Presheaves over sieves and posets

Question

I'm looking for a proof of the claim that given a poset P, the topos of presheaves over P is equivalent to the topos of presheaves over the complete Heyting algebra of sieves on elements of P. I found this claim in John Bell's paper "causal sets and frame valued set theory". I get the impression that the proof is probably pretty trivial, but I can't quite see it at the moment. If anyone could either outline the proof or give a suitable reference, that would be great.

Answer 1

The claim is false. Let $P$ be the trivial poset with one element; then presheaves on $P$ are just sets. But the complete Heyting algebra of sieves in $P$ is the poset $\{ 0 < 1 \}$, and presheaves on this constitute a non-boolean topos. Here is the correct statement:

For any poset $P$, the topos of presheaves on $P$ is equivalent to the topos of sheaves on the complete Heyting algebra of sieves in $P$.

It is a consequence of standard facts in topos theory:

1. The presheaf topos on a poset is localic. [Sketches of an elephant, Example A4.6.2(d)]
2. The subobjects of $1$ in $[P^\mathrm{op}, \mathbf{Set}]$ are the sieves in $P$. [Exercise]
3. A localic topos is the topos of sheaves on the complete Heyting algebra $\mathrm{Sub}(1)$. [Sketches of an elephant, Theorem C1.4.7]