Theorems · Theorem · category theory

CategoryTheory.Functor.map_shiftFunctorComm

∀ {C : Type u_1} {D : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C]
  [inst_1 : CategoryTheory.Category.{v_2, u_2} D] {A : Type u_4} [inst_2 : AddCommMonoid A]
  [inst_3 : CategoryTheory.HasShift C A] [inst_4 : CategoryTheory.HasShift D A] (F : CategoryTheory.Functor C D)
  [inst_5 : F.CommShift A] (X : C) (a b : A),
  F.map ((CategoryTheory.shiftFunctorComm C a b).hom.app X) =
    CategoryTheory.CategoryStruct.comp
      ((CategoryTheory.Functor.commShiftIso F b).hom.app ((CategoryTheory.shiftFunctor C a).obj X))
      (CategoryTheory.CategoryStruct.comp
        ((CategoryTheory.shiftFunctor D b).map ((CategoryTheory.Functor.commShiftIso F a).hom.app X))
        (CategoryTheory.CategoryStruct.comp ((CategoryTheory.shiftFunctorComm D a b).hom.app (F.obj X))
          (CategoryTheory.CategoryStruct.comp
            ((CategoryTheory.shiftFunctor D a).map ((CategoryTheory.Functor.commShiftIso F b).inv.app X))
            ((CategoryTheory.Functor.commShiftIso F a).inv.app ((CategoryTheory.shiftFunctor C b).obj X)))))

Defined in

Mathlib.CategoryTheory.Shift.CommShift

Cited by

1 results in Mathlib

Foundations

Depth 39 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryAddCommMonoidCategoryTheory.HasShiftCategoryTheory.HasShiftCategoryTheory.Functor.CommShift