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I am trying to understand the relation between Lawvere-Tierney topologies and the internal logic of toposes. For a closure operator $\overline{-}$ I am trying to prove that a subobject $A$ of an object $E$ is closed (with respect to the given closure operator) if and only if ${(\forall e, e' \in E)(e \in A \wedge \overline{e = e'} \Rightarrow e'\in A)}$.

Is it true?

(It seems intuitive to me but I cannot prove it.)

Boogie
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    As written, with $e \in A$ appearing twice, the statement is a tautology. Perhaps, the $e \in A$ to the right of the $\Rightarrow$ symbol should be replaced with $e' \in A$. – Geoffrey Trang Dec 17 '21 at 23:46
  • You are right, thanks. – Boogie Dec 17 '21 at 23:49
  • A similar internal logic statement which should be equivalent to $A$ being closed would be: $(\forall e : E)(\overline{e \in A} \rightarrow e \in A)$. – Daniel Schepler Dec 18 '21 at 16:40

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It is not true. Consider the trivial closure operator that sends every subobject of $E$ to $E$ itself. Let $A$ be the empty subobject. Then, vacuously, $e \in A \land \overline{e = e'} \implies e' \in A$. But $A$ is not closed if $E$ is not empty.

Zhen Lin
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  • I wonder if there are any Lawvere-Tierney topologies for which the intuition works. – Boogie Dec 18 '21 at 13:59
  • I think the intuition must be faulty. $\overline{e = e'}$ only tells you about which elements become equal after sheafification; it does not tell you about "new" elements. The closure operator is a shadow of sheafification and this example shows that there is a similar phenomenon of "new" elements in the closure. – Zhen Lin Dec 18 '21 at 14:26
  • I agree that something is wrong with my intuition. Still, what if we assume that ${\overline{empty} = empty}$? – Boogie Dec 19 '21 at 14:33
  • I don’t think that works. Consider the case where $E = 1$: then equality is trivial and your condition reduces to a tautology, no matter what $A$ is. – Zhen Lin Dec 19 '21 at 14:50
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The correct condition for a subobject $A$ of $E$ to be closed is (the somewhat tautological): $$\forall x : E.\ \overline{x \in A} \Rightarrow x \in A.$$

Section 6.4 of these notes of mine summarize a couple of similar properties and constructions, such as the condition to be a sheaf or sheafification, all using the internal language.

Ingo Blechschmidt
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