Editing Subobject classifier (section)

== Introductory example ==

As an example, the set Ω = {0,1} is a subobject classifier in the [[category of sets]] and functions: to every subset ''A'' of ''S'' defined by the inclusion function&nbsp;'' j '' : ''A'' → ''S'' we can assign the function ''χ<sub>A</sub>'' from ''S'' to Ω that maps precisely the elements of ''A'' to 1 (see [[indicator function|characteristic function]]). Every function from ''S'' to Ω arises in this fashion from precisely one subset ''A''.

To be clearer, 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<sub>''A''</sub> : S → {0,1}, which is defined as follows:
:<math>\chi_A(x) = 
\begin{cases} 
  0, & \mbox{if }x\notin A \\
  1, & \mbox{if }x\in A
\end{cases}</math>

(Here we interpret 1 as true and 0 as false.) The role of the characteristic function is to determine which elements belong to the subset ''A''. In fact, χ<sub>''A''</sub> is true precisely on the elements of ''A''.

In this way, the collection of all subsets of ''S'' and the collection of all maps from ''S'' to Ω = {0,1} are [[isomorphic]].

To categorize this notion, recall that, in category theory, a subobject is actually a pair consisting of an object and a [[monomorphism|monic arrow]] (interpreted as the inclusion into another object). Accordingly, '''true''' refers to the element 1, which is selected by the arrow: '''true''': {0} → {0, 1} that maps 0 to 1. The subset ''A'' of ''S'' can now be defined as the [[pullback (category theory)|pullback]] of '''true''' along the characteristic function χ<sub>''A''</sub>, shown on the following diagram:
[[Image:SubobjectClassifier-01.png|center]]
 
Defined that way, χ is a morphism ''Sub''<sub>C</sub>(''S'') →  Hom<sub>C</sub>(S, Ω). By definition, Ω is a '''subobject classifier''' if this morphism is an isomorphism.