Documentation Verification Report

Final

📁 Source: Mathlib/CategoryTheory/Limits/Final.lean

Statistics

MetricCount
DefinitionscoconesEquiv, colimIso, colimitCoconeComp, colimitCoconeOfComp, colimitCompCoyonedaIso, colimitIso, compCreatesColimit, createsColimitOfComp, createsColimitsOfShapeOfFinal, extendCocone, homToLift, induction, isColimitExtendCoconeEquiv, isColimitWhiskerEquiv, Initial, compCreatesLimit, conesEquiv, createsLimitOfComp, createsLimitsOfShapeOfInitial, extendCone, homToLift, induction, isLimitExtendConeEquiv, isLimitWhiskerEquiv, limIso, limitConeComp, limitConeOfComp, limitIso, fiberwiseColimitMapCompEquivalence, structuredArrowToStructuredArrowPre, underPost, overPost
32
Theoremsinitial_pre, coconesEquiv_counitIso, coconesEquiv_functor, coconesEquiv_inverse, coconesEquiv_unitIso, colimitCoconeComp_cocone, colimitCoconeComp_isColimit, colimitCoconeOfComp_cocone, colimitCoconeOfComp_isColimit, colimitIso_hom, colimitIso_inv, colimit_cocone_comp_aux, colimit_pre_isIso, comp_hasColimit, comp_preservesColimit, comp_reflectsColimit, extendCocone_map_hom, extendCocone_obj_pt, extendCocone_obj_ι_app, extendCocone_obj_ι_app', hasColimit_comp_iff, hasColimit_of_comp, hasColimitsOfShape_of_final, instNonemptyStructuredArrow, out, preservesColimit_of_comp, preservesColimitsOfShape_of_final, reflectsColimit_of_comp, reflectsColimitsOfShape_of_final, zigzag_of_eqvGen_colimitTypeRel, ι_colimitIso_hom, ι_colimitIso_hom_assoc, ι_colimitIso_inv, ι_colimitIso_inv_assoc, comp_hasLimit, comp_preservesLimit, comp_reflectsLimit, conesEquiv_counitIso, conesEquiv_functor, conesEquiv_inverse, conesEquiv_unitIso, extendCone_map_hom, extendCone_obj_pt, extendCone_obj_π_app, extendCone_obj_π_app', hasLimit_comp_iff, hasLimit_of_comp, hasLimitsOfShape_of_initial, instNonemptyCostructuredArrow, limitConeComp_cone, limitConeComp_isLimit, limitConeOfComp_cone, limitConeOfComp_isLimit, limitIso_hom, limitIso_inv, limit_cone_comp_aux, limit_pre_isIso, out, preservesLimit_of_comp, preservesLimitsOfShape_of_initial, reflectsLimit_of_comp, reflectsLimitsOfShape_of_initial, final_comp, final_comp_equivalence, final_equivalence_comp, final_fromPUnit_of_isTerminal, final_iff_comp_equivalence, final_iff_comp_final_full_faithful, final_iff_equivalence_comp, final_iff_final_comp, final_iff_isIso_colimit_pre, final_natIso_iff, final_of_adjunction, final_of_colimit_comp_coyoneda_iso_pUnit, final_of_comp_full_faithful, final_of_comp_full_faithful', final_of_equivalence_comp, final_of_final_comp, final_of_initial_op, final_of_isRightAdjoint, final_of_isTerminal_colimit_comp_yoneda, final_of_natIso, final_op_of_initial, initial_comp, initial_comp_equivalence, initial_equivalence_comp, initial_fromPUnit_of_isInitial, initial_iff_comp_equivalence, initial_iff_comp_initial_full_faithful, initial_iff_equivalence_comp, initial_iff_initial_comp, initial_natIso_iff, initial_of_adjunction, initial_of_comp_full_faithful, initial_of_comp_full_faithful', initial_of_equivalence_comp, initial_of_final_op, initial_of_initial_comp, initial_of_isLeftAdjoint, initial_of_natIso, initial_op_of_final, instFinalOppositeLeftOpOfInitial, instFinalOppositeRightOpOfInitial, instInitialOppositeLeftOpOfFinal, instInitialOppositeRightOpOfFinal, final_map, final_pre, of_initial, of_initial, of_final, of_final, initial_ι, final_pre, instFinalProdProd, instInitialCostructuredArrowOverToOver, instInitialProdProd
116
Total148

