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

full_name: HomologicalComplex.IsStrictlySupported
code: /-- If `K : HomologicalComplex C c'`, then `K.IsStrictlySupported e` holds for
an embedding `e : c.Embedding c'` of complex shapes if `K.X i'` is zero
wheneverm `i'` is not of the form `e.f i` for some `i`.-/
class IsStrictlySupported : Prop where
  isZero (i' : ι') (hi' : ∀ i, e.f i ≠ i') : IsZero (K.X i')
goal: Prop

full_name: HomologicalComplex.IsStrictlySupported.isZero
code: /-- If `K : HomologicalComplex C c'`, then `K.IsStrictlySupported e` holds for
an embedding `e : c.Embedding c'` of complex shapes if `K.X i'` is zero
wheneverm `i'` is not of the form `e.f i` for some `i`.-/
class IsStrictlySupported : Prop where
  isZero (i' : ι') (hi' : ∀ i, e.f i ≠ i') : IsZero (K.X i')
goal: CategoryTheory.Limits.IsZero (K.X i')

full_name: HomologicalComplex.isZero_X_of_isStrictlySupported
code: lemma isZero_X_of_isStrictlySupported [K.IsStrictlySupported e]
    (i' : ι') (hi' : ∀ i, e.f i ≠ i') :
    IsZero (K.X i') 
goal: CategoryTheory.Limits.IsZero (K.X i')

full_name: HomologicalComplex.isStrictlySupported_of_iso
code: lemma isStrictlySupported_of_iso [K.IsStrictlySupported e] : L.IsStrictlySupported e 
goal: L.IsStrictlySupported e

full_name: HomologicalComplex.IsSupported
code: /-- If `K : HomologicalComplex C c'`, then `K.IsStrictlySupported e` holds for
an embedding `e : c.Embedding c'` of complex shapes if `K` is exact at `i'`
whenever `i'` is not of the form `e.f i` for some `i`.-/
class IsSupported : Prop where
  exactAt (i' : ι') (hi' : ∀ i, e.f i ≠ i') : K.ExactAt i'
goal: Prop

full_name: HomologicalComplex.IsSupported.exactAt
code: /-- If `K : HomologicalComplex C c'`, then `K.IsStrictlySupported e` holds for
an embedding `e : c.Embedding c'` of complex shapes if `K` is exact at `i'`
whenever `i'` is not of the form `e.f i` for some `i`.-/
class IsSupported : Prop where
  exactAt (i' : ι') (hi' : ∀ i, e.f i ≠ i') : K.ExactAt i'
goal: K.ExactAt i'

full_name: HomologicalComplex.exactAt_of_isSupported
code: lemma exactAt_of_isSupported [K.IsSupported e] (i' : ι') (hi' : ∀ i, e.f i ≠ i') :
    K.ExactAt i' 
goal: K.ExactAt i'

full_name: HomologicalComplex.isSupported_of_iso
code: lemma isSupported_of_iso [K.IsSupported e] : L.IsSupported e 
goal: L.IsSupported e

full_name: HomologicalComplex.instIsSupportedOfIsStrictlySupported
code: instance [K.IsStrictlySupported e] : K.IsSupported e 
goal: K.IsSupported e

full_name: HomologicalComplex.IsStrictlySupportedOutside.isZero
code: /-- If `K : HomologicalComplex C c'`, then `K.IsStrictlySupportedOutside e` holds for
an embedding `e : c.Embedding c'` of complex shapes if `K.X (e.f i)` is zero for all `i`. -/
structure IsStrictlySupportedOutside : Prop where
  isZero (i : ι) : IsZero (K.X (e.f i))
goal: CategoryTheory.Limits.IsZero (K.X (e.f i))

full_name: HomologicalComplex.IsStrictlySupportedOutside
code: /-- If `K : HomologicalComplex C c'`, then `K.IsStrictlySupportedOutside e` holds for
an embedding `e : c.Embedding c'` of complex shapes if `K.X (e.f i)` is zero for all `i`. -/
structure IsStrictlySupportedOutside : Prop where
  isZero (i : ι) : IsZero (K.X (e.f i))
goal: Prop

full_name: HomologicalComplex.IsSupportedOutside.exactAt
code: /-- If `K : HomologicalComplex C c'`, then `K.IsSupportedOutside e` holds for
an embedding `e : c.Embedding c'` of complex shapes if `K` is exact at `e.f i` for all `i`. -/
structure IsSupportedOutside : Prop where
  exactAt (i : ι) : K.ExactAt (e.f i)
goal: K.ExactAt (e.f i)

full_name: HomologicalComplex.IsSupportedOutside
code: /-- If `K : HomologicalComplex C c'`, then `K.IsSupportedOutside e` holds for
an embedding `e : c.Embedding c'` of complex shapes if `K` is exact at `e.f i` for all `i`. -/
structure IsSupportedOutside : Prop where
  exactAt (i : ι) : K.ExactAt (e.f i)
goal: Prop

full_name: HomologicalComplex.IsStrictlySupportedOutside.isSupportedOutside
code: lemma IsStrictlySupportedOutside.isSupportedOutside (h : K.IsStrictlySupportedOutside e) :
    K.IsSupportedOutside e 
goal: K.IsSupportedOutside e

full_name: HomologicalComplex.instIsStrictlySupportedOfNat
code: instance [HasZeroObject C] : (0 : HomologicalComplex C c').IsStrictlySupported e 
goal: HomologicalComplex.IsStrictlySupported 0 e

full_name: HomologicalComplex.isZero_iff_isStrictlySupported_and_isStrictlySupportedOutside
code: lemma isZero_iff_isStrictlySupported_and_isStrictlySupportedOutside :
    IsZero K ↔ K.IsStrictlySupported e ∧ K.IsStrictlySupportedOutside e 
goal: CategoryTheory.Limits.IsZero K ↔ K.IsStrictlySupported e ∧ K.IsStrictlySupportedOutside e

full_name: HomologicalComplex.instIsStrictlySupportedOppositeOpOp
code: instance [K.IsStrictlySupported e] : K.op.IsStrictlySupported e.op 
goal: K.op.IsStrictlySupported e.op

full_name: HomologicalComplex.map_isStrictlySupported
code: instance map_isStrictlySupported [K.IsStrictlySupported e] :
    ((F.mapHomologicalComplex c').obj K).IsStrictlySupported e 
goal: ((F.mapHomologicalComplex c').obj K).IsStrictlySupported e