Theorems · Theorem · category theory

PresheafOfModules.ModuleColimit.homEquiv_symm_apply

∀ {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} (β : M ⟶ (PresheafOfModules.constFunctor cR).obj N)
  {X : Cᵒᵖ} (x : ↑(M.obj X)),
  (CategoryTheory.ConcreteCategory.hom ((PresheafOfModules.ModuleColimit.homEquiv hcR hcM).symm β))
      ((CategoryTheory.ConcreteCategory.hom (cM.ι.app X)) x) =
    (CategoryTheory.ConcreteCategory.hom (β.app X)) x

Defined in

Mathlib.Algebra.Category.ModuleCat.Presheaf.ColimitFunctor

Cited by

1 results in Mathlib

Foundations

Depth 66 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.LocallySmallCategoryTheory.IsCofilteredCategoryTheory.InitiallySmall