Documentation Verification Report

IsSheafFor

πŸ“ Source: Mathlib/CategoryTheory/Sites/IsSheafFor.lean

Statistics

MetricCount
DefinitionsCompatible, familyOfElements, PullbackCompatible, toCompatible, FamilyOfElements, Compatible, IsAmalgamation, PullbackCompatible, SieveCompatible, compPresheafMap, functorPullback, functorPushforward, map, pullback, restrict, sieveExtend, singletonEquiv, IsSeparatedFor, IsSheafFor, amalgamate, extend, YonedaSheafCondition, compatibleEquivGenerateSieveCompatible, instInhabitedFamilyOfElementsBot, natTransEquivCompatibleFamily
25
Theoremsexists_familyOfElements, familyOfElements_compatible, familyOfElements_ofArrows_mk, pullbackCompatible_iff, toCompatible_coe, compPresheafMap, functorPullback, map, pullback, restrict, sieveExtend, to_sieveCompatible, compPresheafMap, map, compPresheafMap_comp, compPresheafMap_id, comp_of_compatible, compatible_singleton_iff, ext, ext_iff, isAmalgamation_iff_ofArrows, isAmalgamation_singleton_iff, map_apply, map_comp, map_id, restrict_map, singletonEquiv_apply, singletonEquiv_symm_apply, singletonEquiv_symm_apply_self, ext, isSheafFor, functorInclusion_comp_extend, functorInclusion_comp_extend_assoc, hom_ext, isAmalgamation, isSeparatedFor, unique_extend, valid_glue, compatibleEquivGenerateSieveCompatible_apply_coe, compatibleEquivGenerateSieveCompatible_symm_apply_coe, compatible_iff_sieveCompatible, extend_agrees, extend_restrict, extension_iff_amalgamation, isAmalgamation_restrict, isAmalgamation_sieveExtend, isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor, isSeparatedFor_iff_generate, isSeparatedFor_iso, isSeparatedFor_singleton, isSeparatedFor_top, isSheafFor_arrows_iff, isSheafFor_arrows_iff_pullbacks, isSheafFor_bind, isSheafFor_iff_generate, isSheafFor_iff_of_iso, isSheafFor_iff_yonedaSheafCondition, isSheafFor_iso, isSheafFor_ofArrows_iff_bijective_toCompabible, isSheafFor_of_nat_equiv, isSheafFor_over_map_op_comp_iff, isSheafFor_over_map_op_comp_ofArrows_iff, isSheafFor_pullback_iff, isSheafFor_singleton, isSheafFor_singleton_iso, isSheafFor_subsieve, isSheafFor_subsieve_aux, isSheafFor_top, isSheafFor_top_sieve, isSheafFor_trans, is_compatible_of_exists_amalgamation, pullbackCompatible_iff, restrict_extend, restrict_inj
74
Total99

CategoryTheory.Presieve

Definitions

NameCategoryTheorems
FamilyOfElements πŸ“–CompOp
10 mathmath: CategoryTheory.Equalizer.firstObjEqFamily_inv, CategoryTheory.Equalizer.firstObjEqFamily_hom, FamilyOfElements.singletonEquiv_symm_apply_self, FamilyOfElements.singletonEquiv_apply, CategoryTheory.Equalizer.Presieve.compatible_iff, FamilyOfElements.singletonEquiv_symm_apply, compatibleEquivGenerateSieveCompatible_apply_coe, CategoryTheory.Equalizer.Sieve.compatible_iff, extension_iff_amalgamation, compatibleEquivGenerateSieveCompatible_symm_apply_coe
IsSeparatedFor πŸ“–MathDef
6 mathmath: isSeparatedFor_singleton, isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor, IsSheafFor.isSeparatedFor, isSeparatedFor_top, CategoryTheory.Presheaf.subsingleton_iff_isSeparatedFor, isSeparatedFor_iff_generate
IsSheafFor πŸ“–MathDef
45 mathmath: isSheafFor_pullback_iff, CategoryTheory.regularTopology.isSheafFor_regular_of_projective, CategoryTheory.Equalizer.Sieve.equalizer_sheaf_condition, isSheaf_pretopology, isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor, isSheafFor_singleton, CategoryTheory.PreZeroHypercover.isSheafFor_iff_of_iso, CategoryTheory.Precoverage.isSheaf_toGrothendieck_iff_of_isStableUnderBaseChange_of_small, isSheafFor_iff_yonedaSheafCondition, CategoryTheory.Presheaf.isLimit_iff_isSheafFor_presieve, isSheafFor_top_sieve, CategoryTheory.Precoverage.isSheaf_toGrothendieck_iff, CategoryTheory.Equalizer.Presieve.isSheafFor_singleton_iff, CategoryTheory.Presheaf.IsSheaf.isSheafFor, IsSheafFor.comp_iff_of_preservesPairwisePullbacks, isSheafFor_arrows_iff, CategoryTheory.GrothendieckTopology.OneHypercover.isSheafFor_presieveβ‚€, CategoryTheory.Equalizer.Presieve.sheaf_condition, CategoryTheory.Pseudofunctor.isPrestackFor_iff_isSheafFor, CategoryTheory.Pseudofunctor.bijective_toDescentData_map_iff, isSheafFor_of_preservesProduct, isSheafFor_iff_generate, IsSheaf.isSheafFor_of_mem_precoverage, CategoryTheory.Equalizer.Presieve.Arrows.sheaf_condition, isSheafFor_over_map_op_comp_ofArrows_iff, isSheafFor_over_map_op_comp_iff, isSheafFor_singleton_iff_of_iso, isSheaf_coverage, CategoryTheory.Pseudofunctor.IsPrestackFor.isSheafFor', CategoryTheory.isSheafFor_extensive_of_preservesFiniteProducts, CategoryTheory.Sieve.forallYonedaIsSheaf_iff_colimit, isSheafFor_arrows_iff_pullbacks, CategoryTheory.isSheaf_coherent, IsSeparatedFor.isSheafFor, CategoryTheory.presheafHom_isSheafFor, isSheafFor_iff_of_iso, CategoryTheory.Presheaf.isLimit_iff_isSheafFor, isSheafFor_singleton_iso, CategoryTheory.Pseudofunctor.isPrestackFor_iff_isSheafFor', isSheafFor_top, CategoryTheory.Equalizer.Presieve.isSheafFor_singleton_iff_of_hasPullback, IsSheaf.isSheafFor, isSheafFor_ofArrows_iff_bijective_toCompabible, isSheafFor_ofArrows_comp_iff, CategoryTheory.Precoverage.isSheaf_toGrothendieck_iff_of_isStableUnderBaseChange
YonedaSheafCondition πŸ“–MathDef
1 mathmath: isSheafFor_iff_yonedaSheafCondition
compatibleEquivGenerateSieveCompatible πŸ“–CompOp
2 mathmath: compatibleEquivGenerateSieveCompatible_apply_coe, compatibleEquivGenerateSieveCompatible_symm_apply_coe
instInhabitedFamilyOfElementsBot πŸ“–CompOpβ€”
natTransEquivCompatibleFamily πŸ“–CompOp
1 mathmath: extension_iff_amalgamation

