- Subobject classifier
In

category theory , a**subobject classifier**is a special object Ω of a category; intuitively, thesubobject s of an object "X" correspond to the morphisms from "X" to Ω. As the name suggests, what a**subobject classifier**does is to identify/classify subobjects of a given object according to which elements belong to the subobject in question. Because of this role, the subobject classifier is also referred to as the**truth value object**. In fact the way in which the**subobject classifier**classifies subobjects of a given object, is by assigning the values true toelements belonging to the subobject in question, and false to elements not belongingto the subobject. This is why the**subobject classifier**is widely used in the categorical description of logic.**Introductory example**As an example, the set Ω = {0,1} is a subobject classifier in the

category of sets and functions: to every subset "j":"U" → "X" we can assign the function "χ_{j}" from "X" to Ω that maps precisely the elements of "U" to 1 (see characteristic function). Every function from "X" to Ω arises in this fashion from precisely one subset "U".To render this example more clear let us consider a

subset "A" of "S" ("A" ⊆ "S"), where "S" is a set. The notion of being a subset can be expressed mathematically using the so-called characteristic function: χ_{"A"}→{0,1}, which is defined as follows::$chi\_A(x)\; =\; egin\{cases\}\; 0,\; mbox\{if\; \}x\; otin\; A\; \backslash \; 1,\; mbox\{if\; \}xin\; Aend\{cases\}$

(Here we interpret 1 as true and 0 as false.) The role of the characteristic function is to determine which elements belong or not to a certain subset. Since in any category subobjects are identified as monic arrows, we identify the value true with the arrow:

**true**: {0} → {0, 1} which maps 0 to 1. Given this definition it can be easily seen that the subset "A" can be uniquely defined through the characteristic function "A"=χ_{"A"}^{-1}(1). Therefore the diagramis a pullback.The above example of subobject classifier in

**Set**is very useful because it enables us to easily prove the following axiom:**Axiom**: Given a category**C**, then there exists anisomorphism ,:y: Sub_{C}("X") ≅ Hom_{C}(X, Ω) ∀ "X" ∈**C**In

**Set**this axiom can be restated as follows:**Axiom**: The collection of all subsets of S denoted by $mathcal\{P\}(S)$, and the collection of all maps from S to the set {0, 1}=2 denoted by 2^{"S"}areisomorphic i.e. the function $y:mathcal\{P\}(S)\; ightarrow2^S$, which in terms of single elements of $mathcal\{P\}(S)$ is "A" → χ_{"A"}, is abijection .The above axiom implies the alternative definition of a subobject classifier:

**Definition**: Ω is a**subobject classifier**iff there is a one to one correspondence between subobject of "X" andmorphisms from "X" to Ω.**Definition**For the general definition, we start with a category

**C**that has aterminal object , which we denote by 1. The object Ω of**C**is a subobject classifier for**C**if there exists a morphism:1 → Ω

with the following property:

:for each

monomorphism "j": "U" → "X" there is a unique morphism "χ_{ j}": "X" → Ω such that the followingcommutative diagram :is apullback diagram — that is, "U" is the limit of the diagram:The morphism "χ

_{ j}" is then called the**classifying morphism**for the subobject represented by "j".**Further examples**Every

topos has a subobject classifier. For the topos of sheaves of sets on atopological space "X", it can be described in these terms: take thedisjoint union Ω of all theopen set s "U" of "X", and its natural mapping π to "X" coming from all theinclusion map s. Then π is alocal homeomorphism , and the corresponding sheaf is the required subobject classifier (in other words the construction of Ω is by means of itsespace étalé ). One can also consider Ω to be, in a (tautological) sense, the graph of the membership relation between points "x" and open sets "U" of "X". For a small category $C$, the**subobject classifer**in the**topos of presheaves**$mathcal\{S\}^\{C^\{op$ is given as follows. For any $c\; in\; C$, $Omega(c)$ is the set of sieves on $c$.**References***cite book

last = Artin

first = Michael

authorlink = Michael Artin

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title = Practical Foundations of Mathematics

publisher =Cambridge University Press

location = Cambridge

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isbn = 0521631076*"Topos-physics": An explanation of Topos theory and its implementation in Physics: [

*http://topos-physics.org/ Topos-physics, Where Geometry meets Dynamics*]

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