Our aim is to illustrate the meaning of the term “subobject classifier” (Wikipedia, nLab)
by giving some examples of how the set $\bftwo = \{\ladjbot, \radjtop\}$
serves as a (“the”) subobject classifier in the familiar Boolean topos $\Set$.
Notation: The symbol “$\lrcorner$” indicates that (the square in which it is contained) is (a pullback square),
with (the pullback object) being (that which is bounded by the “$\lrcorner$”).
Basic examples of subobject classification
| $\bbox[2ex, border:1px black solid] { \begin{array}{} && \emptyset & \xrightarrow[]{} & \bfone \\ && \llap{\emptyset = 1_\emptyset} \Big\Vert & \mkern-1.8em{\raise2ex\hbox{$\lrcorner$}} & \Big\downarrow\rlap\radjtop & \\ && \emptyset & \xrightarrow[\displaystyle \emptyset]{} &\bftwo \\ \end{array} }$ | $2^0 = {0 \choose 0} = 1$ | |||
| $\bbox[2ex, border:1px black solid] { \begin{array}{} & \emptyset & \xrightarrow[]{} & \bfone \\ & \llap{\emptyset}\Big\downarrow & \mkern-1.8em{\raise2ex\hbox{$\lrcorner$}} & \Big\downarrow\rlap\radjtop & \\ & \bfone & \xrightarrow[\displaystyle \ladjbot]{} &\bftwo \\ \end{array} }$ | $\bbox[2ex, border:1px black solid] { \begin{array}{} & \bfone & \xrightarrow[]{} & \bfone \\ & \llap{1_\bfone} \Big\Vert & \mkern-1.8em{\raise2ex\hbox{$\lrcorner$}} & \Big\downarrow\rlap\radjtop & \\ & \bfone & \xrightarrow[\displaystyle \radjtop]{} &\bftwo \\ \end{array} }$ | $2^1 = {1 \choose 0} + {1 \choose 1} = 1 + 1 = 2$ | ||
| $\bbox[2ex, border:1px black solid] { \begin{array}{} & \emptyset & \xrightarrow[]{} & \bfone \\ & \llap{\emptyset}\Big\downarrow & \mkern-1.8em{\raise2ex\hbox{$\lrcorner$}} & \Big\downarrow\rlap\radjtop & \\ & \bftwo & \xrightarrow[\displaystyle !\ladjbot]{} &\bftwo \\ \end{array} }$ | $\bbox[2ex, border:1px black solid] { \begin{array}{} \{\radjtop\} \cong \bfone & \xrightarrow[]{} & \bfone \\ \llap{\radjtop}\Big\downarrow & \mkern-1.8em{\raise2ex\hbox{$\lrcorner$}} & \Big\downarrow\rlap\radjtop & \\ \bftwo & \xrightarrow[\displaystyle 1_\bftwo]{} &\bftwo \\ \end{array} }$ | $\bbox[2ex, border:1px black solid] { \begin{array}{} \{\ladjbot\} \cong \bfone & \xrightarrow[]{} & \bfone \\ \llap{\ladjbot}\Big\downarrow & \mkern-1.8em{\raise2ex\hbox{$\lrcorner$}} & \Big\downarrow\rlap\radjtop & \\ \bftwo & \xrightarrow[\displaystyle \neg\vphantom{1_\bftwo}]{} &\bftwo \\ \end{array} }$ | $\bbox[2ex, border:1px black solid] { \begin{array}{} & \bftwo & \xrightarrow[]{} & \bfone \\ & \llap{1_\bftwo} \Big\Vert & \mkern-1.8em{\raise2ex\hbox{$\lrcorner$}} & \Big\downarrow\rlap\radjtop & \\ & \bftwo & \xrightarrow[\displaystyle !\radjtop]{} &\bftwo \\ \end{array} }$ | $\begin{array}{l} 2^2 = (1+1)^2 = (1+1)\cdot(1+1) = \\ = 1\cdot 1 + 1\cdot 1 + 1\cdot 1 + 1\cdot 1 = \\ = 1 + 1 + 1 + 1 = \\ = {2\choose 0} + {2\choose 1} + {2\choose 2} = 1+2+1 =4\\ \bftwo^2 \cong (\ladjbot+\radjtop)^2 = (\ladjbot+\radjtop) \times (\ladjbot+\radjtop) \cong \\ {} \cong \ladjbot\times\ladjbot + \ladjbot\times\radjtop + \radjtop\times\ladjbot + \radjtop\times\radjtop \sim \\ {} \sim {!\ladjbot} + 1_\bftwo + {\neg} + {!\radjtop} \\ \end{array}$ |