Theorems

NameKindAssumesProvesValidatesDepends On
compatibleEquivGenerateSieveCompatible_apply_coe πŸ“–mathematicalβ€”FamilyOfElements
CategoryTheory.Sieve.arrows
CategoryTheory.Sieve.generate
FamilyOfElements.Compatible
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
compatibleEquivGenerateSieveCompatible
FamilyOfElements.sieveExtend
β€”β€”
compatibleEquivGenerateSieveCompatible_symm_apply_coe πŸ“–mathematicalβ€”FamilyOfElements
FamilyOfElements.Compatible
DFunLike.coe
Equiv
CategoryTheory.Sieve.arrows
CategoryTheory.Sieve.generate
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
compatibleEquivGenerateSieveCompatible
FamilyOfElements.restrict
CategoryTheory.Sieve.le_generate
β€”β€”
compatible_iff_sieveCompatible πŸ“–mathematicalβ€”FamilyOfElements.Compatible
CategoryTheory.Sieve.arrows
FamilyOfElements.SieveCompatible
β€”CategoryTheory.Sieve.downward_closed
CategoryTheory.FunctorToTypes.map_id_apply
CategoryTheory.Category.id_comp
extend_agrees πŸ“–mathematicalFamilyOfElements.CompatibleFamilyOfElements.sieveExtend
CategoryTheory.Sieve.le_generate
β€”CategoryTheory.Sieve.le_generate
CategoryTheory.Category.id_comp
CategoryTheory.FunctorToTypes.map_id_apply
extend_restrict πŸ“–mathematicalFamilyOfElements.Compatible
CategoryTheory.Sieve.arrows
CategoryTheory.Sieve.generate
FamilyOfElements.sieveExtend
FamilyOfElements.restrict
CategoryTheory.Sieve.le_generate
β€”CategoryTheory.Sieve.le_generate
CategoryTheory.Sieve.downward_closed
compatible_iff_sieveCompatible
extension_iff_amalgamation πŸ“–mathematicalβ€”CategoryTheory.CategoryStruct.comp
CategoryTheory.Functor
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
CategoryTheory.Category.toCategoryStruct
CategoryTheory.Functor.category
CategoryTheory.Sieve.functor
CategoryTheory.Functor.obj
CategoryTheory.yoneda
CategoryTheory.Sieve.functorInclusion
FamilyOfElements.IsAmalgamation
CategoryTheory.Sieve.arrows
FamilyOfElements
FamilyOfElements.Compatible
DFunLike.coe
Equiv
Quiver.Hom
CategoryTheory.CategoryStruct.toQuiver
EquivLike.toFunLike
Equiv.instEquivLike
natTransEquivCompatibleFamily
Opposite.op
CategoryTheory.yonedaEquiv
β€”CategoryTheory.yonedaEquiv_naturality
CategoryTheory.Category.id_comp
CategoryTheory.NatTrans.ext'
CategoryTheory.types_ext
isAmalgamation_restrict πŸ“–mathematicalCategoryTheory.Presieve
Preorder.toLE
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CategoryTheory.instCompleteLatticePresieve
FamilyOfElements.IsAmalgamation
FamilyOfElements.restrictβ€”β€”
isAmalgamation_sieveExtend πŸ“–mathematicalFamilyOfElements.IsAmalgamationCategoryTheory.Sieve.arrows
CategoryTheory.Sieve.generate
FamilyOfElements.sieveExtend
β€”CategoryTheory.FunctorToTypes.map_comp_apply
CategoryTheory.op_comp
isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor πŸ“–mathematicalβ€”IsSeparatedFor
FamilyOfElements.IsAmalgamation
IsSheafFor
β€”IsSeparatedFor.eq_1
existsUnique_of_exists_of_unique
ExistsUnique.unique
is_compatible_of_exists_amalgamation
ExistsUnique.exists
isSeparatedFor_iff_generate πŸ“–mathematicalβ€”IsSeparatedFor
CategoryTheory.Sieve.arrows
CategoryTheory.Sieve.generate
β€”CategoryTheory.Sieve.le_generate
isAmalgamation_restrict
isAmalgamation_sieveExtend
isSeparatedFor_iso πŸ“–β€”IsSeparatedForβ€”β€”CategoryTheory.FunctorToTypes.inv_hom_id_app_apply
FamilyOfElements.IsAmalgamation.map
isSeparatedFor_singleton πŸ“–mathematicalβ€”IsSeparatedFor
singleton
CategoryTheory.Functor.obj
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
Opposite.op
CategoryTheory.Functor.map
Quiver.Hom.op
CategoryTheory.CategoryStruct.toQuiver
CategoryTheory.Category.toCategoryStruct
β€”IsSeparatedFor.eq_1
Equiv.forall_congr_left
FamilyOfElements.singletonEquiv_symm_apply_self
isSeparatedFor_top πŸ“–mathematicalβ€”IsSeparatedFor
Top.top
CategoryTheory.Presieve
OrderTop.toTop
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
CompleteLattice.toLattice
CategoryTheory.instCompleteLatticePresieve
BoundedOrder.toOrderTop
CompleteLattice.toBoundedOrder