CategoryTheory

Theorems

NameKindAssumesProvesValidatesDepends On
instFinalProdProd 📖mathematicalFunctor.Final
prod'
Functor.prod
isConnected_of_equivalent
IsConnected.prod
Functor.Final.out
instInitialCostructuredArrowOverToOver 📖mathematicalFunctor.Initial
CostructuredArrow
instCategoryCostructuredArrow
Over
instCategoryOver
CostructuredArrow.toOver
isConnected_iff_of_equivalence
Functor.Initial.out
instInitialProdProd 📖mathematicalFunctor.Initial
prod'
Functor.prod
isConnected_of_equivalent
IsConnected.prod
Functor.Initial.out

CategoryTheory.CostructuredArrow

Theorems

NameKindAssumesProvesValidatesDepends On
initial_pre 📖mathematicalCategoryTheory.Functor.Initial
CategoryTheory.CostructuredArrow
CategoryTheory.Functor.comp
CategoryTheory.instCategoryCostructuredArrow
pre
CategoryTheory.isConnected_iff_of_equivalence
CategoryTheory.Functor.Initial.out

CategoryTheory.Functor

Definitions

NameCategoryTheorems
Initial 📖CompData
64 mathmath: CategoryTheory.TwoSquare.instInitialStructuredArrowObjStructuredArrowDownwardsOfGuitartExact, initial_of_isCofiltered_pUnit, initial_of_adjunction, CategoryTheory.Limits.IsCofiltered.sequentialFunctor_initial, CategoryTheory.InitiallySmall.instInitialCofilteredInitialModelFromCofilteredInitialModel, CategoryTheory.Comma.initial_fst, initial_of_isLeftAdjoint, CategoryTheory.Limits.parallelPair_initial_mk', initial_natIso_iff, CategoryTheory.InitiallySmall.exists_of_isCofiltered, initial_const_initial, initial_comp, CategoryTheory.TwoSquare.guitartExact_iff_initial, initial_of_exists_of_isCofiltered, CategoryTheory.initial_eval, initial_comp_equivalence, CategoryTheory.initial_fst, instInitialOppositeLeftOpOfFinal, initial_iff_equivalence_comp, CategoryTheory.instInitialCostructuredArrowOverToOver, initial_of_exists_of_isCofiltered_of_fullyFaithful, CategoryTheory.CostructuredArrow.initial_post, CategoryTheory.initial_fromInitialModel, CategoryTheory.initial_snd, initial_op_of_final, initial_diag_of_isFiltered, CategoryTheory.Over.initial_forget, initial_fromPUnit_of_isInitial, CategoryTheory.regularTopology.parallelPair_pullback_initial, initial_const_of_isInitial, CategoryTheory.IsFiltered.initial_fst, initial_iff_isCofiltered_costructuredArrow, initial_iff_comp_equivalence, SimplexCategory.Truncated.initial_inclusion, CategoryTheory.ObjectProperty.initial_ι, CategoryTheory.IsFiltered.initial_snd, CategoryTheory.instInitialCostructuredArrowCompPreOfRepresentablyCoflat, initial_iff_of_isCofiltered, CategoryTheory.instInitialProdProd, CategoryTheory.instInitialDiscreteOfIsConnected, CategoryTheory.Comma.initial_snd, localCohomology.ideal_powers_initial, initial_iff_comp_initial_full_faithful, CategoryTheory.Limits.parallelPair_initial_mk, initial_of_final_op, CategoryTheory.isConnected_iff_initial_of_unique, CategoryTheory.Join.instInitialInclLeftOfIsConnected, LightProfinite.Extend.functor_initial, initial_equivalence_comp, initial_iff_initial_comp, initial_of_natIso, CategoryTheory.CostructuredArrow.initial_map₂_id, initial_of_initial_comp, initial_of_comp_full_faithful, CategoryTheory.CostructuredArrow.initial_proj_of_isCofiltered, CategoryTheory.initial_of_representablyCoflat, Profinite.Extend.functor_initial, initial_of_comp_full_faithful', SimplexCategory.Truncated.initial_incl, initial_of_isCofiltered_costructuredArrow, CategoryTheory.CostructuredArrow.initial_pre, instInitialOppositeRightOpOfFinal, initial_of_equivalence_comp, CategoryTheory.InitiallySmall.initial_smallCategory