| $\bbox[2ex, border:1px black solid] { \begin{array}{lccc} \text{graph} & \radjtop \\ \text{of }\mkern.25em !\ladjbot & \ladjbot & \text{X} & \text{X} \\ \text{output} & & \ladjbot & \ladjbot \\ \text{input} & & \ladjbot & \radjtop \\ \end{array} }$ | $\bbox[2ex, border:1px black solid] { \begin{array}{lccc} \text{graph} & \radjtop & & \class {red} {\text{X}} \\ \text{of }\mkern.25em 1_\bftwo & \ladjbot & \text{X} \\ \text{output} & & \ladjbot & \class {red} {\radjtop} \\ \text{input} & & \ladjbot & \radjtop \\ \end{array} }$ | $\bbox[2ex, border:1px black solid] { \begin{array}{lccc} \text{graph} & \radjtop & \class {red} {\text{X}} \\ \text{of }\mkern.25em \neg & \ladjbot & & \text{X} \\ \text{output} & & \class {red} {\radjtop} & \ladjbot \\ \text{input} & & \ladjbot & \radjtop \\ \end{array} }$ | $\bbox[2ex, border:1px black solid] { \begin{array}{lccc} \text{graph} & \radjtop & \class {red} {\text{X}} & \class {red} {\text{X}} \\ \text{of }\mkern.25em !\radjtop & \ladjbot \\ \text{output} & & \class {red} {\radjtop} & \class {red} {\radjtop} \\ \text{input} & & \ladjbot & \radjtop \\ \end{array} }$ | |
General theorem: 2 is a subobject classifier for Set
For an arbitrary set $\setX\in\Set$, there is a bijection $\Sub(\setX) \cong \hom \setX \Set \bftwo$.The correspondence between (subsets $\setA \subseteq \setX$, which are also denoted $i_\setA : \setA \rightarrowtail \setX$) and ($\bftwo$-valued predicates $\functionA : \setX \to \bftwo$) is given by:
| From subsets to predicates | From predicates to subsets | |
| $\bbox[2ex, border:1px black solid] { \begin{array}{} & \functionA & \xrightarrow[]{\displaystyle !} & \bfone \\ & \llap{i_\setA}\Big\downarrow & \mkern-10em{\raise2ex\hbox{$\lrcorner$}} & \Big\downarrow\rlap\radjtop & \\ & X & \xrightarrow[ \displaystyle \boxed{ \begin{array}{} ?\in\functionA\\ (\exists\elta\in\setA)(?=\elta) \\ (\exists\elta)(\elta\in\setA \wedge {?=\elta} \wedge \radjtop) \\ \int^{\elta\in\setA} \hom ? \setX \elta \otimes {!\radjtop} \\ \Lan_{i_\setA} !\radjtop \\ \end{array} } ]{} &\bftwo \\ \end{array} }$ | $\bbox[2ex, border:1px black solid] { \begin{array}{} & \boxed{ \begin{array}{} \{ \eltx \mid \functionA_\eltx \}\\ \{ \eltx \mid \functionA_\eltx = \radjtop \}\\ \functionA^{-1}\radjtop\\ \end{array} } & \xrightarrow[]{\displaystyle !} & \bfone \\ & \llap{}\Big\downarrow & \mkern-3em{\raise2ex\hbox{$\lrcorner$}} & \Big\downarrow\rlap\radjtop & \\ & X & \xrightarrow[\displaystyle \functionA]{} &\bftwo \\ \end{array} }$ |
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