β€”CategoryTheory.FunctorToTypes.map_id_apply
isSheafFor_arrows_iff πŸ“–mathematicalβ€”IsSheafFor
ofArrows
ExistsUnique
CategoryTheory.Functor.obj
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
Opposite.op
β€”Arrows.Compatible.familyOfElements_compatible
isSheafFor_arrows_iff_pullbacks πŸ“–mathematicalβ€”IsSheafFor
ofArrows
ExistsUnique
CategoryTheory.Functor.obj
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
Opposite.op
β€”β€”
isSheafFor_bind πŸ“–mathematicalIsSheafFor
CategoryTheory.Sieve.arrows
IsSeparatedFor
CategoryTheory.Sieve.pullback
CategoryTheory.Sieve.bindβ€”bind_comp
CategoryTheory.Category.assoc
Mathlib.Tactic.Reassoc.eq_whisker'
IsSheafFor.isAmalgamation
compatible_iff_sieveCompatible
IsSeparatedFor.ext
CategoryTheory.Sieve.downward_closed
IsSheafFor.isSeparatedFor
CategoryTheory.FunctorToTypes.map_comp_apply
CategoryTheory.op_comp
IsSheafFor.valid_glue
isSheafFor_iff_generate πŸ“–mathematicalβ€”IsSheafFor
CategoryTheory.Sieve.arrows
CategoryTheory.Sieve.generate
β€”isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor
isSeparatedFor_iff_generate
CategoryTheory.Sieve.le_generate
extend_restrict
isAmalgamation_sieveExtend
FamilyOfElements.Compatible.restrict
restrict_extend
isAmalgamation_restrict
FamilyOfElements.Compatible.sieveExtend
isSheafFor_iff_of_iso πŸ“–mathematicalβ€”IsSheafForβ€”isSheafFor_iso
isSheafFor_iff_yonedaSheafCondition πŸ“–mathematicalβ€”IsSheafFor
CategoryTheory.Sieve.arrows
YonedaSheafCondition
β€”IsSheafFor.eq_1
YonedaSheafCondition.eq_1
Equiv.forall_congr_left
Equiv.apply_symm_apply
isSheafFor_iso πŸ“–β€”IsSheafForβ€”β€”isSheafFor_of_nat_equiv
CategoryTheory.NatTrans.naturality
isSheafFor_ofArrows_iff_bijective_toCompabible πŸ“–mathematicalβ€”IsSheafFor
ofArrows
Function.Bijective
CategoryTheory.Functor.obj
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
Opposite.op
Arrows.Compatible
Arrows.toCompatible
β€”isSheafFor_arrows_iff
ExistsUnique.unique
isSheafFor_of_nat_equiv πŸ“–β€”DFunLike.coe
Equiv
CategoryTheory.Functor.obj
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
Opposite.op
EquivLike.toFunLike
Equiv.instEquivLike
CategoryTheory.Functor.map
Quiver.Hom.op
CategoryTheory.CategoryStruct.toQuiver
CategoryTheory.Category.toCategoryStruct
IsSheafFor
β€”β€”Equiv.injective
Equiv.apply_symm_apply
EquivLike.toEmbeddingLike
Equiv.symm_apply_apply
IsSheafFor.isAmalgamation
IsSheafFor.isSeparatedFor
isSheafFor_over_map_op_comp_iff πŸ“–mathematicalβ€”IsSheafFor
CategoryTheory.Over
CategoryTheory.instCategoryOver
CategoryTheory.Functor.comp
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
CategoryTheory.Functor.op
CategoryTheory.Over.map
CategoryTheory.Sieve.arrows
CategoryTheory.Sieve.pullback
CategoryTheory.Functor.obj
CategoryTheory.Iso.inv
CategoryTheory.Sieve.functorPushforward
β€”CategoryTheory.Sieve.exists_eq_ofArrows
isSheafFor_iff_generate
isSheafFor_pullback_iff
CategoryTheory.Iso.isIso_inv
isSheafFor_over_map_op_comp_ofArrows_iff
le_antisymm
CategoryTheory.Category.assoc
CategoryTheory.Over.w_assoc
CategoryTheory.Over.w
CategoryTheory.Functor.map_comp
CategoryTheory.Over.OverMorphism.ext
isSheafFor_over_map_op_comp_ofArrows_iff πŸ“–mathematicalβ€”IsSheafFor
CategoryTheory.Over
CategoryTheory.instCategoryOver
CategoryTheory.Functor.comp
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
CategoryTheory.Functor.op
CategoryTheory.Over.map
ofArrows
CategoryTheory.Functor.obj
CategoryTheory.Functor.map
β€”CategoryTheory.Functor.congr_map
CategoryTheory.Over.w
CategoryTheory.Category.assoc
Mathlib.Tactic.Reassoc.eq_whisker'
CategoryTheory.Over.OverMorphism.ext
CategoryTheory.Category.id_comp
CategoryTheory.FunctorToTypes.map_comp_apply
Function.Bijective.of_comp_iff'
Equiv.bijective
isSheafFor_pullback_iff πŸ“–mathematicalβ€”IsSheafFor
CategoryTheory.Sieve.arrows
CategoryTheory.Sieve.pullback
β€”CategoryTheory.Sieve.exists_eq_ofArrows
CategoryTheory.Sieve.pullback_ofArrows_of_iso
CategoryTheory.cancel_mono
CategoryTheory.mono_of_strongMono