Theorems

NameKindAssumesProvesValidatesDepends On
final_comp 📖mathematicalFinal
comp
final_iff_comp_equivalence
CategoryTheory.Equivalence.isEquivalence_functor
final_iff_equivalence_comp
CategoryTheory.Equivalence.isEquivalence_inverse
final_natIso_iff
CategoryTheory.Limits.Types.hasColimit
UnivLE.small
UnivLE.self
final_iff_isIso_colimit_pre
CategoryTheory.Limits.colimit.pre_pre
CategoryTheory.IsIso.comp_isIso
final_comp_equivalence 📖mathematicalFinal
comp
final_of_natIso
final_of_comp_full_faithful
IsEquivalence.full
isEquivalence_inv
IsEquivalence.faithful
final_equivalence_comp 📖mathematicalFinal
comp
CategoryTheory.isConnected_of_equivalent
CategoryTheory.StructuredArrow.isEquivalence_pre
Final.out
final_fromPUnit_of_isTerminal 📖mathematicalFinal
CategoryTheory.Discrete
CategoryTheory.discreteCategory
fromPUnit
CategoryTheory.isConnected_of_nonempty_and_subsingleton
CategoryTheory.StructuredArrow.obj_ext
CategoryTheory.Discrete.instSubsingleton
CategoryTheory.Limits.IsTerminal.hom_ext
final_iff_comp_equivalence 📖mathematicalFinal
comp
final_comp_equivalence
final_of_comp_full_faithful
IsEquivalence.full
IsEquivalence.faithful
final_iff_comp_final_full_faithful 📖mathematicalFinal
comp
final_comp
final_of_comp_full_faithful
final_iff_equivalence_comp 📖mathematicalFinal
comp
final_equivalence_comp
final_of_equivalence_comp
final_iff_final_comp 📖mathematicalFinal
comp
final_comp
final_of_final_comp
final_iff_isIso_colimit_pre 📖mathematicalFinal
CategoryTheory.IsIso
CategoryTheory.types
CategoryTheory.Limits.colimit
comp
CategoryTheory.Limits.Types.hasColimit
UnivLE.small
UnivLE.self
CategoryTheory.Limits.colimit.pre
CategoryTheory.Limits.Types.hasColimit
UnivLE.small
UnivLE.self
Final.colimit_pre_isIso
final_of_colimit_comp_coyoneda_iso_pUnit
final_natIso_iff 📖mathematicalFinalfinal_of_natIso
final_of_adjunction 📖mathematicalFinalCategoryTheory.zigzag_isConnected
Relation.ReflTransGen.trans
Relation.ReflTransGen.single
CategoryTheory.Adjunction.homEquiv_counit
map_comp
CategoryTheory.Adjunction.unit_naturality_assoc
CategoryTheory.Adjunction.right_triangle_components
CategoryTheory.Category.comp_id
final_of_colimit_comp_coyoneda_iso_pUnit 📖mathematicalFinalCategoryTheory.Limits.Types.hasColimit
UnivLE.small
UnivLE.self
CategoryTheory.Limits.Types.jointly_surjective'
CategoryTheory.zigzag_isConnected
Equiv.injective
CategoryTheory.Limits.Types.colimit_eq
Final.zigzag_of_eqvGen_colimitTypeRel
final_of_comp_full_faithful 📖mathematicalFinalCategoryTheory.isConnected_of_equivalent
CategoryTheory.StructuredArrow.isEquivalence_post
Final.out
final_of_comp_full_faithful' 📖mathematicalFinalfinal_of_comp_full_faithful
final_of_final_comp
final_of_equivalence_comp 📖mathematicalFinalCategoryTheory.isConnected_of_equivalent
CategoryTheory.StructuredArrow.isEquivalence_pre
Final.out
final_of_final_comp 📖mathematicalFinalfinal_iff_comp_equivalence
CategoryTheory.Equivalence.isEquivalence_functor
final_iff_equivalence_comp
CategoryTheory.Equivalence.isEquivalence_inverse
CategoryTheory.Limits.Types.hasColimit
UnivLE.small
UnivLE.self
final_iff_isIso_colimit_pre
final_natIso_iff
CategoryTheory.IsIso.of_isIso_comp_left
CategoryTheory.Limits.colimit.pre_pre
final_of_initial_op 📖mathematicalFinalCategoryTheory.isConnected_of_isConnected_op
CategoryTheory.isConnected_of_equivalent
Initial.out
final_of_isRightAdjoint 📖mathematicalFinalfinal_of_adjunction
final_of_isTerminal_colimit_comp_yoneda 📖mathematicalFinalCategoryTheory.Limits.functorCategoryHasColimit
CategoryTheory.Limits.Types.hasColimit
UnivLE.small
UnivLE.self
final_of_colimit_comp_coyoneda_iso_pUnit
CategoryTheory.Limits.evaluation_preservesLimit
CategoryTheory.Limits.Types.hasLimit
CategoryTheory.instSmallDiscrete
univLE_of_max
