isConstant_iff_forget — Mathlib

English

If certain faithful/full forgetful properties hold and the composed object is constant, then F is constant; also the converse direction holds when the composition reflects isomorphisms.

Русский

При выполнении условии полности/верности забывающего функторa и отражения изоморфизмов составной объекты константен; наоборот тоже верно.

LaTeX

$$$IsConstant(J,F) \iff IsConstant(J, (\text{sheafCompose } J U).obj F)$$$

Lean4

theorem isConstant_iff_forget [constantSheaf J D |>.Faithful] [constantSheaf J D |>.Full]
    [constantSheaf J B |>.Faithful] [constantSheaf J B |>.Full] [(sheafCompose J U).ReflectsIsomorphisms] {T : C}
    (hT : IsTerminal T) : F.IsConstant J ↔ ((sheafCompose J U).obj F).IsConstant J :=
  ⟨fun _ ↦ inferInstance, fun _ ↦ Sheaf.isConstant_of_forget _ U F hT⟩

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