CategoryTheory.strongMono_of_isIso
CategoryTheory.Category.assoc
CategoryTheory.IsIso.inv_hom_id
CategoryTheory.Category.comp_id
CategoryTheory.eq_whisker
Function.Bijective.of_comp_iff'
Equiv.bijective
Function.Bijective.of_comp_iff
CategoryTheory.Functor.map_isIso
CategoryTheory.isIso_op
CategoryTheory.op_inv
CategoryTheory.Functor.map_inv
CategoryTheory.FunctorToTypes.map_comp_apply
isSheafFor_singleton πŸ“–mathematicalβ€”IsSheafFor
singleton
ExistsUnique
CategoryTheory.Functor.obj
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
Opposite.op
CategoryTheory.Functor.map
Quiver.Hom.op
CategoryTheory.CategoryStruct.toQuiver
CategoryTheory.Category.toCategoryStruct
β€”IsSheafFor.eq_1
Equiv.forall_congr_left
FamilyOfElements.singletonEquiv_symm_apply_self
isSheafFor_singleton_iso πŸ“–mathematicalβ€”IsSheafFor
singleton
CategoryTheory.CategoryStruct.id
CategoryTheory.Category.toCategoryStruct
β€”singleton_self
CategoryTheory.FunctorToTypes.map_id_apply
isSheafFor_subsieve πŸ“–β€”CategoryTheory.Presieve
Preorder.toLE
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CategoryTheory.instCompleteLatticePresieve
CategoryTheory.Sieve.arrows
IsSheafFor
CategoryTheory.Sieve.pullback
β€”β€”isSheafFor_subsieve_aux
CategoryTheory.Sieve.pullback_id
IsSheafFor.isSeparatedFor
isSheafFor_subsieve_aux πŸ“–β€”CategoryTheory.Presieve
Preorder.toLE
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CategoryTheory.instCompleteLatticePresieve
CategoryTheory.Sieve.arrows
IsSheafFor
IsSeparatedFor
CategoryTheory.Sieve.pullback
β€”β€”isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor
IsSheafFor.isSeparatedFor
isAmalgamation_restrict
FamilyOfElements.Compatible.restrict
IsSeparatedFor.ext
CategoryTheory.FunctorToTypes.map_comp_apply
CategoryTheory.op_comp
IsSheafFor.valid_glue
FamilyOfElements.restrict.eq_1
CategoryTheory.Category.id_comp
CategoryTheory.FunctorToTypes.map_id_apply
isSheafFor_top πŸ“–mathematicalβ€”IsSheafFor
Top.top
CategoryTheory.Presieve
OrderTop.toTop
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
CompleteLattice.toLattice
CategoryTheory.instCompleteLatticePresieve
BoundedOrder.toOrderTop
CompleteLattice.toBoundedOrder
β€”CategoryTheory.Sieve.arrows_top
CategoryTheory.Sieve.generate_of_singleton_isSplitEpi
CategoryTheory.IsSplitEpi.of_iso
CategoryTheory.IsIso.id
isSheafFor_iff_generate
isSheafFor_singleton_iso
isSheafFor_top_sieve πŸ“–mathematicalβ€”IsSheafFor
Top.top
CategoryTheory.Presieve
OrderTop.toTop
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
CompleteLattice.toLattice
CategoryTheory.instCompleteLatticePresieve
BoundedOrder.toOrderTop
CompleteLattice.toBoundedOrder
β€”isSheafFor_top
isSheafFor_trans πŸ“–β€”IsSheafFor
CategoryTheory.Sieve.arrows
IsSeparatedFor
CategoryTheory.Sieve.pullback
β€”β€”isSheafFor_subsieve_aux
isSheafFor_bind
CategoryTheory.Sieve.pullback_comp
IsSheafFor.isSeparatedFor
CategoryTheory.Sieve.downward_closed
CategoryTheory.Sieve.ext
CategoryTheory.Sieve.pullback_apply
CategoryTheory.Category.id_comp
is_compatible_of_exists_amalgamation πŸ“–mathematicalFamilyOfElements.IsAmalgamationFamilyOfElements.Compatibleβ€”CategoryTheory.FunctorToTypes.map_comp_apply
CategoryTheory.op_comp
pullbackCompatible_iff πŸ“–mathematicalβ€”FamilyOfElements.Compatible
FamilyOfElements.PullbackCompatible
β€”HasPairwisePullbacks.has_pullbacks
CategoryTheory.Limits.pullback.condition
CategoryTheory.Limits.pullback.lift_fst
CategoryTheory.op_comp
CategoryTheory.FunctorToTypes.map_comp_apply
CategoryTheory.Limits.pullback.lift_snd
restrict_extend πŸ“–mathematicalFamilyOfElements.CompatibleFamilyOfElements.restrict
CategoryTheory.Sieve.arrows
CategoryTheory.Sieve.generate
CategoryTheory.Sieve.le_generate
FamilyOfElements.sieveExtend
β€”CategoryTheory.Sieve.le_generate
extend_agrees
restrict_inj πŸ“–β€”FamilyOfElements.Compatible
CategoryTheory.Sieve.arrows
CategoryTheory.Sieve.generate
FamilyOfElements.restrict
CategoryTheory.Sieve.le_generate
β€”β€”CategoryTheory.Sieve.le_generate
extend_restrict

