Theorems · Theorem · category theory

PresheafOfModules.colimitAdjunction_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) {F : PresheafOfModules R} {G : ModuleCat ↑cR.pt}
  (β : F ⟶ (PresheafOfModules.constFunctor cR).obj G) {X : Cᵒᵖ} (m : ↑(F.obj X)),
  (CategoryTheory.ConcreteCategory.hom (((PresheafOfModules.colimitAdjunction hcR).homEquiv F G).symm β))
      (PresheafOfModules.ModuleColimit.ιM m) =
    (CategoryTheory.ConcreteCategory.hom (β.app X)) m

Defined in

Mathlib.Algebra.Category.ModuleCat.Presheaf.ColimitFunctor

Cited by

0 results in Mathlib

Foundations

Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.LocallySmallCategoryTheory.IsCofilteredCategoryTheory.InitiallySmall