Theorems · Theorem · category theory

PresheafOfModules.ModuleColimit.homEquiv_naturality_left_symm

∀ {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) {M' : PresheafOfModules R} {cM' : CategoryTheory.Limits.Cocone M'.presheaf}
  (hcM' : CategoryTheory.Limits.IsColimit cM') {N : ModuleCat ↑cR.pt} (f : M ⟶ M')
  (g : M' ⟶ (PresheafOfModules.constFunctor cR).obj N),
  (PresheafOfModules.ModuleColimit.homEquiv hcR hcM).symm (CategoryTheory.CategoryStruct.comp f g) =
    CategoryTheory.CategoryStruct.comp (ModuleCat.ofHom (PresheafOfModules.ModuleColimit.map hcR hcM hcM' f))
      ((PresheafOfModules.ModuleColimit.homEquiv hcR hcM').symm g)

Defined in

Mathlib.Algebra.Category.ModuleCat.Presheaf.ColimitFunctor

Cited by

0 results in Mathlib

Foundations

Depth 67 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.LocallySmallCategoryTheory.IsCofilteredCategoryTheory.InitiallySmall