I'm trying to understand what is meant exactly by a "truth value" in a topos. Take for example the topos of irreflexive graphs. It is known that the classifying morphism can take nodes to 2 different values in the subobject classifier, and edges to 5 different ones. So one may say we have as many truth values.
However I've seen a definition wherein truth values are the global elements of the subobject classifier $\Omega$, i.e maps from the terminal object to $\Omega$ in which case there is only three truth values in the topos of graphs.
The same goes for the category of sets equipped with the action of a monoid $M$. In one case there are as many truth values as there are ideals in $M$. With the other definition, there is only two truth values.
Can someone clarify what is the proper definition ? Also, truth values apparently have the structure of a Heyting lattice, but is it for the former approach or the latter ?