Examples of Pullback


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Examples of Pullback

  1. If A, C, D and B were sets, then
    $ D=A \times_C B=\{(a,b)\in A \times B \vert f(a)=g(b)\}\subseteq A \times B$ with maps $ k:A \times_C B\rightarrow B$, ( $ (a,b)\longmapsto b$) and $ h:A \times_C B\rightarrow A$, ( $ (a,b)\longmapsto a$) satisfies the conditions of being a pullback
    Proof
    Given a set E with maps $ j:E\rightarrow B$ and $ i:E\rightarrow A$, then the map $ l:E\rightarrow A \times_C B$, ( $ e \longmapsto (j(e),i(e))$) would make the diagram

    $\displaystyle \xymatrix{ E\ar@/^/[rrrrd]^j\ar@{-->}[rrd]^l\ar@/_/[rrddd]_i&&&&\\ &&D\ar[rr]^k\ar[dd]^h&&B\ar[dd]^g\\ &&&&\\ &&A\ar[rr]_f&&C\\ }$

    commute. In fact we have the following identities for all $ e\in E$:
    $ h\circ l(e)=i(e)$ and $ k\circ l(e)=j(e)$.
    Moreover l is unique since, given any other map $ m:E\rightarrow A \times_C B$ such that $ h\circ m=i$ and $ k\circ m=j$, then for all $ e\in E$ the following hold :
    l(e)=(j(e),i(e))=(h(m(e)),k(m(e)))=m(e)=(h(a,b),k(a,b))=(a,b).
    Therefore l is unique
  2. Pullback exists in any (functor 3.2) category of presheaves $ Sets^{C^{op}}$. In fact, if $ X,Y,B\in Sets^{C^{op}}$, then $ P\in Sets^{C^{op}}$ is a pullback in $ Sets^{C^{op}}$ iff


    \begin{displaymath}\xymatrix{ P(C)\ar[rr]^{k}\ar[dd]_{h}&&Y(C)\ar[dd]^{g}\\ &&\\ X(C)\ar[rr]_l&&B(C)\\ }\end{displaymath}


    is a pullback in set. This implies that $ P=(X \times_B Y)C\cong X(C)\times_{B(C)} Y(C)$ The fact that a pullback in $ Sets^{C^{op}}$ is defined in terms of a pullback in Sets implies that the former always exists since the latter does. In fact, given any object $ C\in\mathscr{C}$ and any three functors 3.2 X, Y, Z, it is always possible to construct (in Set) a diagram as the one above. This implies that it is always possible to define a functor 3.2 $ P:\mathscr{C}\rightarrow Set$ which assigns, for each $ C\in\mathscr{C}$ an object P(C) (a set), and for each arrow $ f:A\rightarrow C$ in $ \mathscr{C}$ the unique arrow $ P(f):P(C)\rightarrow P(A)$ in $ \mathcal{S}^{\mathscr{C}^{op}}$ which makes the following cube a pullback in $ \mathcal{S}^{\mathscr{C}^{op}}$

    $\displaystyle \xymatrix{ P(C)\ar[rrr]^{k}\ar[rd]^{P(f)}\ar[dd]^{h}&&&Y(C)\ar[dd]^{g}\ar[rd]^{Y(f)}&\\ &P(A)\ar[dd]\ar[rrr]&&&Y(A)\ar[dd]\\ X(C)\ar[rrr]^l\ar[rd]^{X(f)}&&&B(C)\ar[rd]^{B(f)}&\\ &X(A)\ar[rrr]^j&&&B(A)\\ }$


Cecilia Flori 2007-02-04