CategoryTheory.Limits.evaluation_preservesColimit
final_of_natIso 📖mathematicalFinalCategoryTheory.isConnected_of_equivalent
Final.out
final_op_of_initial 📖mathematicalFinal
Opposite
CategoryTheory.Category.opposite
op
CategoryTheory.isConnected_of_equivalent
CategoryTheory.isConnected_op
Initial.out
initial_comp 📖mathematicalInitial
comp
final_comp
final_op_of_initial
initial_of_final_op
initial_comp_equivalence 📖mathematicalInitial
comp
initial_of_natIso
initial_of_comp_full_faithful
IsEquivalence.full
isEquivalence_inv
IsEquivalence.faithful
initial_equivalence_comp 📖mathematicalInitial
comp
CategoryTheory.isConnected_of_equivalent
CategoryTheory.CostructuredArrow.isEquivalence_pre
Initial.out
initial_fromPUnit_of_isInitial 📖mathematicalInitial
CategoryTheory.Discrete
CategoryTheory.discreteCategory
fromPUnit
CategoryTheory.isConnected_of_nonempty_and_subsingleton
CategoryTheory.CostructuredArrow.obj_ext
CategoryTheory.Discrete.instSubsingleton
CategoryTheory.Limits.IsInitial.hom_ext
initial_iff_comp_equivalence 📖mathematicalInitial
comp
initial_comp_equivalence
initial_of_comp_full_faithful
IsEquivalence.full
IsEquivalence.faithful
initial_iff_comp_initial_full_faithful 📖mathematicalInitial
comp
initial_comp
initial_of_comp_full_faithful
initial_iff_equivalence_comp 📖mathematicalInitial
comp
initial_equivalence_comp
initial_of_equivalence_comp
initial_iff_initial_comp 📖mathematicalInitial
comp
initial_comp
initial_of_initial_comp
initial_natIso_iff 📖mathematicalInitialinitial_of_natIso
initial_of_adjunction 📖mathematicalInitialCategoryTheory.zigzag_isConnected
Relation.ReflTransGen.trans
Relation.ReflTransGen.single
CategoryTheory.Adjunction.homEquiv_unit
map_comp
CategoryTheory.Category.assoc
CategoryTheory.Adjunction.counit_naturality
CategoryTheory.Adjunction.left_triangle_components_assoc
initial_of_comp_full_faithful 📖mathematicalInitialCategoryTheory.isConnected_of_equivalent
CategoryTheory.CostructuredArrow.isEquivalence_post
Initial.out
initial_of_comp_full_faithful' 📖mathematicalInitialinitial_of_comp_full_faithful
initial_of_initial_comp
initial_of_equivalence_comp 📖mathematicalInitialCategoryTheory.isConnected_of_equivalent
CategoryTheory.CostructuredArrow.isEquivalence_pre
Initial.out
initial_of_final_op 📖mathematicalInitialCategoryTheory.isConnected_of_isConnected_op
CategoryTheory.isConnected_of_equivalent
Final.out
initial_of_initial_comp 📖mathematicalInitialfinal_op_of_initial
final_of_final_comp
initial_of_final_op
initial_of_isLeftAdjoint 📖mathematicalInitialinitial_of_adjunction
initial_of_natIso 📖mathematicalInitialCategoryTheory.isConnected_of_equivalent
Initial.out
initial_op_of_final 📖mathematicalInitial
Opposite
CategoryTheory.Category.opposite
op
CategoryTheory.isConnected_of_equivalent
CategoryTheory.isConnected_op
Final.out
instFinalOppositeLeftOpOfInitial 📖mathematicalFinal
Opposite
CategoryTheory.Category.opposite
leftOp
final_comp
final_op_of_initial
final_of_isRightAdjoint
isRightAdjoint_of_isEquivalence
CategoryTheory.Equivalence.isEquivalence_functor
instFinalOppositeRightOpOfInitial 📖mathematicalFinal
Opposite
CategoryTheory.Category.opposite
rightOp
final_comp
final_of_isRightAdjoint
isRightAdjoint_of_isEquivalence
CategoryTheory.Equivalence.isEquivalence_inverse
final_op_of_initial
instInitialOppositeLeftOpOfFinal 📖mathematicalInitial
Opposite
CategoryTheory.Category.opposite
leftOp
initial_comp
initial_op_of_final
initial_of_isLeftAdjoint
isLeftAdjoint_of_isEquivalence
CategoryTheory.Equivalence.isEquivalence_functor
instInitialOppositeRightOpOfFinal 📖mathematicalInitial
Opposite
CategoryTheory.Category.opposite
rightOp
initial_comp
initial_of_isLeftAdjoint
isLeftAdjoint_of_isEquivalence
CategoryTheory.Equivalence.isEquivalence_inverse
initial_op_of_final

