# 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. 1.

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

2. 2.

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

 $(\Psi\circ\Phi)^{\ast}=\Phi^{\ast}\circ\Psi^{\ast}.$
3. 3.

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

 $\big{(}\Phi^{\ast}\big{)}^{-1}=\big{(}\Phi^{-1}\big{)}^{\ast}.$
4. 4.

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 Pullback 2013-03-22 13:50:04 2013-03-22 13:50:04 matte (1858) matte (1858) 14 matte (1858) Definition msc 03-00 InclusionMapping RestrictionOfAFunction PullbackOfAKForm