pullback

Definition Suppose $X, Y, Z$ 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 $f$ 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 $M(X,Y)$ 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 $f$ is defined on from $Y$ to $X$ . This is illustrated in the below diagram.

$\xymatrix{X\ar[r]^{\Phi}\ar[dr]_{\Phi^{\ast}f}&Y\ar[d]_{f}\\ &Z}$

0.0.1 Properties

1.

For any set $X$ , $(\operatorname{id}_{X})^{\ast}=\operatorname{id}_{M(X,X)}$ .

Suppose we have maps

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

between sets $X, Y, Z$ . Then

$(\Psi\circ\Phi)^{\ast}=\Phi^{\ast}\circ\Psi^{\ast}.$

If $\Phi\colon X\to Y$ is a bijection, then $\Phi^{\ast}$ is a bijection and

$\big{(}\Phi^{\ast}\big{)}^{-1}=\big{(}\Phi^{-1}\big{)}^{\ast}.$

Suppose $X, Y$ 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

$\iota^{\ast}f=f|_{X},$

where $f|_{X}$ is the restriction (http://planetmath.org/RestrictionOfAFunction) of $f$ to $X$ .

Title	pullback
Canonical name	Pullback
Date of creation	2013-03-22 13:50:04
Last modified on	2013-03-22 13:50:04
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	14
Author	matte (1858)
Entry type	Definition
Classification	msc 03-00
Related topic	InclusionMapping
Related topic	RestrictionOfAFunction
Related topic	PullbackOfAKForm