TL;DR: just read the last sentence of the question.
I'm watching a video series on intro level topos theory. It introduces sheaves on a locale as part of the motivation. It claims the following fact:
A locale, viewed as a category (with relation ≤ as morphism), is cartesian closed.
The construction is given explicitly:
$$ (v ⇒ w) := \bigvee_{u ∧ v ≤ w} u $$
It is described as the "largest open set whose intersection with $v$ is ≤ $w$".
I tried to use my intuition on topological spaces to understand this construction: consider the discrete topology, and the largest such open set is just the union of the following:
- $v ∧ w$, and
- the complement of $v ∨ w$.
This really seems to be symmetric, say, exchanging $v$ and $w$ really gives the same set, but it may also not be the case because I only considered the discrete topology. So here's my question:
Consider the internal hom $v ⇒ w$ in a locale viewed as a category. Is it equal to $w ⇒ v$?
It seems that, as was pointed out in the comments, you made a little mistake which explains your confusion: the definition of $(v⇒ w)$ implies that it is the union of $v\land w$ and the complement of $v$, rather than the complement of $v\lor w$.
For instance, if $v$ and $w$ are disjoint, then $(v⇒ w)$ is just $\overline{v}$ (this is clearly the largest open whose intersection with $v$ is $\leqslant w$).