Documentation Verification Report

HomologicalComplex

📁 Source: Mathlib/CategoryTheory/Idempotents/HomologicalComplex.lean

Statistics

MetricCount
Definitionsmap, obj, map, obj, counitIso, functor, inverse, unitIso, karoubiChainComplexEquivalence, karoubiCochainComplexEquivalence, karoubiHomologicalComplexEquivalence, HomologicalComplex
12
Theoremscomp_p_d, comp_p_d_assoc, p_comm_f, p_comm_f_assoc, p_comp_d, p_comp_d_assoc, p_idem, p_idem_assoc, map_f_f, obj_X_X, obj_X_p, obj_d_f, map_f_f, obj_X_X, obj_X_d, obj_p_f, counitIso_hom, counitIso_inv, functor_map, functor_obj, inverse_map, inverse_obj, unitIso_hom_app_f_f, unitIso_inv_app_f_f, instIsIdempotentCompleteHomologicalComplex, karoubiChainComplexEquivalence_counitIso_hom, karoubiChainComplexEquivalence_counitIso_inv, karoubiChainComplexEquivalence_functor_map_f_f, karoubiChainComplexEquivalence_functor_obj_X_X, karoubiChainComplexEquivalence_functor_obj_X_p, karoubiChainComplexEquivalence_functor_obj_d_f, karoubiChainComplexEquivalence_inverse_map_f_f, karoubiChainComplexEquivalence_inverse_obj_X_X, karoubiChainComplexEquivalence_inverse_obj_X_d, karoubiChainComplexEquivalence_inverse_obj_p_f, karoubiChainComplexEquivalence_unitIso_hom_app_f_f, karoubiChainComplexEquivalence_unitIso_inv_app_f_f, karoubiCochainComplexEquivalence_counitIso_hom, karoubiCochainComplexEquivalence_counitIso_inv, karoubiCochainComplexEquivalence_functor_map_f_f, karoubiCochainComplexEquivalence_functor_obj_X_X, karoubiCochainComplexEquivalence_functor_obj_X_p, karoubiCochainComplexEquivalence_functor_obj_d_f, karoubiCochainComplexEquivalence_inverse_map_f_f, karoubiCochainComplexEquivalence_inverse_obj_X_X, karoubiCochainComplexEquivalence_inverse_obj_X_d, karoubiCochainComplexEquivalence_inverse_obj_p_f, karoubiCochainComplexEquivalence_unitIso_hom_app_f_f, karoubiCochainComplexEquivalence_unitIso_inv_app_f_f, karoubiHomologicalComplexEquivalence_counitIso, karoubiHomologicalComplexEquivalence_functor, karoubiHomologicalComplexEquivalence_inverse, karoubiHomologicalComplexEquivalence_unitIso
53
Total65

CategoryTheory.Idempotents

Definitions

NameCategoryTheorems
karoubiChainComplexEquivalence 📖CompOp
13 mathmath: karoubiChainComplexEquivalence_inverse_obj_X_d, karoubiChainComplexEquivalence_functor_obj_X_X, karoubiChainComplexEquivalence_functor_obj_X_p, karoubiChainComplexEquivalence_inverse_obj_p_f, karoubiChainComplexEquivalence_functor_map_f_f, karoubiChainComplexEquivalence_functor_obj_d_f, AlgebraicTopology.DoldKan.compatibility_N₂_N₁_karoubi, karoubiChainComplexEquivalence_inverse_obj_X_X, karoubiChainComplexEquivalence_inverse_map_f_f, karoubiChainComplexEquivalence_counitIso_inv, karoubiChainComplexEquivalence_unitIso_inv_app_f_f, karoubiChainComplexEquivalence_unitIso_hom_app_f_f, karoubiChainComplexEquivalence_counitIso_hom
karoubiCochainComplexEquivalence 📖CompOp
12 mathmath: karoubiCochainComplexEquivalence_functor_map_f_f, karoubiCochainComplexEquivalence_inverse_obj_p_f, karoubiCochainComplexEquivalence_unitIso_hom_app_f_f, karoubiCochainComplexEquivalence_counitIso_hom, karoubiCochainComplexEquivalence_functor_obj_X_p, karoubiCochainComplexEquivalence_inverse_map_f_f, karoubiCochainComplexEquivalence_functor_obj_d_f, karoubiCochainComplexEquivalence_inverse_obj_X_X, karoubiCochainComplexEquivalence_counitIso_inv, karoubiCochainComplexEquivalence_unitIso_inv_app_f_f, karoubiCochainComplexEquivalence_functor_obj_X_X, karoubiCochainComplexEquivalence_inverse_obj_X_d
karoubiHomologicalComplexEquivalence 📖CompOp
4 mathmath: karoubiHomologicalComplexEquivalence_functor, karoubiHomologicalComplexEquivalence_unitIso, karoubiHomologicalComplexEquivalence_inverse, karoubiHomologicalComplexEquivalence_counitIso

