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Homesubobject classifier

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# subobject classifier

# Motivation

Consider a set $A$ and a subset $B\subseteq A$. $B$ can be thought of as a property of $A$: there is a function $\chi_{B}:A\to\{0,1\}$, such that $\chi_{B}(x)=1$ iff $x\in B$. This function can be seen to be uniquely determined by the subset $B$, and conversely. If we denote $P(A)$ the set of all subsets of $A$ (the power set of $A$), and $2^{A}$ the set of all functions from $A$ to $2:=\{0,1\}$, then $P(A)\cong 2^{A}$.

In fact, we have established a commutative diagram

$\xymatrix@+=4pc{B\ar[r]^{{k}}\ar[d]_{{inc}}&\{1\}\ar[d]^{i}\\ A\ar[r]_{{\chi_{B}}}&2}$ |

where $inc$ and $i$ are inclusion functions and $k$ is the unique constant function. Any function $A\to 2$ gives rise to a unique set $B$ making the above diagram commute.

# Definition

In category theory, a *subobject classifier* is the generalization of the above example, where $A$ is an object of a certain given category $\mathcal{C}$ and $B$ is a subobject of $A$, $\{1\}$ is replaced by a terminal object, and $2$ is replaced by what is known as a *subobject classifier*, or a *truth object*. If we think of the category Set, $2$ “classifies” elements of a given set as to whether they belong to a certain subset or not, via a characteristic function. If the value of the function is $1$, then the element is in that subset, otherwise it is not.

Formally, let $\mathcal{C}$ be a category with a terminal object $1$. A *subobject classifier* is an object $\Omega$ in $\mathcal{C}$ such that, for any monomorphism $f:B\to A$, there exists a *unique* morphism $\chi_{B}$ such that

$\xymatrix@+=4pc{B\ar[r]\ar[d]_{f}&1\ar[d]^{{\top}}\\ A\ar@{.>}[r]^{{\chi_{B}}}&\Omega}$ |

is a pullback diagram. $\chi_{B}$ is called the *characteristic morphism* of $f$ and $\top$ is a *truth morphism*.

In a category with a terminal object 1, a subobject classifier may or may not exist. If it does, it is unique up to isomorphism. Suppose $C$ has a terminal object $1$, has pullbacks, and has a subobject $\Omega$. Then for any object $X$ in $\mathcal{C}$, any morphism $f:X\to\Omega$ gives rise to a unique monomorphism $g:A\to X$ via the pull back of $f$ and $\top$:

$\xymatrix@+=4pc{A\ar[r]\ar[d]_{g}&1\ar[d]^{{\top}}\\ X\ar[r]^{f}&\Omega}$ |

Since $\Omega$ is a subobject classifier, $g$ determines $f$ uniquely as well. So what we have is a one-to-one correspondence

${\mathrm{Sub}}(X)\cong\hom(X,\Omega)$ |

between the subobject functor and hom functor. It can be verified that the bijection is actually a natural isomorphism, so that ${\mathrm{Sub}}$ is a representable functor. Conversely, it may be shown that if ${\mathrm{Sub}}$ is representable, then $C$ has a subobject classifier.

## Mathematics Subject Classification

18B25*no label found*

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