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pullback_._category_theory [2015/03/16 20:27] nikolaj |
pullback_._category_theory [2015/03/16 20:39] nikolaj |
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| * 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 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, $\alpha\circ\gamma=\beta\circ\delta\implies\gamma=\delta$. | ||
| - | * In ${\bf{Set}}$, if $\alpha=\beta$, the condition reads $\alpha(x)=\alpha(y)$, i.e. here the pullback object is the full collection of pairs $\langle x,y\rangle$ of term with the same $\alpha$-value. If moreover $\pi_a$ is iso (saying any $\langle x,y\rangle$ is determined by $x$ alone), we get $\alpha(x)=\alpha(y)\implies x=y$, which is the definition of an injection. | + | * 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 a $y$ and the universal property says $\alpha(x)=\alpha(y)\implies x=y$. This is just the definition of an injection. |
| - | * Back to a general catgory, consider the case where $\pi_a$ is 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, consider the case where $\pi_a$ is iso AND $\alpha=\beta$. The condition is $\alpha\circ\gamma=\alpha\circ\delta\implies\gamma=\delta$ and we call such an $\alpha$ a [[monomorphism]] |
| {{ monomorphism-diagram.png?X300}} | {{ monomorphism-diagram.png?X300}} | ||
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| === Examples === | === Examples === | ||
| - | In ${\bf{Set}}$: | + | A finite pullback in ${\bf{Set}}$ that I just made up: |
| * Generally: If $F(f_b)$ is the inclusion of a subset $Fb\subseteq{Fz}$ in ${Fz}$, the pullback is iso to (i.e. in bijection with) $F(f_a^{-1})Fb$. Further, if $F(f_a)$ is an inclusion too, this is in bijection with $Fa\cap{Fb}$. If the subset-interpretation doesn't apply, the function $F(f_b)$ from $Fb$ to $Fz$ should be viewed as defining fibre bundle over $Fz$ and the pullback gives a fibre bundle from $Fa\times_{Fz} Fb$ to $Fa$. | * Generally: If $F(f_b)$ is the inclusion of a subset $Fb\subseteq{Fz}$ in ${Fz}$, the pullback is iso to (i.e. in bijection with) $F(f_a^{-1})Fb$. Further, if $F(f_a)$ is an inclusion too, this is in bijection with $Fa\cap{Fb}$. If the subset-interpretation doesn't apply, the function $F(f_b)$ from $Fb$ to $Fz$ should be viewed as defining fibre bundle over $Fz$ and the pullback gives a fibre bundle from $Fa\times_{Fz} Fb$ to $Fa$. | ||
| * A concrete example: Let | * A concrete example: Let | ||
