sheafHomHasNSMul — Mathlib

English

For a Type-valued presheaf P, IsSheaf and isSheafFor are equivalent under the isSheaf_of_type equivalence.

Русский

Для прешейфа, значения которого в Type, условия IsSheaf и IsSheafFor эквивалентны через эквивалентность isSheaf_of_type.

LaTeX

$$$\\text{isSheafFor} \\quad \\Leftrightarrow \\text{isSheaf (Type-valued)}$$$

Lean4

instance sheafHomHasNSMul : SMul ℕ (P ⟶ Q) where
  smul n
    f :=
    Sheaf.Hom.mk
      { app := fun U => n • f.1.app U
        naturality := fun U V i => by
          induction n with
          | zero => simp only [zero_smul, comp_zero, zero_comp]
          | succ n ih => simp only [add_smul, ih, one_nsmul, comp_add, NatTrans.naturality, add_comp] }

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