In the mathematical field of category theory, a subobject classifier is a special object Ω {\displaystyle \Omega } of a category such that, informally, the subobjects of any object X {\displaystyle X} correspond to the morphisms from X {\displaystyle X} to Ω {\displaystyle \Omega } . This provides an analogue of the set of Booleans { 0 , 1 } {\displaystyle \{0,1\}} in categories other than the category of sets. The main use of subobject classifiers is in topos theory, where an elementary topos is defined as a category with a subobject classifier and certain additional requirements. In the internal language of an elementary topos, the subobject classifier is used to interpret truth values, hence the alternative name “object of truth values”.
Introduction Let X {\displaystyle X} be a set. A subset Y ⊆ X {\displaystyle Y\subseteq X} can be equivalently described by its indicator function
χ Y : X → { 0 , 1 } x ↦ { 1 if x ∈ Y 0 if x ∉ Y {\displaystyle {\begin{aligned}\chi _{Y}:X&\to \{0,1\}\\x&\mapsto {\begin{cases}1{\text{ if }}x\in Y\\0{\text{ if }}x\notin Y\end{cases}}\end{aligned}}}
Informally, subsets of X {\displaystyle X} can be identified with functions X → { 0 , 1 } {\displaystyle X\to \{0,1\}} . A subobject classifier Ω {\displaystyle \Omega } of a category C {\displaystyle {\mathcal {C}}} is an object which plays a similar role as { 0 , 1 } {\displaystyle \{0,1\}} does in the category of sets: subobjects of an object X {\displaystyle X} can be identified with morphisms from X {\displaystyle X} to the subobject classifier. To recover the subset with indicator function χ {\displaystyle \chi } in a “purely categorical way”, one can take a pullback
Y → { 1 } ↓ ↓ X → χ { 0 , 1 } {\displaystyle {\begin{array}{lcl}&Y&\rightarrow &\{1\}&\\&\downarrow &&\downarrow \\&X&{\underset {\chi }{\rightarrow }}&\{0,1\}&\\\end{array}}}
where the function from { 1 } {\displaystyle \{1\}} to { 0 , 1 } {\displaystyle \{0,1\}} is the inclusion map. Indeed, the subset Y := { x ∈ X ∣ χ ( x ) = 1 } {\displaystyle Y:=\{x\in X\mid \chi (x)=1\}} , equipped with the inclusion map Y → X {\displaystyle Y\to X} (and the unique, constant map Y → { 1 } {\displaystyle Y\to \{1\}} ) is such a pullback because it has the correct universal property since a map into X {\displaystyle X} which gives the constant function 1 when composed with χ {\displaystyle \chi } is the same as a map into Y {\displaystyle Y} .
… excerpt ends here. Continue reading the full article.
