Theorems · Theorem · category theory

PresheafOfModules.ModuleColimit.homEquiv_app_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}
  (α : ModuleCat.of (↑cR.pt) (PresheafOfModules.ModuleColimit hcR hcM) ⟶ N) {X : Cᵒᵖ} (x : ↑(M.obj X)),
  (CategoryTheory.ConcreteCategory.hom (((PresheafOfModules.ModuleColimit.homEquiv hcR hcM) α).app X)) x =
    (CategoryTheory.ConcreteCategory.hom α) ((CategoryTheory.ConcreteCategory.hom (cM.ι.app X)) x)

Defined in

Mathlib.Algebra.Category.ModuleCat.Presheaf.ColimitFunctor

Cited by

0 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.LocallySmallCategoryTheory.IsCofilteredCategoryTheory.InitiallySmall