Theorems · Theorem · category theory

PresheafOfModules.ModuleColimit.map_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) {M' : PresheafOfModules R} {cM' : CategoryTheory.Limits.Cocone M'.presheaf}
  (hcM' : CategoryTheory.Limits.IsColimit cM') (f : M ⟶ M') {U : Cᵒᵖ} (m : ↑(M.obj U)),
  (PresheafOfModules.ModuleColimit.map hcR hcM hcM' f) (PresheafOfModules.ModuleColimit.ιM m) =
    PresheafOfModules.ModuleColimit.ιM ((CategoryTheory.ConcreteCategory.hom (f.app U)) m)

Defined in

Mathlib.Algebra.Category.ModuleCat.Presheaf.ColimitFunctor

Cited by

3 results in Mathlib

Foundations

Depth 63 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.LocallySmallCategoryTheory.IsCofilteredCategoryTheory.InitiallySmall