Theorems

NameKindAssumesProvesValidatesDepends On
instIsIdempotentCompleteHomologicalComplex 📖mathematical—CategoryTheory.IsIdempotentComplete
HomologicalComplex
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
HomologicalComplex.instCategory
—isIdempotentComplete_iff_of_equivalence
CategoryTheory.Functor.preservesZeroMorphisms_of_additive
toKaroubiEquivalence_functor_additive
instIsIdempotentCompleteKaroubi
karoubiChainComplexEquivalence_counitIso_hom 📖mathematical—CategoryTheory.Iso.hom
CategoryTheory.Functor
HomologicalComplex
Karoubi
Karoubi.instCategory
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
instPreadditiveKaroubi
ComplexShape.down
AddSemigroup.toAdd
AddRightCancelSemigroup.toAddSemigroup
AddRightCancelSemigroup.toIsRightCancelAdd
HomologicalComplex.instCategory
CategoryTheory.Functor.category
CategoryTheory.Functor.comp
KaroubiHomologicalComplexEquivalence.inverse
KaroubiHomologicalComplexEquivalence.functor
CategoryTheory.Functor.id
CategoryTheory.Equivalence.counitIso
ChainComplex
karoubiChainComplexEquivalence
CategoryTheory.eqToHom
CategoryTheory.Category.toCategoryStruct
—AddRightCancelSemigroup.toIsRightCancelAdd
karoubiChainComplexEquivalence_counitIso_inv 📖mathematical—CategoryTheory.Iso.inv
CategoryTheory.Functor
HomologicalComplex
Karoubi
Karoubi.instCategory
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
instPreadditiveKaroubi
ComplexShape.down
AddSemigroup.toAdd
AddRightCancelSemigroup.toAddSemigroup
AddRightCancelSemigroup.toIsRightCancelAdd
HomologicalComplex.instCategory
CategoryTheory.Functor.category
CategoryTheory.Functor.comp
KaroubiHomologicalComplexEquivalence.inverse
KaroubiHomologicalComplexEquivalence.functor
CategoryTheory.Functor.id
CategoryTheory.Equivalence.counitIso
ChainComplex
karoubiChainComplexEquivalence
CategoryTheory.eqToHom
CategoryTheory.Category.toCategoryStruct
—AddRightCancelSemigroup.toIsRightCancelAdd
karoubiChainComplexEquivalence_functor_map_f_f 📖mathematical—Karoubi.Hom.f
HomologicalComplex.X
Karoubi
Karoubi.instCategory
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
instPreadditiveKaroubi
ComplexShape.down
AddSemigroup.toAdd
AddRightCancelSemigroup.toAddSemigroup
AddRightCancelSemigroup.toIsRightCancelAdd
KaroubiHomologicalComplexEquivalence.Functor.obj
HomologicalComplex.Hom.f
CategoryTheory.Functor.map
HomologicalComplex
HomologicalComplex.instCategory
CategoryTheory.Equivalence.functor
ChainComplex
karoubiChainComplexEquivalence
Karoubi.X
—AddRightCancelSemigroup.toIsRightCancelAdd
karoubiChainComplexEquivalence_functor_obj_X_X 📖mathematical—Karoubi.X
HomologicalComplex.X
Karoubi
Karoubi.instCategory
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
instPreadditiveKaroubi
ComplexShape.down
AddSemigroup.toAdd
AddRightCancelSemigroup.toAddSemigroup
AddRightCancelSemigroup.toIsRightCancelAdd
CategoryTheory.Functor.obj
HomologicalComplex
HomologicalComplex.instCategory
CategoryTheory.Equivalence.functor
ChainComplex
karoubiChainComplexEquivalence
—AddRightCancelSemigroup.toIsRightCancelAdd
karoubiChainComplexEquivalence_functor_obj_X_p 📖mathematical—Karoubi.p
HomologicalComplex.X
Karoubi
Karoubi.instCategory
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
instPreadditiveKaroubi
ComplexShape.down
AddSemigroup.toAdd
AddRightCancelSemigroup.toAddSemigroup
AddRightCancelSemigroup.toIsRightCancelAdd
CategoryTheory.Functor.obj
HomologicalComplex
HomologicalComplex.instCategory
CategoryTheory.Equivalence.functor
ChainComplex
karoubiChainComplexEquivalence
HomologicalComplex.Hom.f
Karoubi.X
—AddRightCancelSemigroup.toIsRightCancelAdd
karoubiChainComplexEquivalence_functor_obj_d_f 📖mathematical—Karoubi.Hom.f
HomologicalComplex.X
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
ComplexShape.down
AddSemigroup.toAdd
AddRightCancelSemigroup.toAddSemigroup
AddRightCancelSemigroup.toIsRightCancelAdd
Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
HomologicalComplex.Hom.f
Karoubi.p
HomologicalComplex.d
Karoubi
Karoubi.instCategory
instPreadditiveKaroubi
CategoryTheory.Functor.obj
CategoryTheory.Equivalence.functor
ChainComplex
karoubiChainComplexEquivalence
CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
—AddRightCancelSemigroup.toIsRightCancelAdd
HomologicalComplex.Hom.comm
karoubiChainComplexEquivalence_inverse_map_f_f 📖mathematical—HomologicalComplex.Hom.f
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
ComplexShape.down
AddSemigroup.toAdd
AddRightCancelSemigroup.toAddSemigroup
AddRightCancelSemigroup.toIsRightCancelAdd
Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
KaroubiHomologicalComplexEquivalence.Inverse.obj
Karoubi.Hom.f
CategoryTheory.Functor.map
Karoubi
Karoubi.instCategory
instPreadditiveKaroubi
CategoryTheory.Equivalence.inverse
ChainComplex
karoubiChainComplexEquivalence
HomologicalComplex.X
—AddRightCancelSemigroup.toIsRightCancelAdd
karoubiChainComplexEquivalence_inverse_obj_X_X 📖mathematical—HomologicalComplex.X
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
ComplexShape.down
AddSemigroup.toAdd
AddRightCancelSemigroup.toAddSemigroup
AddRightCancelSemigroup.toIsRightCancelAdd
Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
CategoryTheory.Functor.obj
Karoubi
Karoubi.instCategory
instPreadditiveKaroubi
CategoryTheory.Equivalence.inverse
ChainComplex
karoubiChainComplexEquivalence
—AddRightCancelSemigroup.toIsRightCancelAdd
karoubiChainComplexEquivalence_inverse_obj_X_d 📖mathematical—HomologicalComplex.d
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
ComplexShape.down
AddSemigroup.toAdd
AddRightCancelSemigroup.toAddSemigroup
AddRightCancelSemigroup.toIsRightCancelAdd
Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
CategoryTheory.Functor.obj
Karoubi
Karoubi.instCategory
instPreadditiveKaroubi
CategoryTheory.Equivalence.inverse
ChainComplex
karoubiChainComplexEquivalence
Karoubi.Hom.f
HomologicalComplex.X
—AddRightCancelSemigroup.toIsRightCancelAdd
karoubiChainComplexEquivalence_inverse_obj_p_f 📖mathematical—HomologicalComplex.Hom.f
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
ComplexShape.down
AddSemigroup.toAdd
AddRightCancelSemigroup.toAddSemigroup
AddRightCancelSemigroup.toIsRightCancelAdd
Karoubi.X
HomologicalComplex.X
Karoubi
Karoubi.instCategory
instPreadditiveKaroubi
Karoubi.Hom.f
HomologicalComplex.d
Karoubi.p
HomologicalComplex
HomologicalComplex.instCategory
CategoryTheory.Functor.obj
CategoryTheory.Equivalence.inverse
ChainComplex
karoubiChainComplexEquivalence
—AddRightCancelSemigroup.toIsRightCancelAdd
karoubiChainComplexEquivalence_unitIso_hom_app_f_f 📖mathematical—HomologicalComplex.Hom.f
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
ComplexShape.down
AddSemigroup.toAdd
AddRightCancelSemigroup.toAddSemigroup
AddRightCancelSemigroup.toIsRightCancelAdd
Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
CategoryTheory.Functor.obj
Karoubi
Karoubi.instCategory
CategoryTheory.Functor.id
CategoryTheory.Functor.comp
instPreadditiveKaroubi
KaroubiHomologicalComplexEquivalence.functor
KaroubiHomologicalComplexEquivalence.inverse
Karoubi.Hom.f
CategoryTheory.NatTrans.app
CategoryTheory.Iso.hom
CategoryTheory.Functor
CategoryTheory.Functor.category
CategoryTheory.Equivalence.unitIso
ChainComplex
karoubiChainComplexEquivalence
Karoubi.p
—AddRightCancelSemigroup.toIsRightCancelAdd
karoubiChainComplexEquivalence_unitIso_inv_app_f_f 📖mathematical—HomologicalComplex.Hom.f
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
ComplexShape.down
AddSemigroup.toAdd
AddRightCancelSemigroup.toAddSemigroup
AddRightCancelSemigroup.toIsRightCancelAdd
Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
CategoryTheory.Functor.obj
Karoubi
Karoubi.instCategory
CategoryTheory.Functor.comp
instPreadditiveKaroubi
KaroubiHomologicalComplexEquivalence.functor
KaroubiHomologicalComplexEquivalence.inverse
CategoryTheory.Functor.id
Karoubi.Hom.f
CategoryTheory.NatTrans.app
CategoryTheory.Iso.inv
CategoryTheory.Functor
CategoryTheory.Functor.category
CategoryTheory.Equivalence.unitIso
ChainComplex
karoubiChainComplexEquivalence
Karoubi.p
—AddRightCancelSemigroup.toIsRightCancelAdd
karoubiCochainComplexEquivalence_counitIso_hom 📖mathematical—CategoryTheory.Iso.hom
CategoryTheory.Functor
HomologicalComplex
Karoubi
Karoubi.instCategory
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
instPreadditiveKaroubi
ComplexShape.up
AddSemigroup.toAdd
AddRightCancelSemigroup.toAddSemigroup
AddRightCancelSemigroup.toIsRightCancelAdd
HomologicalComplex.instCategory
CategoryTheory.Functor.category
CategoryTheory.Functor.comp
KaroubiHomologicalComplexEquivalence.inverse
KaroubiHomologicalComplexEquivalence.functor
CategoryTheory.Functor.id
CategoryTheory.Equivalence.counitIso
CochainComplex
karoubiCochainComplexEquivalence
CategoryTheory.eqToHom
CategoryTheory.Category.toCategoryStruct
—AddRightCancelSemigroup.toIsRightCancelAdd
karoubiCochainComplexEquivalence_counitIso_inv 📖mathematical—CategoryTheory.Iso.inv
CategoryTheory.Functor
HomologicalComplex
Karoubi
Karoubi.instCategory
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
instPreadditiveKaroubi
ComplexShape.up
AddSemigroup.toAdd
AddRightCancelSemigroup.toAddSemigroup
AddRightCancelSemigroup.toIsRightCancelAdd
HomologicalComplex.instCategory
CategoryTheory.Functor.category
CategoryTheory.Functor.comp
KaroubiHomologicalComplexEquivalence.inverse
KaroubiHomologicalComplexEquivalence.functor
CategoryTheory.Functor.id
CategoryTheory.Equivalence.counitIso
CochainComplex
karoubiCochainComplexEquivalence
CategoryTheory.eqToHom
CategoryTheory.Category.toCategoryStruct
—AddRightCancelSemigroup.toIsRightCancelAdd
karoubiCochainComplexEquivalence_functor_map_f_f 📖mathematical—Karoubi.Hom.f
HomologicalComplex.X
Karoubi
Karoubi.instCategory
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
instPreadditiveKaroubi
ComplexShape.up
AddSemigroup.toAdd
AddRightCancelSemigroup.toAddSemigroup
AddRightCancelSemigroup.toIsRightCancelAdd
KaroubiHomologicalComplexEquivalence.Functor.obj
HomologicalComplex.Hom.f
CategoryTheory.Functor.map
HomologicalComplex
HomologicalComplex.instCategory
CategoryTheory.Equivalence.functor
CochainComplex
karoubiCochainComplexEquivalence
Karoubi.X
—AddRightCancelSemigroup.toIsRightCancelAdd
karoubiCochainComplexEquivalence_functor_obj_X_X 📖mathematical—Karoubi.X
HomologicalComplex.X
Karoubi
Karoubi.instCategory
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
instPreadditiveKaroubi
ComplexShape.up
AddSemigroup.toAdd
AddRightCancelSemigroup.toAddSemigroup
AddRightCancelSemigroup.toIsRightCancelAdd
CategoryTheory.Functor.obj
HomologicalComplex
HomologicalComplex.instCategory
CategoryTheory.Equivalence.functor
CochainComplex
karoubiCochainComplexEquivalence
—AddRightCancelSemigroup.toIsRightCancelAdd
karoubiCochainComplexEquivalence_functor_obj_X_p 📖mathematical—Karoubi.p
HomologicalComplex.X
Karoubi
Karoubi.instCategory
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
instPreadditiveKaroubi
ComplexShape.up
AddSemigroup.toAdd
AddRightCancelSemigroup.toAddSemigroup
AddRightCancelSemigroup.toIsRightCancelAdd
CategoryTheory.Functor.obj
HomologicalComplex
HomologicalComplex.instCategory
CategoryTheory.Equivalence.functor
CochainComplex
karoubiCochainComplexEquivalence
HomologicalComplex.Hom.f
Karoubi.X
—AddRightCancelSemigroup.toIsRightCancelAdd
karoubiCochainComplexEquivalence_functor_obj_d_f 📖mathematical—Karoubi.Hom.f
HomologicalComplex.X
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
ComplexShape.up
AddSemigroup.toAdd
AddRightCancelSemigroup.toAddSemigroup
AddRightCancelSemigroup.toIsRightCancelAdd
Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
HomologicalComplex.Hom.f
Karoubi.p
HomologicalComplex.d
Karoubi
Karoubi.instCategory
instPreadditiveKaroubi
CategoryTheory.Functor.obj
CategoryTheory.Equivalence.functor
CochainComplex
karoubiCochainComplexEquivalence
CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
—AddRightCancelSemigroup.toIsRightCancelAdd
HomologicalComplex.Hom.comm
karoubiCochainComplexEquivalence_inverse_map_f_f 📖mathematical—HomologicalComplex.Hom.f
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
ComplexShape.up
AddSemigroup.toAdd
AddRightCancelSemigroup.toAddSemigroup
AddRightCancelSemigroup.toIsRightCancelAdd
Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
KaroubiHomologicalComplexEquivalence.Inverse.obj
Karoubi.Hom.f
CategoryTheory.Functor.map
Karoubi
Karoubi.instCategory
instPreadditiveKaroubi
CategoryTheory.Equivalence.inverse
CochainComplex
karoubiCochainComplexEquivalence
HomologicalComplex.X
—AddRightCancelSemigroup.toIsRightCancelAdd
karoubiCochainComplexEquivalence_inverse_obj_X_X 📖mathematical—HomologicalComplex.X
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
ComplexShape.up
AddSemigroup.toAdd
AddRightCancelSemigroup.toAddSemigroup
AddRightCancelSemigroup.toIsRightCancelAdd
Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
CategoryTheory.Functor.obj
Karoubi
Karoubi.instCategory
instPreadditiveKaroubi
CategoryTheory.Equivalence.inverse
CochainComplex
karoubiCochainComplexEquivalence
—AddRightCancelSemigroup.toIsRightCancelAdd
karoubiCochainComplexEquivalence_inverse_obj_X_d 📖mathematical—HomologicalComplex.d
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
ComplexShape.up
AddSemigroup.toAdd
AddRightCancelSemigroup.toAddSemigroup
AddRightCancelSemigroup.toIsRightCancelAdd
Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
CategoryTheory.Functor.obj
Karoubi
Karoubi.instCategory
instPreadditiveKaroubi
CategoryTheory.Equivalence.inverse
CochainComplex
karoubiCochainComplexEquivalence
Karoubi.Hom.f
HomologicalComplex.X
—AddRightCancelSemigroup.toIsRightCancelAdd
karoubiCochainComplexEquivalence_inverse_obj_p_f 📖mathematical—HomologicalComplex.Hom.f
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
ComplexShape.up
AddSemigroup.toAdd
AddRightCancelSemigroup.toAddSemigroup
AddRightCancelSemigroup.toIsRightCancelAdd
Karoubi.X
HomologicalComplex.X
Karoubi
Karoubi.instCategory
instPreadditiveKaroubi
Karoubi.Hom.f
HomologicalComplex.d
Karoubi.p
HomologicalComplex
HomologicalComplex.instCategory
CategoryTheory.Functor.obj
CategoryTheory.Equivalence.inverse
CochainComplex
karoubiCochainComplexEquivalence
—AddRightCancelSemigroup.toIsRightCancelAdd
karoubiCochainComplexEquivalence_unitIso_hom_app_f_f 📖mathematical—HomologicalComplex.Hom.f
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
ComplexShape.up
AddSemigroup.toAdd
AddRightCancelSemigroup.toAddSemigroup
AddRightCancelSemigroup.toIsRightCancelAdd
Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
CategoryTheory.Functor.obj
Karoubi
Karoubi.instCategory
CategoryTheory.Functor.id
CategoryTheory.Functor.comp
instPreadditiveKaroubi
KaroubiHomologicalComplexEquivalence.functor
KaroubiHomologicalComplexEquivalence.inverse
Karoubi.Hom.f
CategoryTheory.NatTrans.app
CategoryTheory.Iso.hom
CategoryTheory.Functor
CategoryTheory.Functor.category
CategoryTheory.Equivalence.unitIso
CochainComplex
karoubiCochainComplexEquivalence
Karoubi.p
—AddRightCancelSemigroup.toIsRightCancelAdd
karoubiCochainComplexEquivalence_unitIso_inv_app_f_f 📖mathematical—HomologicalComplex.Hom.f
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
ComplexShape.up
AddSemigroup.toAdd
AddRightCancelSemigroup.toAddSemigroup
AddRightCancelSemigroup.toIsRightCancelAdd
Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
CategoryTheory.Functor.obj
Karoubi
Karoubi.instCategory
CategoryTheory.Functor.comp
instPreadditiveKaroubi
KaroubiHomologicalComplexEquivalence.functor
KaroubiHomologicalComplexEquivalence.inverse
CategoryTheory.Functor.id
Karoubi.Hom.f
CategoryTheory.NatTrans.app
CategoryTheory.Iso.inv
CategoryTheory.Functor
CategoryTheory.Functor.category
CategoryTheory.Equivalence.unitIso
CochainComplex
karoubiCochainComplexEquivalence
Karoubi.p
—AddRightCancelSemigroup.toIsRightCancelAdd
karoubiHomologicalComplexEquivalence_counitIso 📖mathematical—CategoryTheory.Equivalence.counitIso
Karoubi
HomologicalComplex
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
HomologicalComplex.instCategory
Karoubi.instCategory
instPreadditiveKaroubi
karoubiHomologicalComplexEquivalence
KaroubiHomologicalComplexEquivalence.counitIso
——
karoubiHomologicalComplexEquivalence_functor 📖mathematical—CategoryTheory.Equivalence.functor
Karoubi
HomologicalComplex
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
HomologicalComplex.instCategory
Karoubi.instCategory
instPreadditiveKaroubi
karoubiHomologicalComplexEquivalence
KaroubiHomologicalComplexEquivalence.functor
——
karoubiHomologicalComplexEquivalence_inverse 📖mathematical—CategoryTheory.Equivalence.inverse
Karoubi
HomologicalComplex
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
HomologicalComplex.instCategory
Karoubi.instCategory
instPreadditiveKaroubi
karoubiHomologicalComplexEquivalence
KaroubiHomologicalComplexEquivalence.inverse
——
karoubiHomologicalComplexEquivalence_unitIso 📖mathematical—CategoryTheory.Equivalence.unitIso
Karoubi
HomologicalComplex
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
HomologicalComplex.instCategory
Karoubi.instCategory
instPreadditiveKaroubi
karoubiHomologicalComplexEquivalence
KaroubiHomologicalComplexEquivalence.unitIso
——

