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Adjoint functors

Suppose we have two categories $\sA$ and $\sB$ and a pair of functors $F\:\sA\to \sB$ , $G\: \sB\to\sA$ . We say that

is right adjoint to

and

is left adjoint to

if there is an isomorphism of sets

$\begin{displaymath} \Hom_{\sA}(G(B), A)\isom \Hom_{\sB}(B, F(A)) \end{displaymath}$

for every $A\in\Ob(\sA)$ , $B\in\Ob(\sB)$ , and the isomorphism is functorial in

and

.
$\begin{example} Let $\sA, \sB =\Sets$. Take a set $X$\ and define \begin{display... ...\varphi\:B\times X\to A \mapsto (B\to \Hom(X, A)) \end{displaymath}\end{example}$
Many examples follow the pattern in (

), but with additional structure.
$\begin{example} Let $\sA, \sB=\Vect$, $V$\ vector space over a field $k$. Define... ...} \Hom_k(B\tensor_k V, A)=\Hom_k(B, \Hom_k(V, A)) \end{displaymath}\end{example}$

$\begin{example} Let $R$\ be a ring, $\sA$\ be the category of left $R$-modules, ... ...om_{\bZ}(B\ox_R V, A)=\Hom_{\bZ}(B, \Hom_R(V, A)) \end{displaymath}\end{example}$

$\begin{example} Define the loop space and the suspension of a topological space ... ...re $\Topp$\ is the category of topological spaces with base point. \end{example}$

Pawel Witkowski 2006-11-07