Theorems · Definition · category theory

SheafOfModules.mapFreeIso

{C : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    {J : CategoryTheory.GrothendieckTopology C} →
      {R : CategoryTheory.Sheaf J RingCat} →
        [inst_1 : CategoryTheory.HasWeakSheafify J AddCommGrpCat] →
          [inst_2 : J.WEqualsLocallyBijective AddCommGrpCat] →
            {C' : Type u₂} →
              [inst_3 : CategoryTheory.Category.{v₂, u₂} C'] →
                {J' : CategoryTheory.GrothendieckTopology C'} →
                  {S : CategoryTheory.Sheaf J' RingCat} →
                    [inst_4 : CategoryTheory.HasSheafify J' AddCommGrpCat] →
                      [inst_5 : J'.WEqualsLocallyBijective AddCommGrpCat] →
                        (F : CategoryTheory.Functor (SheafOfModules R) (SheafOfModules S)) →
                          (I : Type u) →
                            [CategoryTheory.Limits.PreservesColimitsOfShape (CategoryTheory.Discrete I) F] →
                              (SheafOfModules.unit S ≅ F.obj (SheafOfModules.unit R)) →
                                (SheafOfModules.free I ≅ F.obj (SheafOfModules.free I))

Let F be a functor from the category of sheaves of R-modules to sheaves of S-modules. If F preserves coproducts and unit S ≅ F.obj (unit R), then F preserves free sheaves of modules.

Defined in

Mathlib.Algebra.Category.ModuleCat.Sheaf.Free

Cited by

12 results in Mathlib

Foundations

Depth 105 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.HasWeakSheafifyCategoryTheory.GrothendieckTopology.WEqualsLocallyBijectiveCategoryTheory.CategoryCategoryTheory.HasSheafifyCategoryTheory.GrothendieckTopology.WEqualsLocallyBijectiveCategoryTheory.Limits.PreservesColimitsOfShape