CategoryTheory.Idempotents.Karoubi.HomologicalComplex

Theorems

NameKindAssumesProvesValidatesDepends On
comp_p_d 📖mathematical—CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
HomologicalComplex.X
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
CategoryTheory.Idempotents.Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
HomologicalComplex.Hom.f
CategoryTheory.Idempotents.Karoubi.Hom.f
CategoryTheory.Idempotents.Karoubi.p
—HomologicalComplex.congr_hom
CategoryTheory.Idempotents.Karoubi.comp_p
comp_p_d_assoc 📖mathematical—CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
HomologicalComplex.X
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
CategoryTheory.Idempotents.Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
HomologicalComplex.Hom.f
CategoryTheory.Idempotents.Karoubi.Hom.f
CategoryTheory.Idempotents.Karoubi.p
—CategoryTheory.Category.assoc
Mathlib.Tactic.Reassoc.eq_whisker'
comp_p_d
p_comm_f 📖mathematical—CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
HomologicalComplex.X
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
CategoryTheory.Idempotents.Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
HomologicalComplex.Hom.f
CategoryTheory.Idempotents.Karoubi.p
CategoryTheory.Idempotents.Karoubi.Hom.f
—HomologicalComplex.congr_hom
CategoryTheory.Idempotents.Karoubi.p_comm
p_comm_f_assoc 📖mathematical—CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
HomologicalComplex.X
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
CategoryTheory.Idempotents.Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
HomologicalComplex.Hom.f
CategoryTheory.Idempotents.Karoubi.p
CategoryTheory.Idempotents.Karoubi.Hom.f
—CategoryTheory.Category.assoc
Mathlib.Tactic.Reassoc.eq_whisker'
p_comm_f
p_comp_d 📖mathematical—CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
HomologicalComplex.X
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
CategoryTheory.Idempotents.Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
HomologicalComplex.Hom.f
CategoryTheory.Idempotents.Karoubi.p
CategoryTheory.Idempotents.Karoubi.Hom.f
—HomologicalComplex.congr_hom
CategoryTheory.Idempotents.Karoubi.p_comp
p_comp_d_assoc 📖mathematical—CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
HomologicalComplex.X
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
CategoryTheory.Idempotents.Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
HomologicalComplex.Hom.f
CategoryTheory.Idempotents.Karoubi.p
CategoryTheory.Idempotents.Karoubi.Hom.f
—CategoryTheory.Category.assoc
Mathlib.Tactic.Reassoc.eq_whisker'
p_comp_d
p_idem 📖mathematical—CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
HomologicalComplex.X
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
CategoryTheory.Idempotents.Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
HomologicalComplex.Hom.f
CategoryTheory.Idempotents.Karoubi.p
—HomologicalComplex.congr_hom
CategoryTheory.Idempotents.Karoubi.idem
p_idem_assoc 📖mathematical—CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
HomologicalComplex.X
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
CategoryTheory.Idempotents.Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
HomologicalComplex.Hom.f
CategoryTheory.Idempotents.Karoubi.p
—CategoryTheory.Category.assoc
Mathlib.Tactic.Reassoc.eq_whisker'
p_idem

CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence

Definitions

NameCategoryTheorems
counitIso 📖CompOp
3 mathmath: counitIso_inv, CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence_counitIso, counitIso_hom
functor 📖CompOp
15 mathmath: CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence_functor, unitIso_inv_app_f_f, counitIso_inv, CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_unitIso_hom_app_f_f, functor_obj, CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_counitIso_hom, unitIso_hom_app_f_f, CategoryTheory.Idempotents.karoubiChainComplexEquivalence_counitIso_inv, CategoryTheory.Idempotents.karoubiChainComplexEquivalence_unitIso_inv_app_f_f, counitIso_hom, CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_counitIso_inv, CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_unitIso_inv_app_f_f, functor_map, CategoryTheory.Idempotents.karoubiChainComplexEquivalence_unitIso_hom_app_f_f, CategoryTheory.Idempotents.karoubiChainComplexEquivalence_counitIso_hom
inverse 📖CompOp
15 mathmath: unitIso_inv_app_f_f, CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence_inverse, counitIso_inv, CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_unitIso_hom_app_f_f, inverse_map, CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_counitIso_hom, unitIso_hom_app_f_f, CategoryTheory.Idempotents.karoubiChainComplexEquivalence_counitIso_inv, inverse_obj, CategoryTheory.Idempotents.karoubiChainComplexEquivalence_unitIso_inv_app_f_f, counitIso_hom, CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_counitIso_inv, CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_unitIso_inv_app_f_f, CategoryTheory.Idempotents.karoubiChainComplexEquivalence_unitIso_hom_app_f_f, CategoryTheory.Idempotents.karoubiChainComplexEquivalence_counitIso_hom
unitIso 📖CompOp
3 mathmath: unitIso_inv_app_f_f, CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence_unitIso, unitIso_hom_app_f_f

Theorems

NameKindAssumesProvesValidatesDepends On
counitIso_hom 📖mathematical—CategoryTheory.Iso.hom
CategoryTheory.Functor
HomologicalComplex
CategoryTheory.Idempotents.Karoubi
CategoryTheory.Idempotents.Karoubi.instCategory
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
CategoryTheory.Idempotents.instPreadditiveKaroubi
HomologicalComplex.instCategory
CategoryTheory.Functor.category
CategoryTheory.Functor.comp
inverse
functor
CategoryTheory.Functor.id
counitIso
CategoryTheory.eqToHom
CategoryTheory.Category.toCategoryStruct
——
counitIso_inv 📖mathematical—CategoryTheory.Iso.inv
CategoryTheory.Functor
HomologicalComplex
CategoryTheory.Idempotents.Karoubi
CategoryTheory.Idempotents.Karoubi.instCategory
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
CategoryTheory.Idempotents.instPreadditiveKaroubi
HomologicalComplex.instCategory
CategoryTheory.Functor.category
CategoryTheory.Functor.comp
inverse
functor
CategoryTheory.Functor.id
counitIso
CategoryTheory.eqToHom
CategoryTheory.Category.toCategoryStruct
——
functor_map 📖mathematical—CategoryTheory.Functor.map
CategoryTheory.Idempotents.Karoubi
HomologicalComplex
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
HomologicalComplex.instCategory
CategoryTheory.Idempotents.Karoubi.instCategory
CategoryTheory.Idempotents.instPreadditiveKaroubi
functor
Functor.map
——
functor_obj 📖mathematical—CategoryTheory.Functor.obj
CategoryTheory.Idempotents.Karoubi
HomologicalComplex
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
HomologicalComplex.instCategory
CategoryTheory.Idempotents.Karoubi.instCategory
CategoryTheory.Idempotents.instPreadditiveKaroubi
functor
Functor.obj
——
inverse_map 📖mathematical—CategoryTheory.Functor.map
HomologicalComplex
CategoryTheory.Idempotents.Karoubi
CategoryTheory.Idempotents.Karoubi.instCategory
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
CategoryTheory.Idempotents.instPreadditiveKaroubi
HomologicalComplex.instCategory
inverse
Inverse.map
——
inverse_obj 📖mathematical—CategoryTheory.Functor.obj
HomologicalComplex
CategoryTheory.Idempotents.Karoubi
CategoryTheory.Idempotents.Karoubi.instCategory
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
CategoryTheory.Idempotents.instPreadditiveKaroubi
HomologicalComplex.instCategory
inverse
Inverse.obj
——
unitIso_hom_app_f_f 📖mathematical—HomologicalComplex.Hom.f
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
CategoryTheory.Idempotents.Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
CategoryTheory.Functor.obj
CategoryTheory.Idempotents.Karoubi
CategoryTheory.Idempotents.Karoubi.instCategory
CategoryTheory.Functor.id
CategoryTheory.Functor.comp
CategoryTheory.Idempotents.instPreadditiveKaroubi
functor
inverse
CategoryTheory.Idempotents.Karoubi.Hom.f
CategoryTheory.NatTrans.app
CategoryTheory.Iso.hom
CategoryTheory.Functor
CategoryTheory.Functor.category
unitIso
CategoryTheory.Idempotents.Karoubi.p
——
unitIso_inv_app_f_f 📖mathematical—HomologicalComplex.Hom.f
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
CategoryTheory.Idempotents.Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
CategoryTheory.Functor.obj
CategoryTheory.Idempotents.Karoubi
CategoryTheory.Idempotents.Karoubi.instCategory
CategoryTheory.Functor.comp
CategoryTheory.Idempotents.instPreadditiveKaroubi
functor
inverse
CategoryTheory.Functor.id
CategoryTheory.Idempotents.Karoubi.Hom.f
CategoryTheory.NatTrans.app
CategoryTheory.Iso.inv
CategoryTheory.Functor
CategoryTheory.Functor.category
unitIso
CategoryTheory.Idempotents.Karoubi.p
——

CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor

Definitions

NameCategoryTheorems
map 📖CompOp
2 mathmath: map_f_f, CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.functor_map
obj 📖CompOp
7 mathmath: CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_functor_map_f_f, CategoryTheory.Idempotents.karoubiChainComplexEquivalence_functor_map_f_f, map_f_f, CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.functor_obj, obj_X_p, obj_d_f, obj_X_X

Theorems

NameKindAssumesProvesValidatesDepends On
map_f_f 📖mathematical—CategoryTheory.Idempotents.Karoubi.Hom.f
HomologicalComplex.X
CategoryTheory.Idempotents.Karoubi
CategoryTheory.Idempotents.Karoubi.instCategory
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
CategoryTheory.Idempotents.instPreadditiveKaroubi
obj
HomologicalComplex.Hom.f
map
CategoryTheory.Idempotents.Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
——
obj_X_X 📖mathematical—CategoryTheory.Idempotents.Karoubi.X
HomologicalComplex.X
CategoryTheory.Idempotents.Karoubi
CategoryTheory.Idempotents.Karoubi.instCategory
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
CategoryTheory.Idempotents.instPreadditiveKaroubi
obj
HomologicalComplex
HomologicalComplex.instCategory
——
obj_X_p 📖mathematical—CategoryTheory.Idempotents.Karoubi.p
HomologicalComplex.X
CategoryTheory.Idempotents.Karoubi
CategoryTheory.Idempotents.Karoubi.instCategory
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
CategoryTheory.Idempotents.instPreadditiveKaroubi
obj
HomologicalComplex.Hom.f
CategoryTheory.Idempotents.Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
——
obj_d_f 📖mathematical—CategoryTheory.Idempotents.Karoubi.Hom.f
HomologicalComplex.X
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
CategoryTheory.Idempotents.Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
HomologicalComplex.Hom.f
CategoryTheory.Idempotents.Karoubi.p
HomologicalComplex.d
CategoryTheory.Idempotents.Karoubi
CategoryTheory.Idempotents.Karoubi.instCategory
CategoryTheory.Idempotents.instPreadditiveKaroubi
obj
CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
——

CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse

Definitions

NameCategoryTheorems
map 📖CompOp
2 mathmath: map_f_f, CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.inverse_map
obj 📖CompOp
7 mathmath: map_f_f, obj_p_f, obj_X_d, CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_inverse_map_f_f, CategoryTheory.Idempotents.karoubiChainComplexEquivalence_inverse_map_f_f, obj_X_X, CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.inverse_obj

Theorems

NameKindAssumesProvesValidatesDepends On
map_f_f 📖mathematical—HomologicalComplex.Hom.f
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
CategoryTheory.Idempotents.Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
obj
CategoryTheory.Idempotents.Karoubi.Hom.f
map
HomologicalComplex.X
CategoryTheory.Idempotents.Karoubi
CategoryTheory.Idempotents.Karoubi.instCategory
CategoryTheory.Idempotents.instPreadditiveKaroubi
——
obj_X_X 📖mathematical—HomologicalComplex.X
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
CategoryTheory.Idempotents.Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
obj
CategoryTheory.Idempotents.Karoubi
CategoryTheory.Idempotents.Karoubi.instCategory
CategoryTheory.Idempotents.instPreadditiveKaroubi
——
obj_X_d 📖mathematical—HomologicalComplex.d
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
CategoryTheory.Idempotents.Karoubi.X
HomologicalComplex
HomologicalComplex.instCategory
obj
CategoryTheory.Idempotents.Karoubi.Hom.f
HomologicalComplex.X
CategoryTheory.Idempotents.Karoubi
CategoryTheory.Idempotents.Karoubi.instCategory
CategoryTheory.Idempotents.instPreadditiveKaroubi
——
obj_p_f 📖mathematical—HomologicalComplex.Hom.f
CategoryTheory.Preadditive.preadditiveHasZeroMorphisms
CategoryTheory.Idempotents.Karoubi.X
HomologicalComplex.X
CategoryTheory.Idempotents.Karoubi
CategoryTheory.Idempotents.Karoubi.instCategory
CategoryTheory.Idempotents.instPreadditiveKaroubi
CategoryTheory.Idempotents.Karoubi.Hom.f
HomologicalComplex.d
CategoryTheory.Idempotents.Karoubi.p
HomologicalComplex
HomologicalComplex.instCategory
obj
——

(root)

