# pull back maps in category of sets

Given $\phi : A\rightarrow X$ and $\psi : B\rightarrow X$ in $\mathcal{C}$ (category), a pull back of $(\phi,\psi)$ is a pair of morphisms $\alpha : Y\rightarrow A$ and $\beta : Y\rightarrow B$ such that $\phi\circ \alpha=\psi \circ \beta$ and that : given $\gamma : Z\rightarrow A$ and $\delta:Z\rightarrow B$ with $\phi\circ\gamma=\psi\circ \delta$ there exists unique $\zeta :Z\rightarrow Y$ with $\gamma=\alpha\circ \zeta$ and $\delta=\beta\circ\zeta$

I am trying to compute pull back in case of category of sets $\mathcal {C}$..

We need to construct $Y$... when ever i see having maps from $Y$ to $A$ and $Y$ to $B$ first thing that comes to my mind is $Y=A\times B$ and that maps are projections.. So, first guess is $Y=\{(a,b) \in A\times B \}$ and $\alpha,\beta$ are projections to $A$ and $B$ respectively...

I need to take care of this condition $(\phi\circ \alpha)(a,b)=(\psi\circ \beta) (a,b)$ also.. which means that $\phi\circ (\alpha(a,b))=\psi\circ (\beta (a,b))$ i.e., $\phi(a)=\psi(b)$

This tells me that i can not take full $A\times B$ as $Y$ but a subset of $A\times B$ namely $Y=(a,b)\in A\times B : \phi(a)=\psi(b)$

I would like to check for universal property:

Suppose i have $\gamma : Z\rightarrow A$ and $\delta:Z\rightarrow B$ with $\phi\circ\gamma=\psi\circ \delta$

I have to get a $\zeta :Z\rightarrow Y$..

For $z\in Z$ i need to define what $\zeta(z)$ is... all i know is $\zeta(z)=(a,b)$ with $\phi(a)=\psi(b)$..

I also have a condition that $(\phi\circ \gamma)(z)=(\psi\circ \delta)(z)\Rightarrow \phi(\gamma(z))=\psi(\delta(z))$.. See that i have $\zeta(z)=(a,b)$ with $\phi(a)=\psi(b)$.. This made me to choose $\gamma(z)=(\gamma(z),\delta(z))$.. This is working smoothly, i mean if this is my $\zeta$ then i do have $\gamma=\alpha\circ \zeta$ and $\delta=\beta\circ\zeta$...

But i am getting confused when it is coming to prove uniqueness.. please suggest some way to deal with this..

Thank you.

## 1 Answer

Keeping your notations, suppose there exists (another) $\tau\colon Z\to Y$ such that $p_{A}\circ \tau=\gamma$ and $p_{B}\circ \tau=\delta$ ($p_{A}$ and $p_{B}$ are the two projections of $Y$ into $A$ and $B$ respectively, i.e. what you called $\alpha$ and $\beta$ above). Then, for all $z\in Z$, if $\alpha(z)=(z_{1},\ z_{2})\in Y$, those two equations simply say that $z_{1}=\gamma (z)$ and $z_{2}=\delta (z)$. Thus, $\alpha(z)=(\gamma (z),\ \delta(z))=\zeta (z)$, for all $z\in Z$. This gives you uniqueness of $\zeta$.

• I was careless... This should be the obvious one that should possibly come when i define $\zeta(z)$.. In any case, thank you so much :) – user87543 Sep 7 '14 at 14:50
• Done........ :) – user87543 Sep 7 '14 at 14:57