CategoryTheory.Presieve.Arrows

Definitions

NameCategoryTheorems
Compatible πŸ“–MathDef
7 mathmath: CategoryTheory.Pseudofunctor.DescentData.subtypeCompatibleHomEquiv_toCompatible_presheafHomObjHomEquiv, CategoryTheory.Equalizer.Presieve.Arrows.compatible_iff, CategoryTheory.Equalizer.Presieve.Arrows.compatible_iff_of_small, toCompatible_coe, CategoryTheory.GrothendieckTopology.OneHypercover.arrowsCompatible, pullbackCompatible_iff, CategoryTheory.Presieve.isSheafFor_ofArrows_iff_bijective_toCompabible
PullbackCompatible πŸ“–MathDef
1 mathmath: pullbackCompatible_iff
toCompatible πŸ“–CompOp
3 mathmath: CategoryTheory.Pseudofunctor.DescentData.subtypeCompatibleHomEquiv_toCompatible_presheafHomObjHomEquiv, toCompatible_coe, CategoryTheory.Presieve.isSheafFor_ofArrows_iff_bijective_toCompabible

Theorems

NameKindAssumesProvesValidatesDepends On
pullbackCompatible_iff πŸ“–mathematicalβ€”Compatible
PullbackCompatible
β€”CategoryTheory.Presieve.instHasPullbackOfHasPairwisePullbacksOfArrows
CategoryTheory.Limits.pullback.condition
CategoryTheory.Limits.pullback.lift_fst
CategoryTheory.op_comp
CategoryTheory.FunctorToTypes.map_comp_apply
CategoryTheory.Limits.pullback.lift_snd
toCompatible_coe πŸ“–mathematicalβ€”Compatible
toCompatible
CategoryTheory.Functor.map
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
Opposite.op
Quiver.Hom.op
CategoryTheory.CategoryStruct.toQuiver
CategoryTheory.Category.toCategoryStruct
β€”β€”

CategoryTheory.Presieve.Arrows.Compatible

Definitions

NameCategoryTheorems
familyOfElements πŸ“–CompOp
2 mathmath: familyOfElements_ofArrows_mk, familyOfElements_compatible

Theorems

NameKindAssumesProvesValidatesDepends On
exists_familyOfElements πŸ“–β€”CategoryTheory.Presieve.Arrows.Compatibleβ€”β€”CategoryTheory.Category.id_comp
CategoryTheory.FunctorToTypes.map_id_apply
CategoryTheory.Presieve.ofArrows_surj
familyOfElements_compatible πŸ“–mathematicalCategoryTheory.Presieve.Arrows.CompatibleCategoryTheory.Presieve.FamilyOfElements.Compatible
CategoryTheory.Presieve.ofArrows
familyOfElements
β€”β€”
familyOfElements_ofArrows_mk πŸ“–mathematicalCategoryTheory.Presieve.Arrows.CompatiblefamilyOfElementsβ€”exists_familyOfElements