Definitions

NameCategoryTheorems
HomologicalComplex 📖CompData
850 mathmath: HomotopyCategory.spectralObjectMappingCone_δ'_app, CategoryTheory.ShortComplex.ShortExact.instHasSmallLocalizedShiftedHomHomologicalComplexIntUpQuasiIsoX₃CochainComplexMapSingleFunctorOfNatX₁, HomologicalComplex.eqToHom_f, CochainComplex.mappingConeCompTriangleh_comm₁_assoc, HomologicalComplex.extendSingleIso_inv_f, HomologicalComplex.evalCompCoyonedaCorepresentableByDoubleId_homEquiv_apply, HomologicalComplex.singleMapHomologicalComplex_hom_app_ne, homotopyEquivalences_le_quasiIso, CategoryTheory.NatIso.mapHomologicalComplex_inv_app_f, HomologicalComplex.instPreservesLimitsOfShapeSingle, HomologicalComplex.rightUnitor'_inv, ComplexShape.Embedding.truncLEFunctor_obj, CategoryTheory.Functor.mapHomologicalComplexIdIso_hom_app_f, HomotopyCategory.quotient_map_out_comp_out, HomologicalComplex.isZero_single_obj_X, HomologicalComplex₂.totalFlipIso_hom_f_D₁, HomologicalComplex.singleMapHomologicalComplex_hom_app_self, HomologicalComplex.πTruncGE_naturality_assoc, HomologicalComplex.singleObjHomologySelfIso_hom_singleObjOpcyclesSelfIso_hom_assoc, HomologicalComplex.truncGE.rightHomologyMapData_φQ, CategoryTheory.Idempotents.karoubiChainComplexEquivalence_inverse_obj_X_d, HomologicalComplex.singleObjCyclesSelfIso_hom_singleObjOpcyclesSelfIso_hom_assoc, HomologicalComplex.opcyclesMap_comp, HomologicalComplex.truncLEMap_comp_assoc, CategoryTheory.Functor.mapBifunctorHomologicalComplexObj_map_f_f, CategoryTheory.Functor.mapHomologicalComplexUpToQuasiIsoFactorsh_hom_app_assoc, HomologicalComplex.instPreservesColimitsOfShapeSingle, HomologicalComplex.pOpcycles_singleObjOpcyclesSelfIso_inv, quasiIsoAt_iff_comp_right, HomologicalComplex.isZero_single_obj_homology, CategoryTheory.InjectiveResolution.isoRightDerivedToHomotopyCategoryObj_inv_naturality_assoc, HomologicalComplex.cyclesFunctor_map, CategoryTheory.Idempotents.karoubiChainComplexEquivalence_functor_obj_X_X, HomologicalComplex.singleObjOpcyclesSelfIso_hom, HomologicalComplex.singleObjCyclesSelfIso_inv_iCycles, HomologicalComplex.instIsStrictlySupportedOfNat, HomologicalComplex₂.totalShift₂Iso_hom_naturality_assoc, CategoryTheory.InjectiveResolution.ι'_f_zero, CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_functor_map_f_f, HomologicalComplex.opcyclesMapIso_inv, HomologicalComplex.biprod_inr_desc_f, CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.map_f_f, HomologicalComplex.biprod_lift_fst_f_assoc, HomologicalComplex.homologyι_singleObjOpcyclesSelfIso_inv_assoc, ComplexShape.Embedding.truncGE'Functor_obj, Homotopy.nullHomotopicMap_comp, HomologicalComplex.shortComplexFunctor'_obj_X₁, HomologicalComplex.homotopyCofiber.inrCompHomotopy_hom_desc_hom_assoc, HomologicalComplex₂.shiftFunctor₂XXIso_refl, CochainComplex.IsKInjective.nonempty_homotopy_zero, HomologicalComplex.unopFunctor_obj, CategoryTheory.ProjectiveResolution.iso_hom_naturality_assoc, CochainComplex.instIsStrictlyLEObjHomologicalComplexIntUpSingle, CategoryTheory.InjectiveResolution.iso_hom_naturality, HomologicalComplex₂.comm_f, DerivedCategory.instCommShiftHomologicalComplexIntUpHomFunctorQuotientCompQhIso, CategoryTheory.InjectiveResolution.toRightDerivedZero'_naturality_assoc, CategoryTheory.Functor.mapHomologicalComplex_linear, CategoryTheory.InjectiveResolution.isoRightDerivedToHomotopyCategoryObj_inv_naturality, CategoryTheory.ProjectiveResolution.homotopyEquiv_inv_π_assoc, HomotopyCategory.isZero_quotient_obj_iff, CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence_functor, HomologicalComplex.restrictionMap_comp, CategoryTheory.Functor.mapHomotopyEquiv_inv, AlgebraicTopology.DoldKan.homotopyPInftyToId_hom, CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.unitIso_inv_app_f_f, CochainComplex.mappingCone.homologySequenceδ_triangleh, CategoryTheory.Functor.mapHomologicalComplex_obj_d, HomologicalComplex₂.total.mapIso_hom, HomologicalComplex.instPreservesFiniteLimitsEvalOfHasFiniteLimits, HomologicalComplex.opcyclesOpIso_inv_naturality_assoc, HomologicalComplex.inl_biprodXIso_inv_assoc, HomologicalComplex.cyclesMap_comp_assoc, CochainComplex.mapBifunctorShift₂Iso_hom_naturality₂_assoc, HomologicalComplex.instIsStableUnderRetractsQuasiIso, HomologicalComplex.biprod_inr_snd_f_assoc, CategoryTheory.Functor.mapHomologicalComplex_map_f, CochainComplex.IsKProjective.homotopyZero_def, HomologicalComplex.biprod_inl_snd_f_assoc, CategoryTheory.ProjectiveResolution.homotopyEquiv_hom_π, HomologicalComplex₂.totalAux.d₂_eq, HomologicalComplex.instIsNormalEpiCategory, HomologicalComplex₂.D₂_totalShift₁XIso_hom_assoc, CategoryTheory.Functor.mapHomotopyEquiv_homotopyHomInvId, HomologicalComplex.eval_map, HomologicalComplex.map_isStrictlySupported, HomologicalComplex.stupidTruncMap_comp, HomologicalComplex.to_single_hom_ext_iff, CochainComplex.HomComplex.Cochain.map_v, CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_inverse_obj_p_f, HomologicalComplex.biprodXIso_hom_fst, HomologicalComplex₂.D₂_totalShift₂XIso_hom_assoc, HomologicalComplex₂.d_f_comp_d_f, HomologicalComplexUpToQuasiIso.instIsLocalizationHomologicalComplexCompHomotopyCategoryQuotientQhQuasiIso, HomologicalComplex₂.flipEquivalence_unitIso, HomologicalComplex.add_f_apply, HomologicalComplex.extend_op_d_assoc, CategoryTheory.InjectiveResolution.toRightDerivedZero'_comp_iCycles, CategoryTheory.Functor.map_homogical_complex_additive, CategoryTheory.Idempotents.Karoubi.HomologicalComplex.comp_p_d, HomologicalComplex₂.d₁_eq_zero', HomologicalComplex.singleObjCyclesSelfIso_inv_homologyπ, CochainComplex.mappingCone.rotateHomotopyEquiv_comm₂_assoc, CochainComplex.mapBifunctorShift₁Iso_hom_naturality₁_assoc, HomologicalComplex.unopInverse_map, CategoryTheory.NatTrans.mapHomologicalComplex_app_f, CochainComplex.shiftShortComplexFunctorIso_add'_hom_app, CochainComplex.HomComplex.Cochain.map_zero, HomologicalComplex.natIsoSc'_inv_app_τ₂, CochainComplex.instIsCompatibleWithShiftHomologicalComplexIntUpQuasiIso, CochainComplex.HomComplex.Cochain.toSingleMk_v, HomologicalComplex.unopEquivalence_functor, CochainComplex.IsKInjective.rightOrthogonal, HomologicalComplex.single_obj_d, HomotopyCategory.instAdditiveHomologicalComplexQuotientHomotopicFunctor, HomotopyCategory.quotient_inverts_homotopyEquivalences, CochainComplex.IsKInjective.Qh_map_bijective, HomologicalComplex.instFaithfulGradedObjectForget, HomologicalComplex.biprod_lift_fst_f, HomologicalComplex.ιTruncLE_naturality, HomologicalComplex₂.toGradedObjectFunctor_obj, HomologicalComplex.forget_map, CategoryTheory.NatTrans.mapHomologicalComplex_id, HomologicalComplex₂.ιTotalOrZero_map_assoc, HomologicalComplex.asFunctor_obj_X, HomologicalComplex₂.ιTotal_map, CategoryTheory.NatTrans.mapHomotopyCategory_app, HomologicalComplex₂.ι_totalShift₁Iso_hom_f_assoc, CategoryTheory.Functor.mapProjectiveResolution_π, HomologicalComplex.cyclesMap_id, HomologicalComplex.complexOfFunctorsToFunctorToComplex_obj, HomologicalComplex.opcyclesOpIso_inv_naturality, CategoryTheory.instIsIsoToRightDerivedZero', HomologicalComplex₂.ι_D₁_assoc, HomologicalComplex.opEquivalence_unitIso, CochainComplex.instIsStrictlyGEObjHomologicalComplexIntUpSingle, HomologicalComplex.forgetEval_hom_app, HomologicalComplex.homologyι_singleObjOpcyclesSelfIso_inv, HomologicalComplex₂.ιTotal_totalFlipIso_f_inv_assoc, HomologicalComplex.dgoToHomologicalComplex_obj_d, HomotopyCategory.quotient_map_eq_zero_iff, HomologicalComplex.shortComplexFunctor'_map_τ₂, HomologicalComplex.isZero_zero, CategoryTheory.Abelian.LeftResolution.exactAt_map_chainComplex_succ, HomotopyCategory.instFullHomologicalComplexQuotient, CategoryTheory.Idempotents.karoubiChainComplexEquivalence_functor_obj_X_p, HomologicalComplex₂.d₂_eq_zero, HomologicalComplex₂.ι_totalShift₁Iso_inv_f, HomologicalComplex.extendSingleIso_inv_f_assoc, HomologicalComplex.natIsoSc'_inv_app_τ₃, HomologicalComplex₂.ι_totalDesc_assoc, CategoryTheory.ProjectiveResolution.isoLeftDerivedObj_hom_naturality_assoc, HomologicalComplex.truncGEMap_comp_assoc, HomotopyCategory.isoOfHomotopyEquiv_hom, CategoryTheory.Idempotents.Karoubi.HomologicalComplex.p_comm_f_assoc, HomologicalComplex.dgoEquivHomologicalComplex_unitIso, CategoryTheory.ProjectiveResolution.isoLeftDerivedToHomotopyCategoryObj_inv_naturality_assoc, CategoryTheory.Functor.mapHomologicalComplex_commShiftIso_eq, HomologicalComplex.mapBifunctorFlipIso_hom_naturality_assoc, HomologicalComplex.homotopyCofiber.inrCompHomotopy_hom, CategoryTheory.Equivalence.mapHomologicalComplex_unitIso, CategoryTheory.Idempotents.karoubiChainComplexEquivalence_inverse_obj_p_f, HomologicalComplex.opFunctor_obj, CategoryTheory.ShortComplex.ShortExact.singleTriangleIso_hom_hom₁, CategoryTheory.Functor.mapBifunctorHomologicalComplexObj_obj_X_d, HomologicalComplex.natIsoSc'_hom_app_τ₃, HomologicalComplex.instQuasiIsoAtMapOppositeSymmUnopFunctorOp, HomologicalComplex.truncGE'Map_comp, CategoryTheory.InjectiveResolution.toRightDerivedZero_eq, CategoryTheory.Idempotents.instIsIdempotentCompleteHomologicalComplex, ChainComplex.single₀ObjXSelf, HomologicalComplex.instEpiGShortComplexTruncLE, ComplexShape.Embedding.restrictionFunctor_map, HomologicalComplex.cyclesOpIso_inv_naturality_assoc, HomologicalComplex.singleObjHomologySelfIso_hom_singleObjOpcyclesSelfIso_hom, CategoryTheory.ProjectiveResolution.isoLeftDerivedObj_inv_naturality_assoc, CochainComplex.mappingConeCompHomotopyEquiv_comm₂_assoc, HomologicalComplex.Hom.sqFrom_comp, CochainComplex.mappingConeCompHomotopyEquiv_hom_inv_id, HomologicalComplexUpToQuasiIso.Q_inverts_homotopyEquivalences, HomologicalComplex.shortComplexFunctor_obj_X₂, HomologicalComplex.singleObjCyclesSelfIso_hom_naturality, HomologicalComplex.cylinder.πCompι₀Homotopy.nullHomotopicMap_eq, HomologicalComplex.singleObjCyclesSelfIso_inv_naturality_assoc, HomologicalComplex.sub_f_apply, instIsLocalizationHomologicalComplexIntUpHomotopyCategoryQuotientHomotopyEquivalences, HomotopyCategory.eq_of_homotopy, HomologicalComplex.homotopyCofiber.descSigma_ext_iff, HomologicalComplex.instRespectsIsoQuasiIso, Homotopy.map_nullHomotopicMap', CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence_unitIso, HomologicalComplex₂.flipEquivalenceUnitIso_hom_app_f_f, CategoryTheory.Idempotents.Karoubi.HomologicalComplex.p_comp_d_assoc, HomologicalComplex.opcyclesMap_id, CategoryTheory.Functor.mapProjectiveResolution_complex, CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence_inverse, HomologicalComplex₂.ι_totalShift₂Iso_inv_f_assoc, HomologicalComplex₂.d_comm, Rep.standardComplex.