PlanetMath: pullback

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pullback

(Definition)

Definition Suppose are sets, and we have maps

$\displaystyle f\colon Y$	$\displaystyle \to$	$\displaystyle Z,$
$\displaystyle \Phi\colon X$	$\displaystyle \to$	$\displaystyle Y.$

Then the pullback of

under $\Phi$ is the mapping

$\displaystyle \Phi^\ast f\colon X$	$\displaystyle \to$	$\displaystyle Z,$
$\displaystyle x$	$\displaystyle \mapsto$	$\displaystyle (f\circ\Phi)(x).$

Let us denote by the set of all mappings $f\colon X\to Y$ . We then see that $\Phi^\ast$ is a mapping $M(Y,Z)\to M(X,Z)$ . In other words, $\Phi^\ast$ pulls back the set where is defined on from to . This is illustrated in the below diagram.

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ X \ar[r]^\Phi\ar[dr]_{\Phi^\ast f} & Y \ar[d]_{f} \ & Z } } \end{xy}$

Properties

For any set , $(\operatorname{id}_X)^\ast = \operatorname{id}_{M(X,X)}$ .
Suppose we have maps

$\displaystyle \Phi\colon X$ $\displaystyle \to$ $\displaystyle Y,$

$\displaystyle \Psi\colon Y$ $\displaystyle \to$ $\displaystyle Z$

between sets . Then
$\displaystyle (\Psi\circ \Phi)^\ast = \Phi^\ast \circ \Psi^\ast.$
If $\Phi\colon X\to Y$ is a bijection, then $\Phi^\ast$ is a bijection and
$\displaystyle \big(\Phi^\ast\big)^{-1} = \big(\Phi^{-1}\big)^\ast.$
Suppose are sets with $X\subset Y$ . Then we have the inclusion map $\iota:X\hookrightarrow Y$ , and for any $f\colon Y\to Z$ , we have
$\displaystyle \iota^\ast f = f\vert _X,$
where $f\vert _X$ is the restriction of to .

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"pullback" is owned by matte.

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Attachments:: $\Psi$ is surjective if and only if $\Psi^\ast$ is injective (Theorem) by matte

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Cross-references: inclusion map, bijection, maps
There are 16 references to this entry.

This is version 11 of pullback, born on 2003-08-08, modified 2005-10-26.
Object id is 4569, canonical name is Pullback2.
Accessed 3602 times total.

Classification:

AMS MSC:

03-00 (Mathematical logic and foundations :: General reference works )

Pending Errata and Addenda

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