CategoryTheory.Presieve.FamilyOfElements

Definitions

NameCategoryTheorems
Compatible πŸ“–MathDef
16 mathmath: isCompatible_map_smul, CategoryTheory.Presieve.is_compatible_of_exists_amalgamation, compatible_singleton_iff, CategoryTheory.Presieve.pullbackCompatible_iff, CategoryTheory.Equalizer.Presieve.compatible_iff, CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily_compatible, CategoryTheory.Presieve.Arrows.Compatible.familyOfElements_compatible, CategoryTheory.Sieve.yonedaFamily_fromCocone_compatible, CategoryTheory.Presieve.compatibleEquivGenerateSieveCompatible_apply_coe, CategoryTheory.Subfunctor.Subpresheaf.family_of_elements_compatible, CategoryTheory.Equalizer.Sieve.compatible_iff, CategoryTheory.Presieve.compatible_iff_sieveCompatible, CategoryTheory.Presieve.extension_iff_amalgamation, CategoryTheory.Presheaf.FamilyOfElementsOnObjects.IsCompatible.familyOfElements_isCompatible, CategoryTheory.Subfunctor.family_of_elements_compatible, CategoryTheory.Presieve.compatibleEquivGenerateSieveCompatible_symm_apply_coe
IsAmalgamation πŸ“–MathDef
7 mathmath: CategoryTheory.Presieve.IsSheafFor.isAmalgamation, CategoryTheory.Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor, isAmalgamation_iff_ofArrows, isAmalgamation_singleton_iff, isAmalgamation_map_localPreimage, CategoryTheory.Presieve.extension_iff_amalgamation, CategoryTheory.PresheafHom.isAmalgamation_iff
PullbackCompatible πŸ“–MathDef
1 mathmath: CategoryTheory.Presieve.pullbackCompatible_iff
SieveCompatible πŸ“–MathDef
2 mathmath: CategoryTheory.Presieve.compatible_iff_sieveCompatible, Compatible.to_sieveCompatible
compPresheafMap πŸ“–CompOpβ€”
functorPullback πŸ“–CompOp
1 mathmath: Compatible.functorPullback
functorPushforward πŸ“–CompOp
2 mathmath: Compatible.functorPushforward, CategoryTheory.CompatiblePreserving.apply_map
map πŸ“–CompOp
11 mathmath: Compatible.map, map_apply, IsAmalgamation.compPresheafMap, compPresheafMap_comp, isAmalgamation_map_localPreimage, restrict_map, compPresheafMap_id, IsAmalgamation.map, map_comp, Compatible.compPresheafMap, map_id
pullback πŸ“–CompOp
1 mathmath: Compatible.pullback
restrict πŸ“–CompOp
6 mathmath: CategoryTheory.Presieve.extend_restrict, Compatible.restrict, CategoryTheory.Presieve.restrict_extend, restrict_map, CategoryTheory.Presieve.isAmalgamation_restrict, CategoryTheory.Presieve.compatibleEquivGenerateSieveCompatible_symm_apply_coe
sieveExtend πŸ“–CompOp
6 mathmath: CategoryTheory.Presieve.isAmalgamation_sieveExtend, CategoryTheory.Presieve.extend_restrict, CategoryTheory.Presieve.extend_agrees, CategoryTheory.Presieve.compatibleEquivGenerateSieveCompatible_apply_coe, CategoryTheory.Presieve.restrict_extend, Compatible.sieveExtend
singletonEquiv πŸ“–CompOp
3 mathmath: singletonEquiv_symm_apply_self, singletonEquiv_apply, singletonEquiv_symm_apply

