A little something on Topoi

A Topos E is a mathematical object called a locally small Category, which has limits and a power object.  That is to say that for any object A of E the functor Sub($- \times A$) of objects which factor through $- \times A$ is  non-empty member of the category $Sets$.  Moreover there is an object of E called the power object of A for which Hom(-,PA)$\cong$  Sub(-,-$\times$ A).

Note: recall since E is locally small Hom(-PA) is a set so the equivalence of objets above is a bijection in Sets.

Some examples of Topoi would be the category sets; here for any object X PX is $\mathscr{P}$(X) its power set.  With the natural equivalence of $\phi$: Hom(-,PA)$\cong$  Sub(-,-$\times$ A) to be $\phi$:Hom(1,PB)$\rightarrow$Sub(B).