## Subobject classifier

### Collection

context | ${\bf C}$ … category with terminal object $*$ and all pullbacks |

definiendum | $\top$ in it |

inclusion | $\top:{\bf C}[*,\Omega]$ |

for all | $m_S:{\bf C}[S,X]$ … monomorphism |

postulate | There is a unique arrow $\chi_S$, so that the mono $m_S$ is the pullback of $\top$ along $\chi_S$ |

#### 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 $\Omega$ of $\top$, such that the following completes to a pullback diagram

$$ \require{AMScd} \begin{CD} S @>{!_S}>> * \\ @V{m_S}VV @VV{\top}V \\ X @. \Omega \end{CD} $$

I.e. $S$ is a the pullback object $(X\times_\Omega *)$ associated with the unique arrow $\chi_S$, which $S$ “knows everything about”.

#### Theorems

Consider a locally small category ${\bf C}$, i.e. one where hom-classes are sets.

The hom-functor

${\mathrm{Hom}}(-,\Omega):{\bf C}\longrightarrow{\bf{Set}}$

maps objects $X$ to the set ${\mathrm{Hom}}(X,\Omega)$, which by construction is in bijection with the subobjects of $X$.

#### Example

In ${\bf{Set}}$, monos are injections and a subset $S\subset X$ exactly corresponds to the inverse image of the characteristic function. Hence, $\Omega$ is given by a two-element set.

For example, we may choose $*=\{0\}$, $\Omega=\{0,1\}$ and $\top(0):=1$.

$$ \require{AMScd} \begin{CD} S @>{!_S}>> \{0\} \\ @V{m_S}VV @VV{\top}V \\ X @>{\chi_S}>> \{0,1\} \end{CD} $$

Going further: The information of the inclusion of elements in a subset $S\subseteq X$ constitutes a relation $\varepsilon_X$ via

$x\in S\leftrightarrow \langle S,x\rangle\in \varepsilon_X$,

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:

$$ \require{AMScd} \begin{CD} \varepsilon_X @>{!}>> \{0\} \\ @VVV @VV{\top}V \\ \{0,1\}^X\times X @>{\mathrm{eval}}>> \{0,1\} \end{CD} $$

#### Reference

Wikipedia: Subobject classifier