Theorems · Definition · algebraic geometry

SheafOfModules.Presentation.mapGenerators

{C : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    {J : CategoryTheory.GrothendieckTopology C} →
      {R : CategoryTheory.Sheaf J RingCat} →
        [inst_1 : CategoryTheory.HasSheafify 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] →
                        {M : SheafOfModules R} →
                          (P : M.Presentation) →
                            (F : CategoryTheory.Functor (SheafOfModules R) (SheafOfModules S)) →
                              [CategoryTheory.Limits.PreservesColimitsOfSize.{u, u, max u u₁, max u u₂,
                                      max (max (u + 1) u₁) v₁, max (max (u + 1) u₂) v₂}
                                    F] →
                                (SheafOfModules.unit S ≅ F.obj (SheafOfModules.unit R)) →
                                  (SheafOfModules.free P.generators.I ⟶ F.obj M)

Let F be a functor from sheaf of R-module to sheaf of S-module, if F preserves colimits and F.obj (unit R) ≅ unit S, given a P : Presentation M, then we will obtain generators of Presentation (F.obj M).

Defined in

Mathlib.Algebra.Category.ModuleCat.Sheaf.Quasicoherent

Cited by

4 results in Mathlib

Foundations

Depth 110 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.HasSheafifyCategoryTheory.GrothendieckTopology.WEqualsLocallyBijectiveCategoryTheory.CategoryCategoryTheory.HasSheafifyCategoryTheory.GrothendieckTopology.WEqualsLocallyBijectiveCategoryTheory.Limits.PreservesColimitsOfSize