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dependent_product_functor [2015/12/22 17:58]
nikolaj
dependent_product_functor [2015/12/22 18:01] (current)
nikolaj
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 \end{CD} \end{CD}
 $$ $$
-Essentially,​ we get the section space $\prod_X p$ that we are after, if we just replace the projection $\pi_2:​X\times Y\to X$ with the more general $p$. A section of this space must in general then not map an $x$ in $X$ to $x$ in $Y\times X$, but instead merely map it somehow into $p^{-1}(x)$. ​+Essentially,​ we get the section space $\prod_X p$ that we are after, if we just replace the projection $\pi_2:​X\times Y\to X$ with the more general $p$. The function space we obtain need the not just be $Y^X$, but can be some more general so called section space. A section of this space must in general then not map an $x$ in $X$ to $x$ in $Y\times X$, but instead merely map it somehow into $p^{-1}(x)$. ​
  
 So for an auxiliary object $A$, consider the projection $\pi_2:​(A\times X)\to X$, formally defined as the arrow obtained by pulling back $!_A:A\to*$ along $!_X:​X\to*$,​ i.e. $\pi_2:​=!_X{}^*!_A$. This definition is appearent from the commuting square defining the product: ​ So for an auxiliary object $A$, consider the projection $\pi_2:​(A\times X)\to X$, formally defined as the arrow obtained by pulling back $!_A:A\to*$ along $!_X:​X\to*$,​ i.e. $\pi_2:​=!_X{}^*!_A$. This definition is appearent from the commuting square defining the product: ​
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