| Version 3 |
Version 2 |
| Let $\mathcal{C},\mathcal{D}$ be categories and $(T,S,\nu)$ be an adjunction from $\mathcal{C}$ to $\mathcal{D}$. For every pair of objects $C\in\mathcal{C}$ and $D\in\mathcal{D}$, we have a bijection |
Let $\mathcal{C},\mathcal{D}$ be categories and $(T,S,\nu)$ be an adjunction from $\mathcal{C}$ to $\mathcal{D}$. For every pair of objects $C\in\mathcal{C}$ and $D\in\mathcal{D}$, we have a bijection |
| $$\nu_{C,D}:\hom_{\mathcal{D}}(T(C),D) \longrightarrow \hom_{\mathcal{C}}(C,S(D))$$ |
$$\nu_{C,D}:\hom_{\mathcal{D}}(T(C),D) \longrightarrow \hom_{\mathcal{C}}(C,S(D))$$ |
| that is natural in each variable. |
that is natural in each variable. |
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| If we set $D=T(C)$, and write $\nu_C$ for $\nu_{C,T(C)}$, then we get a bijection |
If we set $D=T(C)$, and write $\nu_C$ for $\nu_{C,T(C)}$, then we get a bijection |
| $$\nu_C:\hom_{\mathcal{D}}(T(C),T(C)) \longrightarrow \hom_{\mathcal{C}}(C,ST(C))$$ where $ST$ is the abbreviation of $S\circ T$. |
$$\nu_C:\hom_{\mathcal{D}}(T(C),T(C)) \longrightarrow \hom_{\mathcal{C}}(C,ST(C))$$ where $ST$ is the abbreviation of $S\circ T$. |
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| As $1_{T(C)}$ is the identity morphism in $\hom_{\mathcal{D}}(T(C),T(C))$, define $$\mu_C:=\nu_C(1_{T(C)}).$$ Note that $\mu_C$ is a morphism in $\mathcal{C}$ from $C$ to $ST(C)$. |
As $1_{T(C)}$ is the identity morphism in $\hom_{\mathcal{D}}(T(C),T(C))$, define $$\mu_C:=\nu_C(1_{T(C)}).$$ Note that $\mu_C$ is a morphism in $\mathcal{C}$ from $C$ to $ST(C)$. |
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| \begin{thm} $(T(C),\mu_C)$ is a universal arrow from $C$ to $S$. \end{thm} |
\begin{thm} $(T(C),\mu_C)$ is a universal arrow from $C$ to $S$. \end{thm} |
| \begin{proof} Let $Y$ be an object in $\mathcal{D}$ and $f:C\to S(Y)$ a morphism in $\mathcal{C}$. We want to find a morphism $g:T(C)\to Y$ in $\mathcal{D}$ such that |
\begin{proof} Let $Y$ be an object in $\mathcal{D}$ and $f:C\to S(Y)$ a morphism in $\mathcal{C}$. We want to find a morphism $g:T(C)\to Y$ in $\mathcal{D}$ such that |
| $$ |
$$ |
| \xymatrix{ |
\xymatrix{ |
| & C \ar[dr]^{f} \ar[dl]_{\mu_C} & \\ |
& C \ar[dr]^{f} \ar[dl]_{\mu_C} & \\ |
| ST(C) \ar[rr]_{S(g)} && S(Y) }$$ |
ST(C) \ar[rr]_{S(g)} && S(Y) }$$ |
| is a commutative diagram. The existence and uniqueness of $g$ is guaranteed by the bijection |
is a commutative diagram. The existence and uniqueness of $g$ is guaranteed by the bijection |
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$$\nu_{C,Y}:\hom_{\mathcal{D}}(T(C),Y) \longrightarrow \hom_{\mathcal{C}}(C,S(Y)),$$ where $f=\nu_{C,Y}(g)$, and the commutativity of the triangle above is guaranteed by the naturality in the second variable
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$$\nu_{C,Y}:\hom_{\mathcal{D}}(T(C),Y) \longrightarrow \hom_{\mathcal{C}}(C,S(Y)),$$ and the commutativity of the triangle above is guaranteed by the naturality in the second variable
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| $$ |
$$ |
| \xymatrix{ |
\xymatrix{ |
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\hom_{\mathcal{D}}(T(C),T(C)) \ar[d]_{\hom(T(C),g)} \ar[r]^{\nu_C} & \hom_{\mathcal{C}}(C,ST(C)) \ar[d]^{\hom(C,S(g))} \\
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\hom_{\mathcal{D}}(T(C),T(C)) \ar[d]_g \ar[r]^{\nu_C} & \hom_{\mathcal{C}}(C,ST(C)) \ar[d]^{S(g)} \\
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| \hom_{\mathcal{D}}(T(C),Y) \ar[r]_{\nu_{C,Y}} & \hom_{\mathcal{C}}(C,S(Y)), }$$ |
\hom_{\mathcal{D}}(T(C),Y) \ar[r]_{\nu_{C,Y}} & \hom_{\mathcal{C}}(C,S(Y)), }$$ |
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as $$\hom(C,S(g))\circ \nu_C(1_{T(C)})=\hom(C,S(g))\circ \mu_C=S(g)\circ \mu_C$$ on the one hand, and $$\nu_{C,Y}\circ \hom(T(C),g)(1_{T(C)})=\nu_{C,Y}(g\circ 1_{T(C)})=\nu_{C,Y}(g)=f$$ on the other, and the two are equal.
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as $S(g)\circ \nu_C(1_{T(C)})=S(g)\circ \mu_C$ on the one hand, and $\nu_{C,Y}\circ g(1_{T(C)})=\nu_{C,Y}(g)=f$ on the other, and the two are equal.
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| \end{proof} |
\end{proof} |
| \begin{thm} $\mu: C \mapsto \mu_C$ is a natural transformation from the identity functor $I_{\mathcal{C}}$ to $ST$. \end{thm} |
\begin{thm} $\mu: C \mapsto \mu_C$ is a natural transformation from the identity functor $I_{\mathcal{C}}$ to $ST$. \end{thm} |
| \begin{proof} |
\begin{proof} |
| \end{proof} |
\end{proof} |
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| more to come... |
more to come... |
