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subobject_classifier [2015/04/12 15:38] nikolaj |
subobject_classifier [2016/05/29 17:16] 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 ${\mathcal P}(X)$ 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 | ||
