Suppose I have a map $j : \text{Sub}(1) \to \text{Sub}(1)$ from subterminal objects of a topos to themselves which satisfies analogous axioms to those of a Lawvere-Tierney topology, namely $j(1) = 1$, $j(a \wedge b) = j(a) \wedge j(b)$, $a \leq j(a)$ and $jj(a) \leq j(a)$. I'm having trouble defining what the closure of a subobject would be in terms of such a $j$. Is there a way to define $j$-sheaves for this, or to upgrade it into a bona fide LT-topology?
2026-03-30 05:16:28.1774847788
"External" Lawvere-Tierney Topologies?
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