File: Mathlib/Algebra/Homology/Embedding/Restriction.lean

full_name: HomologicalComplex.restriction_d
code: /-- Given `K : HomologicalComplex C c'` and `e : c.Embedding c'` (satisfying `[e.IsRelIff]`),
this is the homological complex in `HomologicalComplex C c` obtained by restriction. -/
@[simps]
def restriction : HomologicalComplex C c 
goal: ∀ (x x_1 : ι), (K.restriction e).d x x_1 = K.d (e.f x) (e.f x_1)

full_name: HomologicalComplex.restriction_X
code: /-- Given `K : HomologicalComplex C c'` and `e : c.Embedding c'` (satisfying `[e.IsRelIff]`),
this is the homological complex in `HomologicalComplex C c` obtained by restriction. -/
@[simps]
def restriction : HomologicalComplex C c 
goal: (K.restriction e).X i = K.X (e.f i)

full_name: HomologicalComplex.restriction
code: /-- Given `K : HomologicalComplex C c'` and `e : c.Embedding c'` (satisfying `[e.IsRelIff]`),
this is the homological complex in `HomologicalComplex C c` obtained by restriction. -/
@[simps]
def restriction : HomologicalComplex C c 
goal: HomologicalComplex C c

full_name: HomologicalComplex.restrictionXIso
code: /-- The isomorphism `(K.restriction e).X i ≅ K.X i'` when `e.f i = i'`. -/
def restrictionXIso {i : ι} {i' : ι'} (h : e.f i = i') :
    (K.restriction e).X i ≅ K.X i' 
goal: (K.restriction e).X i ≅ K.X i'

full_name: HomologicalComplex.restriction_d_eq
code: @[reassoc]
lemma restriction_d_eq {i j : ι} {i' j' : ι'} (hi : e.f i = i') (hj : e.f j = j') :
    (K.restriction e).d i j = (K.restrictionXIso e hi).hom ≫ K.d i' j' ≫
      (K.restrictionXIso e hj).inv 
goal: (K.restriction e).d i j = (K.restrictionXIso e hi).hom ≫ K.d i' j' ≫ (K.restrictionXIso e hj).inv

full_name: HomologicalComplex.restrictionMap
code: /-- The morphism `K.restriction e ⟶ L.restriction e` induced by a morphism `φ : K ⟶ L`. -/
@[simps]
def restrictionMap : K.restriction e ⟶ L.restriction e 
goal: K.restriction e ⟶ L.restriction e

full_name: HomologicalComplex.restrictionMap_f
code: /-- The morphism `K.restriction e ⟶ L.restriction e` induced by a morphism `φ : K ⟶ L`. -/
@[simps]
def restrictionMap : K.restriction e ⟶ L.restriction e 
goal: (HomologicalComplex.restrictionMap φ e).f i = φ.f (e.f i)

full_name: HomologicalComplex.restrictionMap_f'
code: @[reassoc]
lemma restrictionMap_f' {i : ι} {i' : ι'} (hi : e.f i = i') :
    (restrictionMap φ e).f i = (K.restrictionXIso e hi).hom ≫
      φ.f i' ≫ (L.restrictionXIso e hi).inv 
goal: (HomologicalComplex.restrictionMap φ e).f i = (K.restrictionXIso e hi).hom ≫ φ.f i' ≫ (L.restrictionXIso e hi).inv

full_name: HomologicalComplex.restrictionMap_id
code: @[simp]
lemma restrictionMap_id : restrictionMap (𝟙 K) e = 𝟙 _ 
goal: HomologicalComplex.restrictionMap (𝟙 K) e = 𝟙 (K.restriction e)

full_name: HomologicalComplex.restrictionMap_comp
code: @[simp, reassoc]
lemma restrictionMap_comp :
    restrictionMap (φ ≫ φ') e = restrictionMap φ e ≫ restrictionMap φ' e 
goal: HomologicalComplex.restrictionMap (φ ≫ φ') e =
  HomologicalComplex.restrictionMap φ e ≫ HomologicalComplex.restrictionMap φ' e

full_name: ComplexShape.Embedding.restrictionFunctor
code: /-- Given `e : ComplexShape.Embedding c c'`, this is the restriction
functor `HomologicalComplex C c' ⥤ HomologicalComplex C c`. -/
@[simps]
noncomputable def restrictionFunctor [HasZeroMorphisms C] :
    HomologicalComplex C c' ⥤ HomologicalComplex C c 
goal: HomologicalComplex C c' ⥤ HomologicalComplex C c

full_name: ComplexShape.Embedding.restrictionFunctor_obj
code: /-- Given `e : ComplexShape.Embedding c c'`, this is the restriction
functor `HomologicalComplex C c' ⥤ HomologicalComplex C c`. -/
@[simps]
noncomputable def restrictionFunctor [HasZeroMorphisms C] :
    HomologicalComplex C c' ⥤ HomologicalComplex C c 
goal: (e.restrictionFunctor C).obj K = K.restriction e

full_name: ComplexShape.Embedding.restrictionFunctor_map
code: /-- Given `e : ComplexShape.Embedding c c'`, this is the restriction
functor `HomologicalComplex C c' ⥤ HomologicalComplex C c`. -/
@[simps]
noncomputable def restrictionFunctor [HasZeroMorphisms C] :
    HomologicalComplex C c' ⥤ HomologicalComplex C c 
goal: ∀ {X Y : HomologicalComplex C c'} (φ : X ⟶ Y), (e.restrictionFunctor C).map φ = HomologicalComplex.restrictionMap φ e

full_name: ComplexShape.Embedding.instPreservesZeroMorphismsHomologicalComplexRestrictionFunctor
code: instance [HasZeroMorphisms C] : (e.restrictionFunctor C).PreservesZeroMorphisms 
goal: (e.restrictionFunctor C).PreservesZeroMorphisms

full_name: ComplexShape.Embedding.instAdditiveHomologicalComplexRestrictionFunctor
code: instance [Preadditive C] : (e.restrictionFunctor C).Additive 
goal: (e.restrictionFunctor C).Additive