Theorems · Theorem · category theory

CategoryTheory.Functor.CommShift.OfComp.map_iso_hom_app_assoc

∀ {C : Type u_1} {D : Type u_2} {E : Type u_3} [inst : CategoryTheory.Category.{v_1, u_1} C]
  [inst_1 : CategoryTheory.Category.{v_2, u_2} D] [inst_2 : CategoryTheory.Category.{v_3, u_3} E]
  {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D E} {H : CategoryTheory.Functor C E} (e : F.comp G ≅ H)
  [inst_3 : G.Full] [inst_4 : G.Faithful] {A : Type u_5} [inst_5 : AddMonoid A] [inst_6 : CategoryTheory.HasShift C A]
  [inst_7 : CategoryTheory.HasShift D A] [inst_8 : CategoryTheory.HasShift E A] [inst_9 : G.CommShift A]
  [inst_10 : H.CommShift A] (a : A) (X : C) {Z : E} (h : G.obj ((CategoryTheory.shiftFunctor D a).obj (F.obj X)) ⟶ Z),
  CategoryTheory.CategoryStruct.comp (G.map ((CategoryTheory.Functor.CommShift.OfComp.iso e a).hom.app X)) h =
    CategoryTheory.CategoryStruct.comp (e.hom.app ((CategoryTheory.shiftFunctor C a).obj X))
      (CategoryTheory.CategoryStruct.comp ((CategoryTheory.Functor.commShiftIso H a).hom.app X)
        (CategoryTheory.CategoryStruct.comp ((CategoryTheory.shiftFunctor E a).map (e.inv.app X))
          (CategoryTheory.CategoryStruct.comp ((CategoryTheory.Functor.commShiftIso G a).inv.app (F.obj X)) h)))

Defined in

Mathlib.CategoryTheory.Shift.CommShift

Cited by

0 results in Mathlib

Foundations

Depth 33 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.FullCategoryTheory.Functor.FaithfulAddMonoidCategoryTheory.HasShiftCategoryTheory.HasShiftCategoryTheory.HasShiftCategoryTheory.Functor.CommShiftCategoryTheory.Functor.CommShift