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examples of pullbacks
This entry shows some examples of categorical pullbacks.
1. In the category of sets, the pullback of a pair of functions $f:A\to C$ and $g:B\to C$ is given by the set $D:=\{(a,b)\in A\times B\mid f(a)=g(b)\}$, along with the projections $r:D\to A$ and $s:D\to B$. Here’s a sketch of the proof: first, $f\circ r=g\circ s$, and if there are functions $u:E\to A$ and $v:E\to B$ with $f\circ u=g\circ v$, then define a function $w:E\to D$ by $w(e)=(u(e),v(e))$. As $f(u(e))=g(v(e))$, we have that $(u(e),v(e))\in D$, so that $w$ is a welldefined function. Furthermore, $r\circ w(e)=r(u(e),v(e))=u(e)$ and $s\circ w(e)=s(u(e),v(e))=v(e)$. Finally, this $w$ is easily seen to be unique. Therefore, $(D,r:D\to A,s:D\to B)$ is the pullback of $f$ and $g$.
2. In the category of groups, the pullback of a pair of group homomorphisms $f:A\to C$ and $g:B\to C$ is again the group $D=\{(a,b)\in A\times B\mid f(a)=g(b)\}$, where the product is defined componentwise, along with the usual projections. The verification that this is indeed the pullback of $f$ and $g$ is almost like the one above. The only thing that needs to be verified is that $D$ is indeed a group. If $(a,b),(c,d)\in D$, then $f(ac)=f(a)f(c)=g(b)g(d)=g(bd)$, so that $(ac,bd)\in D$. Also, $f(1_{A})=1_{C}=g(1_{B})$, so that $(1_{A},1_{B})\in D$. Finally, if $(x,y)\in D$, then $f(x^{{1}})=f(x)^{{1}}=g(y)^{{1}}=g(y^{{1}})$, or $(x^{{1}},y^{{1}})\in D$. Therefore, $D$ is a group (a subgroup of $A\times B$).
3. In fact, both of the examples above can be obtained by finding the equalizer of $f\circ p_{A}$ and $g\circ p_{B}$, where $p_{A}$ and $p_{B}$ are projections from $A\times B$ to $A$ and $B$ respectively. This is the consequence of the fact that a category with finite products and equalizers also has pullbacks, and the pullbacks are obtained in the manner just described (see proof here).
4. The category of small categories has pullbacks. Given small categories $\mathcal{A},\mathcal{B}$, and $\mathcal{C}$, and functors $F:\mathcal{A}\to\mathcal{C}$ and $G:\mathcal{B}\to\mathcal{C}$, consider the subcategory $\mathcal{D}$ of the comma category $(F\downarrow G)$, where

objects are $(A,B,f)$ where $F(A)=G(B)$ and $f=1_{{F(A)}}$, and

morphisms are $(x,y):(A,B,1_{{F(A)}})\to(C,D,1_{{F(C)}})$ where $F(x)=G(y)$.
Then it can be shown that $\mathcal{D}$, along with the the functors

$H_{{\mathcal{A}}}:\mathcal{D}\to\mathcal{A}$ with $H_{{\mathcal{A}}}(A,B,f)=A$ and $H_{{\mathcal{A}}}(x,y)=x$, and

$H_{{\mathcal{B}}}:\mathcal{D}\to\mathcal{B}$ with $H_{{\mathcal{B}}}(A,B,f)=B$ and $H_{{\mathcal{B}}}(x,y)=y$
is the pullback of $F$ and $G$. The proof is similar to the proof on the universal property of a comma category.

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