εToSingle₀_comp_eq, HomologicalComplex.inr_biprodXIso_inv, CategoryTheory.Idempotents.Karoubi.HomologicalComplex.comp_p_d_assoc, HomologicalComplex.stupidTruncMap_id, HomologicalComplex.homotopyCofiber.inrCompHomotopy_hom_desc_hom, HomologicalComplex₂.d_f_comp_d_f_assoc, CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.counitIso_inv, HomologicalComplex.instHasFilteredColimitsOfSize, HomologicalComplex.Hom.isoOfComponents_inv_f, CochainComplex.homologyFunctor_shift, HomologicalComplex.cyclesOpNatIso_inv_app, HomologicalComplex.opcyclesFunctor_map, HomologicalComplex.gradedHomologyFunctor_map, HomologicalComplex.quasiIso_iff_evaluation, quasiIsoAt_comp, AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex_homotopyHomInvId, HomologicalComplex₂.ι_D₂, HomologicalComplex.homologyMap_neg, HomologicalComplex.isZero_stupidTrunc_iff, CategoryTheory.ProjectiveResolution.isoLeftDerivedToHomotopyCategoryObj_hom_naturality_assoc, HomologicalComplex.from_single_hom_ext_iff, Homotopy.nullHomotopicMap'_comp, HomologicalComplex.natTransHomologyπ_app, HomologicalComplex₂.total.mapAux.d₁_mapMap, HomologicalComplex.singleObjHomologySelfIso_hom_singleObjHomologySelfIso_inv_assoc, HomologicalComplex.dgoEquivHomologicalComplexUnitIso_hom_app_f, CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_unitIso_hom_app_f_f, CategoryTheory.NatTrans.mapHomologicalComplex_comp, HomologicalComplex.opcyclesMapIso_hom, CategoryTheory.InjectiveResolution.instIsIsoToRightDerivedZero'Self, HomologicalComplex₂.d₁_eq, HomologicalComplex.cyclesMap_comp, CategoryTheory.Idempotents.karoubiChainComplexEquivalence_functor_map_f_f, HomologicalComplex.homologyπ_singleObjHomologySelfIso_hom_assoc, HomologicalComplex₂.instHasTotalIntObjUpShiftFunctor₂, HomologicalComplex.instQuasiIsoMapOppositeSymmUnopFunctorOp, HomologicalComplex.single_obj_X_self, HomologicalComplex.biprod_inr_snd_f, HomologicalComplex.id_f, HomologicalComplex.singleMapHomologicalComplex_inv_app_self, HomologicalComplex.dgoEquivHomologicalComplex_functor, ComplexShape.Embedding.instFaithfulHomologicalComplexExtendFunctor, HomologicalComplex₂.XXIsoOfEq_hom_ιTotal_assoc, HomologicalComplex.extendSingleIso_hom_f_assoc, HomologicalComplex.homologyMapIso_hom, HomologicalComplex.instHasHomologyObjSingle, HomologicalComplex.instIsCorepresentableCompEvalObjOppositeFunctorTypeCoyonedaOp, CategoryTheory.ProjectiveResolution.isoLeftDerivedObj_hom_naturality, CategoryTheory.Functor.mapHomologicalComplex_obj_X, CochainComplex.mappingCone.rotateHomotopyEquiv_comm₂, HomologicalComplex.ιTruncLE_naturality_assoc, HomologicalComplex.instMonoFShortComplexTruncLE, CochainComplex.HomComplex.Cochain.map_add, CochainComplex.homologySequenceδ_quotient_mapTriangle_obj_assoc, CochainComplex.HomComplex.Cochain.fromSingleMk_v, CategoryTheory.Functor.mapHomologicalComplex_upToQuasiIso_Q_inverts_quasiIso, CategoryTheory.Functor.mapHomologicalComplex_reflects_iso, HomologicalComplex₂.instHasTotalIntObjUpCompShiftFunctor₁ShiftFunctor₂, HomologicalComplex.opcyclesOpIso_hom_naturality_assoc, HomologicalComplex₂.XXIsoOfEq_rfl, CochainComplex.IsKInjective.homotopyZero_def, HomologicalComplex.dgoToHomologicalComplex_obj_X, HomotopyCategory.instEssSurjHomologicalComplexQuotient, HomotopyCategory.quasiIso_eq_quasiIso_map_quotient, HomologicalComplex.extendMap_comp_assoc, HomologicalComplex.restrictionMap_id, HomologicalComplex₂.ι_D₂_assoc, ComplexShape.Embedding.AreComplementary.hom_ext, HomologicalComplex.isIso_homologyMap_shortComplexTruncLE_g, HomotopyCategory.quot_mk_eq_quotient_map, HomologicalComplex.units_smul_f_apply, HomologicalComplex.quasiIsoAt_unopFunctor_map_iff, CategoryTheory.ProjectiveResolution.pOpcycles_comp_fromLeftDerivedZero'_assoc, CategoryTheory.NatIso.mapHomologicalComplex_hom_app_f, CategoryTheory.Functor.mapBifunctorHomologicalComplexObj_obj_X_X, CategoryTheory.ShortComplex.ShortExact.singleTriangleIso_inv_hom₂, ChainComplex.quasiIsoAt₀_iff, HomologicalComplex.singleObjCyclesSelfIso_hom_singleObjOpcyclesSelfIso_hom, CategoryTheory.InjectiveResolution.isoRightDerivedObj_hom_naturality, HomologicalComplex₂.totalFunctor_obj, CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.inverse_map, HomologicalComplex.mono_of_mono_f, groupHomology.isoShortComplexH1_hom, HomologicalComplex.isSeparator_coproduct_separatingFamily, HomologicalComplex₂.totalShift₁Iso_hom_totalShift₂Iso_hom, CochainComplex.instLinearHomologicalComplexIntUpShiftFunctor, CategoryTheory.ProjectiveResolution.homotopyEquiv_inv_π, HomologicalComplex.opFunctor_map_f, HomologicalComplex.instIsMultiplicativeQuasiIso, HomologicalComplex.instIsIsoπTruncGEOfIsStrictlySupported, CategoryTheory.ProjectiveResolution.instIsIsoFromLeftDerivedZero'Self, HomologicalComplex₂.flipFunctor_obj, CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.map_f_f, CategoryTheory.Functor.instCommShiftCochainComplexIntDerivedCategoryHomMapDerivedCategoryFactors, HomologicalComplex₂.totalAux.d₂_eq', CategoryTheory.ProjectiveResolution.π'_f_zero_assoc, CategoryTheory.ProjectiveResolution.iso_hom_naturality, CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.obj_p_f, HomologicalComplex₂.totalAux.d₁_eq, ComplexShape.Embedding.homRestrict_precomp, HomologicalComplex.instPreservesZeroMorphismsEval, HomologicalComplex₂.D₁_totalShift₂XIso_hom_assoc, HomologicalComplex₂.shiftFunctor₁XXIso_refl, HomologicalComplex.homologyMap_id, HomologicalComplex.instHasLimitDiscreteWalkingPairCompPairEval, HomologicalComplex.instQuasiIsoAtOppositeMapSymmOpFunctorOp, HomologicalComplex₂.ι_totalDesc, ComplexShape.Embedding.extendFunctor_map, HomologicalComplexUpToQuasiIso.Q_map_eq_of_homotopy, HomologicalComplex₂.total.mapAux.d₂_mapMap_assoc, CochainComplex.ConnectData.restrictionLEIso_inv_f, HomologicalComplex.ι_mapBifunctorFlipIso_inv_assoc, HomologicalComplex.extend_single_d, ComplexShape.Embedding.stupidTruncFunctor_map, HomologicalComplex.singleObjHomologySelfIso_inv_homologyι_assoc, CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.functor_obj, Homotopy.map_nullHomotopicMap, HomologicalComplex₂.d₁_eq_zero, HomologicalComplex.zsmul_f_apply, HomologicalComplex₂.flip_X_X, CategoryTheory.ShortComplex.ShortExact.singleTriangleIso_inv_hom₃, HomotopyCategory.instFullFunctorHomologicalComplexObjWhiskeringLeftQuotient, HomologicalComplex.biprod_lift_snd_f, CategoryTheory.ProjectiveResolution.pOpcycles_comp_fromLeftDerivedZero', HomologicalComplex.homologyMap_add, HomologicalComplex.Hom.isoOfComponents_hom_f, CochainComplex.quasiIso_shift_iff, HomologicalComplex.cylinder.ι₁_π, HomologicalComplex.singleObjCyclesSelfIso_inv_homologyπ_assoc, HomologicalComplex.biprod_inl_fst_f_assoc, CategoryTheory.Functor.map₂HomologicalComplex_map_app, HomologicalComplex.unopEquivalence_unitIso, HomologicalComplex.nsmul_f_apply, HomologicalComplex.g_shortComplexTruncLEX₃ToTruncGE, HomologicalComplex.homotopyCofiber.inr_desc, CategoryTheory.Idempotents.karoubiChainComplexEquivalence_functor_obj_d_f, CategoryTheory.Functor.mapBifunctorHomologicalComplex_map_app_f_f, HomologicalComplex.instHasLimitsOfShape, CochainComplex.mapBifunctorHomologicalComplexShift₂Iso_inv_f_f, HomologicalComplex.opEquivalence_counitIso, HomotopyCategory.quotient_obj_surjective, HomologicalComplex.inl_biprodXIso_inv, AlgebraicTopology.DoldKan.compatibility_N₂_N₁_karoubi, CochainComplex.mappingCone.map_inr, CategoryTheory.Functor.mapCochainComplexShiftIso_inv_app_f, HomologicalComplex.homotopyCofiber.inr_desc_assoc, HomologicalComplex.mapBifunctorFlipIso_hom_naturality, HomologicalComplex.mono_homologyMap_shortComplexTruncLE_g, HomologicalComplex₂.D₁_totalShift₁XIso_hom_assoc, HomologicalComplex.truncLEMap_comp, HomologicalComplex.cylinder.inrX_π_assoc, CochainComplex.mapBifunctorShift₂Iso_hom_naturality₂, HomologicalComplex.homologyOp_hom_naturality_assoc, HomologicalComplex.homologyMap_comp, HomologicalComplex.homotopyCofiber.inlX_desc_f_assoc, HomologicalComplex.biprod_inl_fst_f, HomologicalComplex.instAdditiveHomologyFunctor, HomologicalComplex.eval_obj, HomologicalComplex.ab5OfSize, HomologicalComplex.shortComplexFunctor_map_τ₁, HomologicalComplex₂.instHasTotalIntObjUpCompShiftFunctor₂ShiftFunctor₁, CategoryTheory.InjectiveResolution.toRightDerivedZero'_naturality, HomologicalComplex.comp_f, HomologicalComplex.cylinder.ι₀_π, CategoryTheory.Functor.mapHomotopy_hom, HomologicalComplex.singleObjOpcyclesSelfIso_inv_naturality, HomologicalComplex.isIso_ιTruncLE_iff, CategoryTheory.Functor.mapBifunctorHomologicalComplex_obj_obj_X_d, HomologicalComplex.cyclesOpIso_inv_naturality, quasiIso_iff_comp_left, HomologicalComplex.biprod_inr_fst_f, HomologicalComplex.shortComplexFunctor'_obj_X₂, CategoryTheory.ProjectiveResolution.fromLeftDerivedZero'_naturality, HomologicalComplex.opcyclesOpIso_hom_naturality, HomologicalComplex.dgoEquivHomologicalComplex_counitIso, HomologicalComplex.homotopyCofiber.eq_desc, HomologicalComplex.truncLE'Map_id, CategoryTheory.Functor.mapHomologicalComplexIdIso_inv_app_f, Homotopy.compRight_hom, ComplexShape.Embedding.homRestrict_comp_extendMap_assoc, CochainComplex.mapBifunctorHomologicalComplexShift₂Iso_hom_f_f, CochainComplex.IsKProjective.nonempty_homotopy_zero, HomologicalComplex.instHasHomologyOppositeObjSymmOpFunctorOp, HomologicalComplex.biprodXIso_hom_fst_assoc, HomologicalComplex₂.flipEquivalenceUnitIso_inv_app_f_f, HomologicalComplex.opcyclesMap_comp_assoc, HomologicalComplex.quasiIsoAt_shortComplexTruncLE_g, HomotopyCategory.instFaithfulFunctorHomologicalComplexObjWhiskeringLeftQuotient, HomologicalComplex₂.d₂_eq_zero', CategoryTheory.ProjectiveResolution.leftDerived_app_eq, HomologicalComplex.shortComplexFunctor_map_τ₃, HomologicalComplex₂.ι_totalShift₁Iso_inv_f_assoc, HomologicalComplex₂.ιTotal_totalFlipIso_f_hom_assoc, HomologicalComplex.shortComplexFunctor_obj_f, HomologicalComplex.zero_f, HomologicalComplex₂.XXIsoOfEq_inv_ιTotal_assoc, HomologicalComplex.cylinder.ι₀_π_assoc, HomologicalComplex.homotopyCofiber.inrCompHomotopy_hom_eq_zero, CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.obj_X_d, HomologicalComplex.homologyFunctor_map, HomologicalComplex.cylinder.ι₁_π_assoc, CategoryTheory.HasExt.hasSmallLocalizedShiftedHom_of_isLE_of_isGE, CochainComplex.quasiIsoAt_shift_iff, HomologicalComplex₂.total.mapIso_inv, HomologicalComplex.shortComplexTruncLE_f, HomologicalComplex.extend_op_d, CategoryTheory.HasExt.instHasSmallLocalizedShiftedHomHomologicalComplexIntUpQuasiIsoOfIsGEOfIsLEOfNat, CategoryTheory.ProjectiveResolution.homotopyEquiv_hom_π_assoc, HomologicalComplex.shortComplexFunctor'_obj_X₃, HomologicalComplex.complexOfFunctorsToFunctorToComplex_map_app_f, CochainComplex.HomComplex.Cochain.map_comp, HomologicalComplex.cylinder.ι₀_desc, HomologicalComplex.opInverse_obj, HomologicalComplex.isGrothendieckAbelian, HomologicalComplex.mkHomToSingle_f, HomologicalComplex.biprodXIso_hom_snd, CategoryTheory.Idempotents.karoubiChainComplexEquivalence_inverse_obj_X_X, CategoryTheory.ProjectiveResolution.fromLeftDerivedZero'_naturality_assoc, CochainComplex.HomComplex.Cochain.map_sub, ComplexShape.Embedding.