Theorems

NameKindAssumesProvesValidatesDepends On
compPresheafMap_comp πŸ“–mathematicalβ€”map
CategoryTheory.CategoryStruct.comp
CategoryTheory.Functor
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
CategoryTheory.Category.toCategoryStruct
CategoryTheory.Functor.category
β€”map_comp
compPresheafMap_id πŸ“–mathematicalβ€”map
CategoryTheory.CategoryStruct.id
CategoryTheory.Functor
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
CategoryTheory.Category.toCategoryStruct
CategoryTheory.Functor.category
β€”map_id
comp_of_compatible πŸ“–mathematicalCompatible
CategoryTheory.Sieve.arrows
CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
CategoryTheory.Sieve.downward_closed
CategoryTheory.Functor.map
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
Opposite.op
Quiver.Hom.op
CategoryTheory.CategoryStruct.toQuiver
β€”CategoryTheory.Sieve.downward_closed
CategoryTheory.FunctorToTypes.map_id_apply
CategoryTheory.Category.id_comp
compatible_singleton_iff πŸ“–mathematicalβ€”Compatible
CategoryTheory.Presieve.singleton
CategoryTheory.Functor.map
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
Opposite.op
Quiver.Hom.op
CategoryTheory.CategoryStruct.toQuiver
CategoryTheory.Category.toCategoryStruct
β€”β€”
ext πŸ“–β€”β€”β€”β€”β€”
ext_iff πŸ“–β€”β€”β€”β€”ext
isAmalgamation_iff_ofArrows πŸ“–mathematicalβ€”IsAmalgamation
CategoryTheory.Presieve.ofArrows
CategoryTheory.Functor.map
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
Opposite.op
Quiver.Hom.op
CategoryTheory.CategoryStruct.toQuiver
CategoryTheory.Category.toCategoryStruct
β€”β€”
isAmalgamation_singleton_iff πŸ“–mathematicalβ€”IsAmalgamation
CategoryTheory.Presieve.singleton
CategoryTheory.Functor.map
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
Opposite.op
Quiver.Hom.op
CategoryTheory.CategoryStruct.toQuiver
CategoryTheory.Category.toCategoryStruct
β€”β€”
map_apply πŸ“–mathematicalβ€”map
CategoryTheory.NatTrans.app
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
Opposite.op
β€”β€”
map_comp πŸ“–mathematicalβ€”map
CategoryTheory.CategoryStruct.comp
CategoryTheory.Functor
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
CategoryTheory.Category.toCategoryStruct
CategoryTheory.Functor.category
β€”β€”
map_id πŸ“–mathematicalβ€”map
CategoryTheory.CategoryStruct.id
CategoryTheory.Functor
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
CategoryTheory.Category.toCategoryStruct
CategoryTheory.Functor.category
β€”β€”
restrict_map πŸ“–mathematicalCategoryTheory.Presieve
Preorder.toLE
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CategoryTheory.instCompleteLatticePresieve
map
restrict
β€”β€”
singletonEquiv_apply πŸ“–mathematicalβ€”DFunLike.coe
Equiv
CategoryTheory.Presieve.FamilyOfElements
CategoryTheory.Presieve.singleton
CategoryTheory.Functor.obj
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
Opposite.op
EquivLike.toFunLike
Equiv.instEquivLike
singletonEquiv
β€”β€”
singletonEquiv_symm_apply πŸ“–mathematicalCategoryTheory.Presieve.singletonDFunLike.coe
Equiv
CategoryTheory.Functor.obj
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
Opposite.op
CategoryTheory.Presieve.FamilyOfElements
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
singletonEquiv
CategoryTheory.Functor.map
Quiver.Hom.op
CategoryTheory.CategoryStruct.toQuiver
CategoryTheory.Category.toCategoryStruct
CategoryTheory.eqToHom
β€”β€”
singletonEquiv_symm_apply_self πŸ“–mathematicalβ€”DFunLike.coe
Equiv
CategoryTheory.Functor.obj
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
Opposite.op
CategoryTheory.Presieve.FamilyOfElements
CategoryTheory.Presieve.singleton
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
singletonEquiv
β€”CategoryTheory.FunctorToTypes.map_id_apply

CategoryTheory.Presieve.FamilyOfElements.Compatible

Theorems

NameKindAssumesProvesValidatesDepends On
compPresheafMap πŸ“–mathematicalCategoryTheory.Presieve.FamilyOfElements.CompatibleCategoryTheory.Presieve.FamilyOfElements.mapβ€”map
functorPullback πŸ“–mathematicalCategoryTheory.Presieve.FamilyOfElements.Compatible
CategoryTheory.Functor.obj
CategoryTheory.Functor.comp
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
CategoryTheory.Functor.op
CategoryTheory.Presieve.functorPullback
CategoryTheory.Presieve.FamilyOfElements.functorPullback
β€”CategoryTheory.Functor.map_comp
map πŸ“–mathematicalCategoryTheory.Presieve.FamilyOfElements.CompatibleCategoryTheory.Presieve.FamilyOfElements.mapβ€”CategoryTheory.FunctorToTypes.naturality
pullback πŸ“–mathematicalCategoryTheory.Presieve.FamilyOfElements.Compatible
CategoryTheory.Sieve.arrows
CategoryTheory.Sieve.pullback
CategoryTheory.Presieve.FamilyOfElements.pullback
β€”CategoryTheory.Sieve.downward_closed
CategoryTheory.Category.assoc
restrict πŸ“–mathematicalCategoryTheory.Presieve
Preorder.toLE
PartialOrder.toPreorder
CompleteSemilatticeInf.toPartialOrder
CompleteLattice.toCompleteSemilatticeInf
CategoryTheory.instCompleteLatticePresieve
CategoryTheory.Presieve.FamilyOfElements.Compatible
CategoryTheory.Presieve.FamilyOfElements.restrictβ€”β€”
sieveExtend πŸ“–mathematicalCategoryTheory.Presieve.FamilyOfElements.CompatibleCategoryTheory.Sieve.arrows
CategoryTheory.Sieve.generate
CategoryTheory.Presieve.FamilyOfElements.sieveExtend
β€”CategoryTheory.FunctorToTypes.map_comp_apply
CategoryTheory.op_comp
CategoryTheory.Category.assoc
to_sieveCompatible πŸ“–mathematicalCategoryTheory.Presieve.FamilyOfElements.Compatible
CategoryTheory.Sieve.arrows
CategoryTheory.Presieve.FamilyOfElements.SieveCompatibleβ€”CategoryTheory.Presieve.compatible_iff_sieveCompatible

