The local state classifier of a slice topos

## Abstract Let E be a topos possessing a local state classifier Ξ = colim(E_mono ↪ E), with universal cocone {ξ_X : X → Ξ}, and let X be an object of E. We prove that the slice topos E/X again possesses a local state classifier, and that it is the principal down-set object D_X = { (s, x) ∈ Ξ × X : s ≤ ξ_X(x) }, that is, the pullback along ξ_X of the projection ⪯ → Ξ onto the larger coordinate, regarded as an object of E/X via the second projection. Informally: the local states of E/X are exactly the local states of E that are at least as unfolded as X itself. The proof is elementary and site-free; it uses only the folding inequality ξ_A ≤ ξ_Y ∘ f, the product formula ξ_{A×B} = ξ_A ∧ ξ_B, and the retraction (s, x) ↦ (s ∧ ξ_X(x), x) of Ξ × X onto D_X. In particular the existence of Ξ_{E/X} is not assumed but proved. We record the induced semilattice structure, verify the formula on group actions and on localic toposes, observe that the naive analogue of the slice-stability Ω_{E/X} ≅ X*Ω of the subobject classifier fails for Ξ, and give a second, site-theoretic proof for presheaf toposes.