ιTruncLENatTrans_app, HomologicalComplex.shortComplexFunctor'_obj_g, HomologicalComplex.natIsoSc'_hom_app_τ₂, CategoryTheory.Idempotents.Karoubi.HomologicalComplex.p_idem, CategoryTheory.Functor.mapHomologicalComplexUpToQuasiIsoLocalizerMorphism_functor, CochainComplex.IsKProjective.Qh_map_bijective, CategoryTheory.Equivalence.mapHomologicalComplex_functor, CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_counitIso_hom, HomologicalComplex.cyclesOpIso_hom_naturality_assoc, HomologicalComplex₂.flipEquivalence_counitIso, HomologicalComplex₂.ι_totalShift₂Iso_hom_f_assoc, CategoryTheory.NatTrans.mapHomologicalComplex_naturality, HomologicalComplex.biprodXIso_hom_snd_assoc, quasiIso_comp, HomologicalComplex.instPreservesZeroMorphismsHomologyFunctor, CategoryTheory.ProjectiveResolution.π'_f_zero, HomologicalComplex.leftUnitor'_inv, HomologicalComplex₂.totalFlipIso_hom_f_D₂, HomologicalComplex.forget_obj, QuasiIsoAt.quasiIso, HomologicalComplex.isSeparating_separatingFamily, Homotopy.comp_nullHomotopicMap, HomologicalComplex.single_map_f_self_assoc, HomologicalComplex.isIso_πTruncGE_iff, HomologicalComplex.homologicalComplexToDGO_map_f, CategoryTheory.Functor.mapHomologicalComplex_commShiftIso_inv_app_f, ComplexShape.Embedding.instFullHomologicalComplexExtendFunctor, CochainComplex.exists_iso_single, groupHomology.isoShortComplexH1_inv, CochainComplex.ConnectData.restrictionLEIso_hom_f, CategoryTheory.Idempotents.Karoubi.HomologicalComplex.p_comp_d, CategoryTheory.ShortComplex.ShortExact.singleTriangleIso_hom_hom₃, HomologicalComplex₂.D₁_totalShift₁XIso_hom, HomologicalComplex.πTruncGE_naturality, HomologicalComplex₂.D₂_totalShift₁XIso_hom, CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_functor_obj_X_p, ComplexShape.Embedding.instAdditiveHomologicalComplexRestrictionFunctor, ComplexShape.Embedding.homEquiv_symm_apply, HomologicalComplex₂.comm_f_assoc, HomologicalComplex.cylinder.inrX_π, CategoryTheory.ShortComplex.ShortExact.singleTriangleIso_inv_hom₁, HomologicalComplex.homologyMap_sub, HomologicalComplex₂.totalFlipIso_hom_f_D₁_assoc, Homotopy.comp_nullHomotopicMap', HomologicalComplex.instPreservesColimitsOfShapeEvalOfHasColimitsOfShape, CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_inverse_map_f_f, ComplexShape.Embedding.homEquiv_apply_coe, HomologicalComplex₂.XXIsoOfEq_inv_ιTotal, HomologicalComplex.cylinder.πCompι₀Homotopy.inrX_nullHomotopy_f, HomologicalComplex.homologyπ_singleObjHomologySelfIso_hom, CategoryTheory.Functor.mapHomotopyEquiv_homotopyInvHomId, CategoryTheory.InjectiveResolution.iso_hom_naturality_assoc, HomologicalComplex₂.total.map_comp, HomologicalComplex.cyclesOpIso_hom_naturality, CochainComplex.mappingConeCompHomotopyEquiv_comm₂, CategoryTheory.Functor.map₂HomologicalComplex_obj_obj, HomologicalComplex.instHasColimitsOfShape, AlgebraicTopology.map_alternatingFaceMapComplex, CategoryTheory.InjectiveResolution.isoRightDerivedObj_hom_naturality_assoc, CategoryTheory.Idempotents.karoubiChainComplexEquivalence_inverse_map_f_f, HomologicalComplex.smul_f_apply, CochainComplex.HomComplex.CohomologyClass.toHom_bijective, CategoryTheory.InjectiveResolution.isoRightDerivedObj_inv_naturality, HomologicalComplex₂.ofGradedObject_X_d, HomologicalComplex.singleObjCyclesSelfIso_inv_iCycles_assoc, HomologicalComplex₂.flipEquivalence_functor, Homotopy.compLeftId_hom, HomologicalComplex.singleObjOpcyclesSelfIso_hom_naturality_assoc, HomologicalComplex.zero_f_apply, CochainComplex.mappingCone.rotateHomotopyEquiv_comm₃_assoc, HomotopyCategory.instIsCompatibleWithShiftHomologicalComplexIntUpHomotopic, HomologicalComplex.natTransOpCyclesToCycles_app, Homotopy.add_hom, CategoryTheory.InjectiveResolution.ι'_f_zero_assoc, Rep.standardComplex.quasiIso_forget₂_εToSingle₀, HomologicalComplex.instMonoιTruncLE, HomologicalComplex.inr_biprodXIso_inv_assoc, HomologicalComplex.singleMapHomologicalComplex_inv_app_ne, CategoryTheory.instHasSmallLocalizedShiftedHomHomologicalComplexIntUpQuasiIsoObjCochainComplexCompSingleFunctorOfNatOfHasExt, HomologicalComplex.truncLE'ToRestriction_naturality_assoc, HomologicalComplex.dgoToHomologicalComplex_map_f, ComplexShape.Embedding.instPreservesZeroMorphismsHomologicalComplexExtendFunctor, HomologicalComplex.shortComplexFunctor'_map_τ₁, HomologicalComplexUpToQuasiIso.Qh_inverts_quasiIso, HomologicalComplex₂.ofGradedObject_X_X, instIsLocalizationHomologicalComplexDownHomotopyCategoryQuotientHomotopyEquivalences, HomologicalComplex.cyclesMap_zero, HomologicalComplex₂.flipEquivalence_inverse, HomologicalComplex.single_map_f_self, CochainComplex.instQuasiIsoIntMapHomologicalComplexUpShiftFunctor, HomologicalComplex.homologyMap_zero, HomologicalComplex.mapBifunctorFlipIso_flip, CategoryTheory.Equivalence.mapHomologicalComplex_counitIso, CategoryTheory.InjectiveResolution.isoRightDerivedToHomotopyCategoryObj_hom_naturality_assoc, CategoryTheory.Functor.leftDerived_map_eq, Homotopy.compRightId_hom, HomologicalComplex.extendMap_add, HomologicalComplex.cyclesFunctor_obj, groupHomology.isoShortComplexH2_hom, HomologicalComplex.shortComplexTruncLE_shortExact, CochainComplex.IsKProjective.leftOrthogonal, HomologicalComplex.instHasTwoOutOfThreePropertyQuasiIso, HomologicalComplex.singleCompEvalIsoSelf_inv_app, HomologicalComplex.asFunctor_obj_d, ComplexShape.Embedding.truncLE'Functor_map, HomologicalComplex.shortComplexFunctor_obj_X₁, HomologicalComplex₂.ιTotalOrZero_map, HomologicalComplex.restrictionMap_comp_assoc, HomotopyCategory.quotient_obj_as, HomologicalComplex₂.totalShift₁Iso_trans_totalShift₂Iso, HomologicalComplex.shortComplexTruncLE_X₂, HomologicalComplex₂.d₂_eq, CategoryTheory.ProjectiveResolution.leftDerivedToHomotopyCategory_app_eq, HomologicalComplex.truncLEMap_id, ComplexShape.Embedding.πTruncGENatTrans_app, HomologicalComplex.instHasFiniteColimits, ComplexShape.Embedding.truncLEFunctor_map, CochainComplex.isKProjective_iff_leftOrthogonal, HomologicalComplex₂.flip_totalFlipIso, HomologicalComplex.quasiIsoAt_iff_evaluation, HomologicalComplex.truncGE.rightHomologyMapData_φH, groupCohomology.isoShortComplexH1_hom, HomologicalComplex₂.total.map_comp_assoc, quasiIsoAt_iff', HomologicalComplex.homologyFunctorSingleIso_inv_app, HomologicalComplex.instPreservesLimitsOfShapeEvalOfHasLimitsOfShape, HomologicalComplex₂.shape_f, homotopy_congruence, ComplexShape.Embedding.truncLE'Functor_obj, HomologicalComplexUpToQuasiIso.instIsLocalizationHomotopyCategoryQhQuasiIso, AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex_homotopyInvHomId, CochainComplex.quasiIsoAt₀_iff, HomologicalComplex.singleObjHomologySelfIso_hom_naturality, HomologicalComplex.gradedHomologyFunctor_obj, CochainComplex.mappingCone.trianglehMapOfHomotopy_hom₃, HomologicalComplex.cylinder.ι₁_desc_assoc, HomologicalComplex.dgoEquivHomologicalComplexCounitIso_hom_app_f, CategoryTheory.Idempotents.Karoubi.HomologicalComplex.p_idem_assoc, HomologicalComplex.instQuasiIsoOppositeMapSymmOpFunctorOp, CochainComplex.ConnectData.restrictionGEIso_inv_f, HomologicalComplex.restrictionToTruncGE'_naturality_assoc, HomologicalComplex.singleObjHomologySelfIso_hom_naturality_assoc, CategoryTheory.NatTrans.mapHomologicalComplex_naturality_assoc, HomologicalComplex.singleObjHomologySelfIso_inv_naturality_assoc, HomologicalComplex.instHasBinaryBiproduct, HomologicalComplex₂.ofGradedObject_d_f, CochainComplex.mappingCone.trianglehMapOfHomotopy_hom₁, HomologicalComplex₂.ι_totalShift₂Iso_inv_f, CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.obj_X_X, CochainComplex.homologySequenceδ_quotient_mapTriangle_obj, CategoryTheory.Functor.mapBifunctorHomologicalComplex_obj_obj_X_X, CochainComplex.mapBifunctorHomologicalComplexShift₁Iso_hom_f_f, HomotopyCategory.homologyFunctor_shiftMap_assoc, HomologicalComplex₂.totalShift₂Iso_hom_naturality, CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.unitIso_hom_app_f_f, CategoryTheory.Functor.mapHomotopyCategory_map, HomologicalComplex.pOpcycles_singleObjOpcyclesSelfIso_inv_assoc, CategoryTheory.Functor.rightDerived_map_eq, HomologicalComplex.truncGE'Map_comp_assoc, HomologicalComplex.mapBifunctorMapHomotopy.comm₁_aux, ComplexShape.Embedding.instAdditiveHomologicalComplexExtendFunctor, HomologicalComplex.epi_homologyMap_shortComplexTruncLE_g, HomologicalComplex.truncGE'Map_id, CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_functor_obj_d_f, Homotopy.smul_hom, CategoryTheory.ProjectiveResolution.isoLeftDerivedToHomotopyCategoryObj_inv_naturality, HomologicalComplex₂.total.mapAux.d₂_mapMap, HomologicalComplex.isZero_iff_isStrictlySupported_and_isStrictlySupportedOutside, CochainComplex.mappingCone.trianglehMapOfHomotopy_hom₂, HomologicalComplex.evalCompCoyonedaCorepresentableByDoubleId_homEquiv_symm_apply, HomologicalComplex₂.ιTotal_totalFlipIso_f_hom, HomologicalComplex.cyclesMapIso_inv, HomologicalComplex.singleObjCyclesSelfIso_hom_assoc, ComplexShape.Embedding.truncGE'Functor_map, CategoryTheory.Functor.mapHomotopyCategory_obj, ComplexShape.Embedding.homRestrict_precomp_assoc, HomologicalComplex.singleObjCyclesSelfIso_inv_naturality, quasiIsoAt_iff_comp_left, HomologicalComplex.opEquivalence_functor, CategoryTheory.Idempotents.karoubiChainComplexEquivalence_counitIso_inv, HomologicalComplex.singleCompEvalIsoSelf_hom_app, HomologicalComplex.homologicalComplexToDGO_obj_d, HomologicalComplex₂.flip_X_d, HomologicalComplex.instHasSeparator, HomologicalComplex.singleObjOpcyclesSelfIso_inv_naturality_assoc, HomologicalComplex.instPreservesFiniteColimitsSingle, HomologicalComplex.quasiIso_opFunctor_map_iff, Homotopy.comp_hom, CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.inverse_obj, HomologicalComplex.biprod_inl_desc_f, HomologicalComplex.singleObjHomologySelfIso_inv_naturality, CategoryTheory.Functor.mapHomologicalComplex_commShiftIso_hom_app_f, HomologicalComplex.restrictionToTruncGE'_naturality, HomologicalComplex.instIsIsoιTruncLEOfIsStrictlySupported, HomologicalComplex₂.totalShift₁Iso_hom_totalShift₂Iso_hom_assoc, ComplexShape.Embedding.restrictionToTruncGE'NatTrans_app, groupCohomology.isoShortComplexH2_hom, CochainComplex.mappingCone.mapHomologicalComplexXIso'_hom, CochainComplex.mapBifunctorHomologicalComplexShift₁Iso_inv_f_f, HomologicalComplex.evalCompCoyonedaCorepresentableBySingle_homEquiv_symm_apply, HomologicalComplex.homologicalComplexToDGO_obj_obj, HomologicalComplex.stupidTruncMap_comp_assoc, HomologicalComplex.homotopyCofiber.desc_f, HomotopyCategory.homologyShiftIso_hom_app, CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.obj_X_p, HomologicalComplex.ι_mapBifunctorFlipIso_hom, AlgebraicTopology.DoldKan.homotopyPToId_eventually_constant, HomologicalComplex.truncLE'Map_comp, HomologicalComplex.biprod_inr_desc_f_assoc, HomologicalComplex.evalCompCoyonedaCorepresentableBySingle_homEquiv_apply, CochainComplex.HomComplex.δ_map, CategoryTheory.Functor.map₂HomologicalComplex_obj_map, HomologicalComplex₂.totalShift₁Iso_hom_naturality_assoc, HomologicalComplex.shortComplexFunctor'_map_τ₃, HomologicalComplex.unopInverse_obj, CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_inverse_obj_X_X, CategoryTheory.InjectiveResolution.iso_inv_naturality_assoc, CategoryTheory.InjectiveResolution.isoRightDerivedToHomotopyCategoryObj_hom_naturality, HomologicalComplex.opEquivalence_inverse, HomologicalComplex.truncLE'Map_comp_assoc, HomologicalComplex₂.