pullback

DefinitionSuppose $X,Y,Z$ are sets, and we have maps

$f:Y$$\to$$Z,$

$\mathrm{\Phi }:X$$\to$$Y.$

Then the pullback of $f$ under $\mathrm{\Phi }$ is the mapping

${\mathrm{\Phi }}^{\ast }f:X$$\to$$Z,$

$x$$\mapsto$$(f\circ \mathrm{\Phi })(x).$

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

0.0.1 Properties

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

2. Suppose we have maps

   $\mathrm{\Phi }:X$$\to$$Y,$

   $\mathrm{\Psi }:Y$$\to$$Z$

   between sets $X,Y,Z$. Then

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

3. If $\mathrm{\Phi }:X\to Y$ is a bijection, then${\mathrm{\Phi }}^{\ast }$ is a bijection and

   $${\left({\mathrm{\Phi }}^{\ast }\right)}^{-1}={\left({\mathrm{\Phi }}^{-1}\right)}^{\ast }.$$

4. Suppose $X,Y$ are sets with $X\subset Y$.Then we have the inclusion map $\iota :X\hookrightarrow Y$, andfor any $f:Y\to Z$, we have

   $${\iota }^{\ast }f={f|}_X,$$

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