Documentation

Mathlib.Algebra.Category.ModuleCat.Sheaf

Sheaves of modules over a sheaf of rings #

In this file, we define the category SheafOfModules R when R : Sheaf J RingCat is a sheaf of rings on a category C equipped with a Grothendieck topology J.

structure SheafOfModules {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {J : CategoryTheory.GrothendieckTopology C} (R : CategoryTheory.Sheaf J RingCat) :
Type (max (max (max u u₁) (v + 1)) v₁)

A sheaf of modules is a presheaf of modules such that the underlying presheaf of abelian groups is a sheaf.

A morphism between sheaves of modules is a morphism between the underlying presheaves of modules.

  • val : X.val Y.val

    a morphism between the underlying presheaves of modules

The forgetful functor SheafOfModules.{v} R ⥤ PresheafOfModules R.val.

Equations

The forget functor SheafOfModules R ⥤ PresheafOfModules R.val is fully faithful.

Equations
  • One or more equations did not get rendered due to their size.

The forget functor SheafOfModules R ⥤ Sheaf J AddCommGrp.

Equations
  • One or more equations did not get rendered due to their size.

The forgetful functor from sheaves of modules over sheaf of ring R to sheaves of R(X)-module when X is initial.

Equations
  • One or more equations did not get rendered due to their size.
@[reducible, inline]

The type of sections of a sheaf of modules.

Equations
@[reducible, inline]

The map M.sections → N.sections induced by a morphisms M ⟶ N of sheaves of modules.

Equations

The functor which sends a sheaf of modules to its type of sections.

Equations
@[reducible, inline]

A morphism of presheaves of modules is locally surjective if the underlying morphism of presheaves of abelian groups is.

Equations
@[reducible, inline]

A morphism of presheaves of modules is locally injective if the underlying morphism of presheaves of abelian groups is.

Equations

The bijection (M₂ ⟶ N) ≃ (M₁ ⟶ N) induced by a locally bijective morphism f : M₁ ⟶ M₂ of presheaves of modules, when N is a sheaf.

Equations
  • One or more equations did not get rendered due to their size.