Theorems · Theorem · category theory

CategoryTheory.SingleFunctors.mk.inj

∀ {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_5} {inst_2 : AddMonoid A}
  {inst_3 : CategoryTheory.HasShift D A} {functor : A → CategoryTheory.Functor C D}
  {shiftIso : (n a a' : A) → n + a = a' → ((functor a').comp (CategoryTheory.shiftFunctor D n) ≅ functor a)}
  {shiftIso_zero : ∀ (a : A), shiftIso 0 a a ⋯ = (functor a).isoWhiskerLeft (CategoryTheory.shiftFunctorZero D A)}
  {shiftIso_add :
    ∀ (n m a a' a'' : A) (ha' : n + a = a') (ha'' : m + a' = a''),
      shiftIso (m + n) a a'' ⋯ =
        (functor a'').isoWhiskerLeft (CategoryTheory.shiftFunctorAdd D m n) ≪≫
          ((functor a'').associator (CategoryTheory.shiftFunctor D m) (CategoryTheory.shiftFunctor D n)).symm ≪≫
            CategoryTheory.Functor.isoWhiskerRight (shiftIso m a' a'' ha'') (CategoryTheory.shiftFunctor D n) ≪≫
              shiftIso n a a' ha'}
  {functor_1 : A → CategoryTheory.Functor C D}
  {shiftIso_1 : (n a a' : A) → n + a = a' → ((functor_1 a').comp (CategoryTheory.shiftFunctor D n) ≅ functor_1 a)}
  {shiftIso_zero_1 : ∀ (a : A), shiftIso_1 0 a a ⋯ = (functor_1 a).isoWhiskerLeft (CategoryTheory.shiftFunctorZero D A)}
  {shiftIso_add_1 :
    ∀ (n m a a' a'' : A) (ha' : n + a = a') (ha'' : m + a' = a''),
      shiftIso_1 (m + n) a a'' ⋯ =
        (functor_1 a'').isoWhiskerLeft (CategoryTheory.shiftFunctorAdd D m n) ≪≫
          ((functor_1 a'').associator (CategoryTheory.shiftFunctor D m) (CategoryTheory.shiftFunctor D n)).symm ≪≫
            CategoryTheory.Functor.isoWhiskerRight (shiftIso_1 m a' a'' ha'') (CategoryTheory.shiftFunctor D n) ≪≫
              shiftIso_1 n a a' ha'},
  { functor := functor, shiftIso := shiftIso, shiftIso_zero := shiftIso_zero, shiftIso_add := shiftIso_add } =
      { functor := functor_1, shiftIso := shiftIso_1, shiftIso_zero := shiftIso_zero_1,
        shiftIso_add := shiftIso_add_1 } →
    functor = functor_1 ∧ shiftIso ≍ shiftIso_1

Defined in

Mathlib.CategoryTheory.Shift.SingleFunctors

Cited by

1 results in Mathlib

Foundations

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