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subobject_classifier [2015/03/22 01:37] nikolaj |
subobject_classifier [2015/10/20 18:26] nikolaj |
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| === Elaboration === | === Elaboration === | ||
| - | The postulate says that for any given mono $m_S$ from $S$ to $X$, there is a unique arrow $\chi_S$ from $X$ to the domain of $\top$, such that the following is a [[pullback . category theory|pullback]] diagram | + | The postulate says that for any given mono $m_S$ from $S$ to $X$, there is a unique arrow $\chi_S$ from $X$ to the domain $\Omega$ of $\top$, such that the following completes to a [[pullback . category theory|pullback]] diagram |
| $$ | $$ | ||
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| $x\in S\leftrightarrow \langle S,x\rangle\in \varepsilon_A$, | $x\in S\leftrightarrow \langle S,x\rangle\in \varepsilon_A$, | ||
| - | 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 $\varepsilon_A \subseteq {\mathcal P}(X)\times X$. 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: |
| $$ | $$ | ||
| Line 61: | Line 61: | ||
| \begin{CD} | \begin{CD} | ||
| - | \varepsilon_A @>{!_S}>> \{0\} | + | \varepsilon_A @>{!}>> \{0\} |
| \\ | \\ | ||
| @VVV @VV{\top}V | @VVV @VV{\top}V | ||
