A universal closure operation on a topos $\mathcal{E}$ is a family of functions $c_{(-)}: Sub(-) \to Sub(-)$ such that for $A$ a subobject of $X$ we have
- $A \leq c_X(A)$
- $c_X(c_X(A)) = c_X(A)$
- For any $f: Y \to X$, we have that $c_Y(f^*(A)) = f^*(c_X(A))$.
I am looking for non-trivial examples. As universal closure operations are in bijection with Lawvere-Tierney topologies, there are no intersting ones on Set. Another archetypical example is the category of presheaves. If we take $\mathcal{C}$ to be a topology regarded as a poset, we have a Grothendieck topology that says a sieve on $U$ (family of subsets of $U$) is covering if and only if those subsets cover $U$. But the associated Lawvere-Tierney topology is given by
$$J_U( (O_i \subseteq U)_{i \in I} ) = \{V \subseteq U \ | \ ((V \subseteq U)^*(O_i))_{i \in I} \textrm{ covers } V \}$$
which is the maximal sieve as $(V \subseteq U)^*(O) = V \cap O$. And the associated closure operation sends a subobject $A$ of $X$ to the identity on $X$, so this one is not interesting either.
Do you have any easy to work with examples? I was surprised I couldn't find one by using google.
Edit: actually, is there a mistake in the previous paragraph? There is a bijection between Grothendieck topologies and Lawvere-Tierney topologies, but the Grothendieck topology that has only the maximal sieve as a covering sieve, gives rise to the same Lawvere-Tierney topology.