CategoryTheory.Functor.Final

Definitions

NameCategoryTheorems
coconesEquiv 📖CompOp
4 mathmath: coconesEquiv_unitIso, coconesEquiv_functor, coconesEquiv_counitIso, coconesEquiv_inverse
colimIso 📖CompOp
colimitCoconeComp 📖CompOp
2 mathmath: colimitCoconeComp_cocone, colimitCoconeComp_isColimit
colimitCoconeOfComp 📖CompOp
2 mathmath: colimitCoconeOfComp_isColimit, colimitCoconeOfComp_cocone
colimitCompCoyonedaIso 📖CompOp
colimitIso 📖CompOp
6 mathmath: ι_colimitIso_hom_assoc, ι_colimitIso_inv, colimitIso_hom, ι_colimitIso_inv_assoc, ι_colimitIso_hom, colimitIso_inv
compCreatesColimit 📖CompOp
createsColimitOfComp 📖CompOp
createsColimitsOfShapeOfFinal 📖CompOp
extendCocone 📖CompOp
9 mathmath: colimitCoconeOfComp_isColimit, coconesEquiv_unitIso, extendCocone_obj_ι_app, coconesEquiv_functor, extendCocone_obj_ι_app', coconesEquiv_counitIso, extendCocone_obj_pt, extendCocone_map_hom, colimitCoconeOfComp_cocone
homToLift 📖CompOp
3 mathmath: extendCocone_obj_ι_app, colimit_cocone_comp_aux, extendCocone_map_hom
induction 📖CompOp
isColimitExtendCoconeEquiv 📖CompOp
1 mathmath: colimitCoconeOfComp_isColimit
isColimitWhiskerEquiv 📖CompOp
2 mathmath: colimitCoconeComp_isColimit, CategoryTheory.TransfiniteCompositionOfShape.ici_isColimit

