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pullback_._category_theory [2015/03/16 20:39]
nikolaj
pullback_._category_theory [2015/03/16 21:06] (current)
nikolaj
Line 27: Line 27:
 == Special cases == == Special cases ==
   * If $\pi_a$ is an iso, then $A\times_Z B\cong A$. As $A$ is already the pullback, it alone fully determines the "full solution"​.   * If $\pi_a$ is an iso, then $A\times_Z B\cong A$. As $A$ is already the pullback, it alone fully determines the "full solution"​.
-  * If moreover $\pi_b$ is an iso too, the projections ​vanish from the diagram and the universal property says that arrows $\gamma,​\delta$ ​(see above) ​can be wholly glued together, i.e., up to iso, $\alpha\circ\gamma=\beta\circ\delta\implies\gamma=\delta$. +  * If moreover $\pi_b$ is an iso too, the projections, we can consider ​the equivalent pullback with $\pi_b=\pi_a=1_A$. The universal property ​now says that arrows $\gamma,​\delta$ can be wholly glued together: Up to iso, $\alpha\circ\gamma=\beta\circ\delta\implies\gamma=\delta$. 
-  * In ${\bf{Set}}$,​ if $\alpha=\beta$,​ the pullback definition says its elements $\langle x,y\rangle$ fulfill $\alpha(x)=\alpha(y)$,​ i.e. here the pullback object is the full collection of pairs of term with give the same $\alpha$ value. If moreover $\pi_a$ is iso, any $x$ determines an $\langle x,y\rangle$ and hence $y$ and the universal property ​says $\alpha(x)=\alpha(y)\implies x=y$. This is just the definition of an injection. +  * In ${\bf{Set}}$,​ if $\alpha=\beta$,​ the pullback definition says that its elements $\langle x,y\rangle$ fulfill $\alpha(x)=\alpha(y)$,​ i.e. here the pullback object is the full collection of pairs of term with give the same $\alpha$ value. If moreover $\pi_a$ is iso, any $x$ determines an $\langle x,y\rangle$ and hence an $y$ and the universal property ​translates to $\alpha(x)=\alpha(y)\implies x=y$. This is just the definition of an injection. 
-  * Back to a general category, consider ​the case where $\pi_ais iso AND $\alpha=\beta$. The condition ​is $\alpha\circ\gamma=\alpha\circ\delta\implies\gamma=\delta$ ​and we call such an $\alpha$ a [[monomorphism]]+  * Back to a general category. If the pullback of $\alphaalong itself ($\alpha=\beta$) is such that a projection $\pi_a$ is iso, we call $\alpha$ a [[monomorphism]]. The associated ​condition ​reads $\alpha\circ\gamma=\alpha\circ\delta\implies\gamma=\delta$.
 {{ monomorphism-diagram.png?​X300}} {{ monomorphism-diagram.png?​X300}}
  
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