Subobjects


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Subobjects

In order to understand what a subobject is, let us consider a specific example in Set which is a type of Category. Given a subset A of S i.e $ A\subseteq S$, the notion of being a subset can be expressed mathematically using the so called characteristic function: $ \chi_A:S\rightarrow\{0,1\}$, which is defined as follows:

$\displaystyle \chi_A(x)=\begin{cases}0& if\hspace{.1in}x\notin A\\ 1& if\hspace{.1in}x\in A\end{cases}$ (1.1)

(here we interpret 1=true and 0=false). The role of the characteristic function is to determine which elements belong to a certain subset.
Remembering that in any category subobjects are identified as monic arrows, we define the value true as follows:
$ true:1=\{0\}\rightarrow 2=\{0,1\}$ which in terms of individual elements becomes $ 0\rightarrow 1$. Given this definition, it can be easily seen that $ A=\chi_A^{-1}(1)$, this is like saying that the diagram

$\displaystyle \xymatrix{ *++{S}\ar@{^{(}->}[rr]\ar[dd]&&A\ar[dd]^{\chi_A}\\ &&\\ 1\ar[rr]_T&&2\\ }$

is a pullback.

Cecilia Flori 2007-01-02