Theorems · Definition · category theory

PresheafOfModules.ModuleColimit.homEquiv

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    [inst_1 : CategoryTheory.LocallySmall.{w, v, u} C] →
      [inst_2 : CategoryTheory.IsCofiltered C] →
        [inst_3 : CategoryTheory.InitiallySmall C] →
          {R : CategoryTheory.Functor Cᵒᵖ RingCat} →
            {cR : CategoryTheory.Limits.Cocone R} →
              (hcR : CategoryTheory.Limits.IsColimit cR) →
                {M : PresheafOfModules R} →
                  {cM : CategoryTheory.Limits.Cocone M.presheaf} →
                    (hcM : CategoryTheory.Limits.IsColimit cM) →
                      {N : ModuleCat ↑cR.pt} →
                        (ModuleCat.of (↑cR.pt) (PresheafOfModules.ModuleColimit hcR hcM) ⟶ N) ≃+
                          (M ⟶ (PresheafOfModules.constFunctor cR).obj N)

This is the universal property of PresheafOfModules.ModuleColimit as a module. See also PresheafOfModules.colimitAdjunction.

Defined in

Mathlib.Algebra.Category.ModuleCat.Presheaf.ColimitFunctor

Cited by

6 results in Mathlib

Foundations

Depth 65 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.LocallySmallCategoryTheory.IsCofilteredCategoryTheory.InitiallySmall