instHasExactColimitsOfShapeOfHasFiniteLimitsOfPreservesColimitsOfShapeFunctorOppositeSheafToPresheaf

English

If J is a Grothendieck topology on C and A is Grothendieck abelian with a sheafification, then the category of sheaves Sheaf J A is Grothendieck abelian.

Русский

Если топология Дж на C — Гротениджевская, и A — Гротенджевская абелева с преобразованиемSheaf, тогда Sheaf J A — Гротениджевская абелева.

LaTeX

$$$\mathrm{IsGrothendieckAbelian}(\mathrm{Shaf J A})$$$

Lean4

instance [HasFiniteLimits A] [HasColimitsOfShape K A] [HasExactColimitsOfShape K A]
    [PreservesColimitsOfShape K (sheafToPresheaf J A)] : HasExactColimitsOfShape K (Sheaf J A) :=
  HasExactColimitsOfShape.domain_of_functor K (sheafToPresheaf J A)

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