# Topos Theory, internal Heyting Algebra

Given a topos $\mathcal{E}$ with subobject classifier $\Omega$.

If we denote by $N\Omega$ the former of all local operators on $\Omega$, that is, Lawvere–Tierney topologies of $\mathcal{E}$, it is know that $N\Omega$ forms a Heyting algebra. Also $\Omega$ is a internal Heyting algebra in $\mathcal{E}$.

There exists a morphism from $\Omega$ to $N\Omega$?

• This question is not very clear (what kind of morphism?), but one morphism you may be interested in can be described as taking $u \in \Omega$ to the operator $u \vee -$, the so-called closed nucleus attached to $u$. (This definition can be internalized.) This gives an internal frame map, but off the bat I don't think it would preserve Heyting implication. Another "canonical" map $\Omega \to N \Omega$ in the topos takes $u$ to $u \Rightarrow -$. – Todd Trimble Nov 23 '15 at 7:35
• @ToddTrimble The first morphism you mentioned preserves implication iff the topos is Boolean (it amounts to the identity $(u\Rightarrow v)\lor w=u\Rightarrow(v\lor w)$ which gives excluded middle when $u=w$, $v=0$). The second reverses order, so it is a Heyting homomorphism only in trivial cases. – მამუკა ჯიბლაძე Nov 23 '15 at 10:44
• @მამუკაჯიბლაძე Thanks for filling that in. The question admits trivial answers of course since we can easily construct maps e.g. $\Omega \to 1 \to N \Omega$ as morphisms in the topos. (Asking about existence of structures can occasionally look like fishing expeditions; here it might be better to propose a map and ask specific questions about its properties.) – Todd Trimble Nov 23 '15 at 14:27
• @ToddTrimble Right, although I presume it is about (internal) Heyting algebra morphisms. At least this is the only interesting interpretation that comes to my mind... – მამუკა ჯიბლაძე Nov 23 '15 at 16:41
• @ToddTrimble thanks you are right the next time I will try to be more specific, later on I realizes that my question is not so clear, also I found the morphisms that you mentioned in the Elephant. – Angel Zaldívar Nov 24 '15 at 4:10