Definition of truth values in a topos.

Question

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 ?

Answer 1

If I recall correctly, the set of right ideals is how we construct $\Omega$, and so that gives number of elements that we see in $\Omega$ for the topos of $M$-sets. However, internally, the topos doesn't know about these elements. The only elements the topos knows about arise from the morphisms $1 \rightarrow \Omega$, and these are the truth values.

Answer 2

Truth values are actually defined as subobjects of the terminal object, because this definition also makes sense in categories without subobject classifiers. The subobjects of the terminal object correspond to global sections of $\Omega$ in a topos, so this is an isomorphic Heyting algebra.