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subobject_classifier [2015/10/20 18:26]
nikolaj
subobject_classifier [2016/05/29 17:16] (current)
nikolaj
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 $$ $$
  
-Going further: The information of the inclusion of elements in a subset $S\subseteq X$ constitutes a relation $\varepsilon_A$ via+Going further: The information of the inclusion of elements in a subset $S\subseteq X$ constitutes a relation $\varepsilon_X$ via
  
-$x\in S\leftrightarrow \langle S,​x\rangle\in \varepsilon_A$, +$x\in S\leftrightarrow \langle S,​x\rangle\in \varepsilon_X$, 
  
-where $\varepsilon_A ​\subseteq {\mathcal P}(X)\times X$. As a power set ${\mathcal P}(X)$ is in bijection with the space of characteristic functions ${\mathrm{Hom}}(X,​\{0,​1\})=\{0,​1\}^X$,​ the "is element of"​-relation can be defined as the following pullback:+where $\varepsilon_X ​\subseteq {\mathcal P}(X)\times X$. As a power set ${\mathcal P}(X)$ is in bijection with the space of characteristic functions ${\mathrm{Hom}}(X,​\{0,​1\})=\{0,​1\}^X$,​ the "is element of"​-relation can be defined as the following pullback:
  
 $$ $$
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 \begin{CD} ​         ​ \begin{CD} ​         ​
-\varepsilon_A ​ ​@>​{!}>> ​     \{0\}                   +\varepsilon_X ​ ​@>​{!}>> ​     \{0\}                   
 \\  \\ 
 @VVV      @VV{\top}V ​   @VVV      @VV{\top}V ​  
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