CategoryTheory.Presieve.FamilyOfElements.IsAmalgamation

Theorems

NameKindAssumesProvesValidatesDepends On
compPresheafMap πŸ“–mathematicalCategoryTheory.Presieve.FamilyOfElements.IsAmalgamationCategoryTheory.Presieve.FamilyOfElements.map
CategoryTheory.NatTrans.app
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
Opposite.op
β€”map
map πŸ“–mathematicalCategoryTheory.Presieve.FamilyOfElements.IsAmalgamationCategoryTheory.Presieve.FamilyOfElements.map
CategoryTheory.NatTrans.app
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
Opposite.op
β€”CategoryTheory.NatTrans.naturality
CategoryTheory.types_comp_apply

CategoryTheory.Presieve.IsSeparatedFor

Theorems

NameKindAssumesProvesValidatesDepends On
ext πŸ“–β€”CategoryTheory.Presieve.IsSeparatedFor
CategoryTheory.Functor.map
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
Opposite.op
Quiver.Hom.op
CategoryTheory.CategoryStruct.toQuiver
CategoryTheory.Category.toCategoryStruct
β€”β€”β€”
isSheafFor πŸ“–mathematicalCategoryTheory.Presieve.IsSeparatedFor
CategoryTheory.Presieve.FamilyOfElements.IsAmalgamation
CategoryTheory.Presieve.IsSheafForβ€”CategoryTheory.Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor

CategoryTheory.Presieve.IsSheafFor

Definitions

NameCategoryTheorems
amalgamate πŸ“–CompOp
3 mathmath: isAmalgamation, CategoryTheory.Functor.IsCoverDense.sheaf_eq_amalgamation, valid_glue
extend πŸ“–CompOp
3 mathmath: unique_extend, functorInclusion_comp_extend, functorInclusion_comp_extend_assoc

Theorems

NameKindAssumesProvesValidatesDepends On
functorInclusion_comp_extend πŸ“–mathematicalCategoryTheory.Presieve.IsSheafFor
CategoryTheory.Sieve.arrows
CategoryTheory.CategoryStruct.comp
CategoryTheory.Functor
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
CategoryTheory.Category.toCategoryStruct
CategoryTheory.Functor.category
CategoryTheory.Sieve.functor
CategoryTheory.Functor.obj
CategoryTheory.yoneda
CategoryTheory.Sieve.functorInclusion
extend
β€”ExistsUnique.exists
CategoryTheory.Presieve.isSheafFor_iff_yonedaSheafCondition
functorInclusion_comp_extend_assoc πŸ“–mathematicalCategoryTheory.Presieve.IsSheafFor
CategoryTheory.Sieve.arrows
CategoryTheory.CategoryStruct.comp
CategoryTheory.Functor
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
CategoryTheory.Category.toCategoryStruct
CategoryTheory.Functor.category
CategoryTheory.Sieve.functor
CategoryTheory.Functor.obj
CategoryTheory.yoneda
CategoryTheory.Sieve.functorInclusion
extend
β€”CategoryTheory.Category.assoc
Mathlib.Tactic.Reassoc.eq_whisker'
functorInclusion_comp_extend
hom_ext πŸ“–β€”CategoryTheory.Presieve.IsSheafFor
CategoryTheory.Sieve.arrows
CategoryTheory.CategoryStruct.comp
CategoryTheory.Functor
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
CategoryTheory.Category.toCategoryStruct
CategoryTheory.Functor.category
CategoryTheory.Sieve.functor
CategoryTheory.Functor.obj
CategoryTheory.yoneda
CategoryTheory.Sieve.functorInclusion
β€”β€”unique_extend
isAmalgamation πŸ“–mathematicalCategoryTheory.Presieve.IsSheafFor
CategoryTheory.Presieve.FamilyOfElements.Compatible
CategoryTheory.Presieve.FamilyOfElements.IsAmalgamation
amalgamate
β€”ExistsUnique.exists
isSeparatedFor πŸ“–mathematicalCategoryTheory.Presieve.IsSheafForCategoryTheory.Presieve.IsSeparatedForβ€”CategoryTheory.Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor
unique_extend πŸ“–mathematicalCategoryTheory.Presieve.IsSheafFor
CategoryTheory.Sieve.arrows
CategoryTheory.CategoryStruct.comp
CategoryTheory.Functor
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
CategoryTheory.Category.toCategoryStruct
CategoryTheory.Functor.category
CategoryTheory.Sieve.functor
CategoryTheory.Functor.obj
CategoryTheory.yoneda
CategoryTheory.Sieve.functorInclusion
extendβ€”ExistsUnique.unique
CategoryTheory.Presieve.isSheafFor_iff_yonedaSheafCondition
functorInclusion_comp_extend
valid_glue πŸ“–mathematicalCategoryTheory.Presieve.IsSheafFor
CategoryTheory.Presieve.FamilyOfElements.Compatible
CategoryTheory.Functor.map
Opposite
CategoryTheory.Category.opposite
CategoryTheory.types
Opposite.op
Quiver.Hom.op
CategoryTheory.CategoryStruct.toQuiver
CategoryTheory.Category.toCategoryStruct
amalgamate
β€”isAmalgamation

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