PlanetMath: unit of adjunction

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

As $1_{T(C)}$ is the identity morphism in $\hom_{\mathcal{D}}(T(C),T(C))$ , define

Proof. Let

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

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ & C \ar[dr]^{f} \ar[dl]_{\eta_C} & \ ST(C) \ar[rr]_{S(g)} && S(Y) } } \end{xy}$

is a commutative diagram. The existence and uniqueness of

is guaranteed by the bijection

$\displaystyle \nu_{C,Y}:\hom_{\mathcal{D}}(T(C),Y) \longrightarrow \hom_{\mathcal{C}}(C,S(Y)),$

where $f=\nu_{C,Y}(g)$ , and the commutativity of the triangle above is guaranteed by the naturality in the second variable

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ \hom_{\mathcal{D}}(T(C),T(C)) \a... ...cal{D}}(T(C),Y) \ar[rr]_{\nu_{C,Y}} && \hom_{\mathcal{C}}(C,S(Y)), } } \end{xy}$

where $\hat{g}:=\hom_{\mathcal{D}}(1_{T(C)},g)$ and $\overline{g}:=\hom_{\mathcal{C}}(1_C,S(g))$ , as

$\displaystyle \overline{g}\circ \nu_C(1_{T(C)})=\hom_{\mathcal{C}}(C,S(g))\circ \eta_C=S(g)\circ \eta_C$

on the one hand, and

$\displaystyle \nu_{C,Y}\circ \hat{g}(1_{T(C)})=\nu_{C,Y}\circ \hom(T(C),g)(1_{T(C)})=\nu_{C,Y}(g\circ 1_{T(C)})=\nu_{C,Y}(g)=f$

on the other, and the two are equal. $\qedsymbol$

Proof. Suppose $f:A\to B$ is a morphism in $\mathcal{C}$ . We want to show that

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ A \ar[d]_{\eta_A} \ar[rr]^f && B \ar[d]^{\mu_B} \ ST(A) \ar[rr]_{ST(f)} && ST(B) } } \end{xy}$

is commutative. To see this, write out the expressions

$\displaystyle \eta_B\circ f$	$\displaystyle = 1_{ST(B)}\circ \eta_B\circ f$	property of identity morphism
	$\displaystyle = S(1_{T(B)})\circ \eta_B\circ f$	property of functor on identity morphism
	$\displaystyle = \nu_A(1_{T(B)}\circ T(f))$	by equation (3) above
	$\displaystyle = \nu_A(T(f)\circ 1_{T(A)})$	$\displaystyle T(f)$ commutes with identity morphisms
	$\displaystyle = \nu_A(T(f)\circ T(1_A))$	property of functor on identity morphism
	$\displaystyle = ST(f)\circ \eta_A\circ 1_A$	by equation (3) above
	$\displaystyle = ST(f)\circ \eta_A$	property of identity morphisms $\displaystyle .$

$\qedsymbol$

Definition. The natural transformation $\eta:I_{\mathcal{C}}\dot{\to} ST$ defined above is called the unit of the adjunction $(T,S,\nu)$ from $\mathcal{C}$ to $\mathcal{D}$ .

Dually, one can find a natural transformation $\epsilon:TS \dot{\to} I_{\mathcal{D}}$ called the counit of the adjunction $(T,S,\nu):\mathcal{C}\to\mathcal{D}$ . To do this, set

and use equation (1) to get a bijection $\nu_D:=\nu_{S(D),D}$ and subsequently define

Bibliography