Pullback


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Pullback

Definition 1.6   A pullback or fibered product of a pair of functions $ f:A\rightarrow B$ and $ g:B\rightarrow C$ in a category $ \mathscr{C}$ is a pair of $ \mathscr{C}$-arrows $ h:D\rightarrow A$ and $ k:D\rightarrow B$, such that the following conditions are satisfied:
  1. $ f o h=g o k$ i.e the following diagram commutes

    $\displaystyle \xymatrix{ D\ar[rr]^{k}\ar[dd]^{h}&&B\ar[dd]^{g}\\ &&\\ A\ar[rr]_f&&C\\ }$

    One usually writes $ D=A\times_C B$

  2. Given two functions $ i:E\rightarrow A$ and $ j:E\rightarrow B$, where $ f o i=g o j$ then, there exists a unique $ \mathscr{C}$-arrow l from E to D such that the outer rectangle of the following diagram commutes

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

    i.e.

    $\displaystyle i=h o l\hspace{.2in}j=k o l$    

    We then say that f (respectively g) has been pulled back along g (respectively f)


Subsections
Cecilia Flori 2007-01-02