathbf{mod}} |
{\Hom(N,-)}:{R\! -\!\mathbf{mod}}\to {R\! -\!\mathbf{mod}} |
| \] |
\] |
| given by |
given by |
| \[ |
\[ |
| L\mapsto\mathrm{Hom}_R(N,L). |
L\mapsto\mathrm{Hom}_R(N,L). |
| \] |
\] |
| 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. |
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. |
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| \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 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. |
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| \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 $U$. |
\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} |
\end{enumerate} |
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| \begin{thebibliography}{99} |
\begin{thebibliography}{99} |
| \bibitem{K} |
\bibitem{K} |
| Daniel~M.~Kan. Adjoint functors. {\it Transactions of the American Mathematical Society}, Vol. 87, No. 2, (1958), 294--329. |
Daniel~M.~Kan. Adjoint functors. {\it Transactions of the American Mathematical Society}, Vol. 87, No. 2, (1958), 294--329. |
| \end{thebibliography} |
\end{thebibliography} |
