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dependent_product_functor [2015/10/13 20:50] nikolaj |
dependent_product_functor [2015/12/22 18:01] nikolaj |
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| - | 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 the second component $x$ to itself, but instead map it 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: | ||