ι_D₁, HomologicalComplex.opcyclesFunctor_obj, HomologicalComplex₂.flipEquivalenceCounitIso_inv_app_f_f, HomologicalComplex.natTransHomologyι_app, HomologicalComplex.homologyFunctorIso_hom_app, ComplexShape.Embedding.restrictionFunctor_obj, HomologicalComplex₂.toGradedObjectFunctor_map, CategoryTheory.Functor.mapHomologicalComplexUpToQuasiIsoFactorsh_hom_app, HomologicalComplex.asFunctor_map_f, HomologicalComplex.singleObjCyclesSelfIso_hom_naturality_assoc, ComplexShape.Embedding.truncLE'ToRestrictionNatTrans_app, CategoryTheory.Idempotents.Karoubi.HomologicalComplex.p_comm_f, HomologicalComplex.mkHomFromSingle_f, CategoryTheory.ShortComplex.ShortExact.singleTriangleIso_hom_hom₂, HomologicalComplex.cyclesOpNatIso_hom_app, HomologicalComplex.shortComplexFunctor_map_τ₂, HomologicalComplex.instPreservesFiniteColimitsEvalOfHasFiniteColimits, CategoryTheory.Idempotents.karoubiChainComplexEquivalence_unitIso_inv_app_f_f, Homotopy.compLeft_hom, HomologicalComplex.shortComplexFunctor_obj_g, groupHomology.isoShortComplexH2_inv, HomologicalComplex₂.ι_totalShift₂Iso_hom_f, HomologicalComplex.exactAt_single_obj, HomologicalComplex.shortComplexTruncLE_X₁, HomologicalComplex₂.instHasTotalIntObjUpShiftFunctor₁, HomologicalComplex.hasExactColimitsOfShape, CochainComplex.HomComplex.Cochain.map_ofHom, CochainComplex.HomComplex.CohomologyClass.toHom_mk_eq_zero_iff, HomologicalComplex.extendMap_comp, HomologicalComplex.quasiIso_unopFunctor_map_iff, ComplexShape.QFactorsThroughHomotopy.areEqualizedByLocalization, HomologicalComplex.instPreservesBinaryBiproductEval, CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence_counitIso, HomologicalComplex₂.XXIsoOfEq_hom_ιTotal, Homotopy.nullHomotopy_hom, CochainComplex.mappingCone.map_eq_mapOfHomotopy, HomologicalComplex.HomologySequence.composableArrows₃Functor_map, HomologicalComplex₂.totalShift₁Iso_hom_naturality, HomologicalComplex.Hom.isoApp_inv, HomotopyCategory.quotient_obj_singleFunctors_obj, HomologicalComplex₂.total.mapAux.d₁_mapMap_assoc, HomologicalComplex₂.d₂_eq', HomologicalComplex.dgoEquivHomologicalComplex_inverse, ComplexShape.quotient_isLocalization, quasiIso_iff_comp_right, CategoryTheory.Functor.mapCochainComplexShiftIso_hom_app_f, CategoryTheory.InjectiveResolution.toRightDerivedZero'_comp_iCycles_assoc, HomologicalComplex.unopEquivalence_inverse, CochainComplex.mapBifunctorShift₁Iso_hom_naturality₁, HomotopyCategory.quotient_obj_mem_subcategoryAcyclic_iff_acyclic, HomologicalComplex.homotopyCofiber.inlX_desc_f, HomotopyCategory.instAdditiveHomologicalComplexQuotient, HomotopyCategory.quotient_map_out, CategoryTheory.ProjectiveResolution.iso_inv_naturality, HomologicalComplex₂.d₁_eq', HomologicalComplex.truncLE'ToRestriction_naturality, HomologicalComplex.instPreservesFiniteLimitsSingle, CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.counitIso_hom, HomologicalComplex.extendMap_zero, quasiIsoAt_iff, HomologicalComplex.quasiIsoAt_map_of_preservesHomology, HomologicalComplex.instAdditiveSingle, HomologicalComplex.shortComplexFunctor'_obj_f, HomotopyCategory.homologyFunctor_shiftMap, HomologicalComplex.HomologySequence.composableArrows₃Functor_obj, HomologicalComplex₂.toGradedObjectMap_apply, HomologicalComplex₂.ιTotalOrZero_eq_zero, CochainComplex.shiftShortComplexFunctorIso_zero_add_hom_app, HomologicalComplex.opFunctor_additive, HomologicalComplex.instPreservesZeroMorphismsCyclesFunctor, HomologicalComplex.instQuasiIsoShortComplexTruncLEX₃ToTruncGE, HomologicalComplex.eval_additive, HomologicalComplex₂.flip_d_f, CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_counitIso_inv, HomologicalComplex₂.D₁_totalShift₂XIso_hom, CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.obj_d_f, CategoryTheory.InjectiveResolution.isoRightDerivedObj_inv_naturality_assoc, CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.obj_X_X, CochainComplex.mappingConeCompTriangleh_comm₁, HomologicalComplex.opcyclesMap_zero, HomologicalComplex₂.D₂_totalShift₂XIso_hom, HomologicalComplex.instEpiπTruncGE, HomologicalComplex.instHasBinaryBiproductObjEval, groupCohomology.isoShortComplexH1_inv, HomologicalComplex.shortExact_iff_degreewise_shortExact, HomologicalComplex.shortComplexTruncLE_shortExact_δ_eq_zero, CategoryTheory.Functor.mapBifunctorHomologicalComplex_obj_map_f_f, CategoryTheory.ProjectiveResolution.isoLeftDerivedToHomotopyCategoryObj_hom_naturality, HomologicalComplex.biprod_lift_snd_f_assoc, CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_unitIso_inv_app_f_f, HomologicalComplex.biprod_inr_fst_f_assoc, ComplexShape.Embedding.extendFunctor_obj, CategoryTheory.Functor.mapBifunctorHomologicalComplexObj_obj_d_f, CochainComplex.isKInjective_iff_rightOrthogonal, CochainComplex.ShiftSequence.shiftIso_inv_app, HomologicalComplex.neg_f_apply, CategoryTheory.ProjectiveResolution.isoLeftDerivedObj_inv_naturality, CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_functor_obj_X_X, HomologicalComplex.truncGEMap_comp, CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.functor_map, HomologicalComplex.dgoEquivHomologicalComplexUnitIso_inv_app_f, HomologicalComplex.unopFunctor_map_f, HomologicalComplex.homologyOp_hom_naturality, HomologicalComplex.locallySmall, CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_inverse_obj_X_d, HomologicalComplex₂.flipEquivalenceCounitIso_hom_app_f_f, HomologicalComplex.homologyFunctor_inverts_quasiIso, HomologicalComplex.instHasColimitDiscreteWalkingPairCompPairEval, HomologicalComplex.instPreservesZeroMorphismsSingle, CategoryTheory.ProjectiveResolution.iso_inv_naturality_assoc, CategoryTheory.InjectiveResolution.rightDerivedToHomotopyCategory_app_eq, HomologicalComplex.singleObjCyclesSelfIso_hom, HomologicalComplex.quasiIsoAt_opFunctor_map_iff, HomotopyCategory.quotient_obj_mem_subcategoryAcyclic_iff_exactAt, HomologicalComplex₂.d_comm_assoc, CategoryTheory.Functor.mapHomotopyEquiv_hom, CochainComplex.mappingCone.rotateHomotopyEquiv_comm₃, CategoryTheory.Equivalence.mapHomologicalComplex_inverse, ComplexShape.Embedding.truncGEFunctor_map, HomologicalComplex₂.totalAux.d₁_eq', CategoryTheory.Idempotents.karoubiChainComplexEquivalence_unitIso_hom_app_f_f, HomologicalComplex.cylinder.ι₁_desc, HomologicalComplex.ι_mapBifunctorFlipIso_hom_assoc, HomologicalComplex.extendSingleIso_hom_f, CochainComplex.HomComplex.CohomologyClass.toHom_mk, HomologicalComplex₂.total.hom_ext_iff, groupCohomology.isoShortComplexH2_inv, HomologicalComplex.instFullSingle, HomotopyCategory.quotient_map_mem_quasiIso_iff, HomologicalComplex.biprod_inl_snd_f, HomologicalComplex.homologyFunctorIso_inv_app, CochainComplex.liftCycles_shift_homologyπ_assoc, HomologicalComplex.comp_f_assoc, HomologicalComplex.unopFunctor_additive, HomologicalComplex.instHasZeroObject, HomologicalComplex.instIsNormalMonoCategory, HomologicalComplex.g_shortComplexTruncLEX₃ToTruncGE_assoc, CategoryTheory.InjectiveResolution.iso_inv_naturality, HomologicalComplex.instFaithfulSingle, HomologicalComplex.unopEquivalence_counitIso, ComplexShape.Embedding.homRestrict_comp_extendMap, HomologicalComplex.truncGEMap_id, HomologicalComplex.opInverse_map, HomologicalComplex₂.flipFunctor_map_f_f, HomologicalComplex.singleObjHomologySelfIso_inv_homologyι, HomologicalComplex.homologyFunctor_obj, HomotopyCategory.isoOfHomotopyEquiv_inv, CategoryTheory.Functor.mapDerivedCategoryFactorsh_hom_app, HomologicalComplex.shortComplexTruncLE_X₃_isSupportedOutside, HomologicalComplex.biprod_inl_desc_f_assoc, CategoryTheory.ProjectiveResolution.fromLeftDerivedZero_eq, HomologicalComplex.extendMap_id, HomologicalComplex.Hom.isoApp_hom, HomologicalComplex.homologyFunctorSingleIso_hom_app, ComplexShape.Embedding.AreComplementary.hom_ext', HomologicalComplex.singleObjOpcyclesSelfIso_hom_assoc, HomologicalComplex.homologyMap_comp_assoc, HomologicalComplex.singleObjHomologySelfIso_hom_singleObjHomologySelfIso_inv, HomologicalComplex.isZero_single_comp_eval, CategoryTheory.instPreservesZeroMorphismsHomologicalComplexMapHomologicalComplex, HomologicalComplex.cylinder.ι₀_desc_assoc, HomologicalComplex.dgoEquivHomologicalComplexCounitIso_inv_app_f, Homotopy.nullHomotopy'_hom, HomologicalComplex.Hom.isIso_of_components, HomologicalComplex.singleObjOpcyclesSelfIso_hom_naturality, CochainComplex.mappingCone.mapHomologicalComplexXIso'_inv, HomologicalComplexUpToQuasiIso.isIso_Q_map_iff_mem_quasiIso, HomologicalComplex.homologyMapIso_inv, Rep.FiniteCyclicGroup.resolution.π_f, HomologicalComplex.natIsoSc'_inv_app_τ₁, HomologicalComplex.instHasFiniteLimits, HomologicalComplex.exact_iff_degreewise_exact, CategoryTheory.InjectiveResolution.rightDerived_app_eq, HomologicalComplex.instHasHomologyObjOppositeSymmUnopFunctorOp, HomologicalComplex.Hom.fAddMonoidHom_apply, HomologicalComplex.epi_of_epi_f, HomologicalComplex₂.total.map_id, HomotopyCategory.instLinearHomologicalComplexQuotient, CochainComplex.single₀ObjXSelf, CochainComplex.mappingConeCompHomotopyEquiv_hom_inv_id_assoc, HomologicalComplex₂.total.forget_map, CochainComplex.liftCycles_shift_homologyπ, HomologicalComplex₂.totalFunctor_map, HomologicalComplex.natIsoSc'_hom_app_τ₁, HomologicalComplex.instPreservesZeroMorphismsOpcyclesFunctor, HomologicalComplex.extendMap_id_f, ComplexShape.Embedding.stupidTruncFunctor_obj, HomologicalComplex.cyclesMapIso_hom, CochainComplex.HomComplex.Cochain.map_neg, CategoryTheory.Functor.mapBifunctorHomologicalComplex_obj_obj_toGradedObject, HomologicalComplex₂.ιTotal_map_assoc, HomologicalComplex.ι_mapBifunctorFlipIso_inv, CochainComplex.HomComplex.CohomologyClass.homAddEquiv_apply, CategoryTheory.Idempotents.karoubiChainComplexEquivalence_counitIso_hom, HomologicalComplex.cylinder.πCompι₀Homotopy.inlX_nullHomotopy_f, ChainComplex.map_chain_complex_of, HomologicalComplex₂.totalFlipIso_hom_f_D₂_assoc, CochainComplex.instAdditiveHomologicalComplexIntUpShiftFunctor, HomologicalComplex₂.ι_totalShift₁Iso_hom_f, HomologicalComplex.forgetEval_inv_app, CategoryTheory.instIsIsoFromLeftDerivedZero', ComplexShape.Embedding.instPreservesZeroMorphismsHomologicalComplexRestrictionFunctor, HomologicalComplex₂.instFaithfulGradedObjectProdToGradedObjectFunctor, ComplexShape.Embedding.truncGEFunctor_obj, CategoryTheory.Functor.mapBifunctorHomologicalComplex_obj_obj_d_f, HomologicalComplex.shortComplexFunctor_obj_X₃, CochainComplex.ShiftSequence.shiftIso_hom_app, HomotopyCategory.instCommShiftHomologicalComplexIntUpHomFunctorMapHomotopyCategoryFactors, CochainComplex.mappingCone.map_δ, HomologicalComplex₂.ιTotal_totalFlipIso_f_inv, CochainComplex.ConnectData.restrictionGEIso_hom_f, HomologicalComplex.quasiIsoAt_map_iff_of_preservesHomology, HomologicalComplex.Hom.sqFrom_id

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