Theorems

NameKindAssumesProvesValidatesDepends On
coconesEquiv_counitIso 📖mathematicalCategoryTheory.Equivalence.counitIso
CategoryTheory.Limits.Cocone
CategoryTheory.Functor.comp
CategoryTheory.Limits.Cocone.category
coconesEquiv
CategoryTheory.NatIso.ofComponents
CategoryTheory.Limits.Cocones.whiskering
extendCocone
CategoryTheory.Functor.id
CategoryTheory.Limits.Cocones.ext
CategoryTheory.Functor.obj
CategoryTheory.Iso.refl
CategoryTheory.Limits.Cocone.pt
coconesEquiv_functor 📖mathematicalCategoryTheory.Equivalence.functor
CategoryTheory.Limits.Cocone
CategoryTheory.Functor.comp
CategoryTheory.Limits.Cocone.category
coconesEquiv
extendCocone
coconesEquiv_inverse 📖mathematicalCategoryTheory.Equivalence.inverse
CategoryTheory.Limits.Cocone
CategoryTheory.Functor.comp
CategoryTheory.Limits.Cocone.category
coconesEquiv
CategoryTheory.Limits.Cocones.whiskering
coconesEquiv_unitIso 📖mathematicalCategoryTheory.Equivalence.unitIso
CategoryTheory.Limits.Cocone
CategoryTheory.Functor.comp
CategoryTheory.Limits.Cocone.category
coconesEquiv
CategoryTheory.NatIso.ofComponents
CategoryTheory.Functor.id
extendCocone
CategoryTheory.Limits.Cocones.whiskering
CategoryTheory.Limits.Cocones.ext
CategoryTheory.Functor.obj
CategoryTheory.Iso.refl
CategoryTheory.Limits.Cocone.pt
colimitCoconeComp_cocone 📖mathematicalCategoryTheory.Limits.ColimitCocone.cocone
CategoryTheory.Functor.comp
colimitCoconeComp
CategoryTheory.Limits.Cocone.whisker
colimitCoconeComp_isColimit 📖mathematicalCategoryTheory.Limits.ColimitCocone.isColimit
CategoryTheory.Functor.comp
colimitCoconeComp
DFunLike.coe
Equiv
CategoryTheory.Limits.IsColimit
CategoryTheory.Limits.ColimitCocone.cocone
CategoryTheory.Limits.Cocone.whisker
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
isColimitWhiskerEquiv
colimitCoconeOfComp_cocone 📖mathematicalCategoryTheory.Limits.ColimitCocone.cocone
colimitCoconeOfComp
CategoryTheory.Functor.obj
CategoryTheory.Limits.Cocone
CategoryTheory.Functor.comp
CategoryTheory.Limits.Cocone.category
extendCocone
colimitCoconeOfComp_isColimit 📖mathematicalCategoryTheory.Limits.ColimitCocone.isColimit
colimitCoconeOfComp
DFunLike.coe
Equiv
CategoryTheory.Limits.IsColimit
CategoryTheory.Functor.comp
CategoryTheory.Limits.ColimitCocone.cocone
CategoryTheory.Functor.obj
CategoryTheory.Limits.Cocone
CategoryTheory.Limits.Cocone.category
extendCocone
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
isColimitExtendCoconeEquiv
colimitIso_hom 📖mathematicalCategoryTheory.Iso.hom
CategoryTheory.Limits.colimit
CategoryTheory.Functor.comp
comp_hasColimit
colimitIso
CategoryTheory.Limits.colimit.pre
comp_hasColimit
colimitIso_inv 📖mathematicalCategoryTheory.Iso.inv
CategoryTheory.Limits.colimit
CategoryTheory.Functor.comp
comp_hasColimit
colimitIso
CategoryTheory.inv
CategoryTheory.Limits.colimit.pre
colimit_pre_isIso
comp_hasColimit
colimit_cocone_comp_aux 📖mathematicalCategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
CategoryTheory.Functor.obj
lift
CategoryTheory.Functor
CategoryTheory.Functor.category
CategoryTheory.Functor.const
CategoryTheory.Limits.Cocone.pt
CategoryTheory.Functor.comp
CategoryTheory.Functor.map
homToLift
CategoryTheory.NatTrans.app
CategoryTheory.Limits.Cocone.ι
CategoryTheory.Functor.map_comp
CategoryTheory.Category.assoc
CategoryTheory.Limits.Cocone.w
colimit_pre_isIso 📖mathematicalCategoryTheory.IsIso
CategoryTheory.Limits.colimit
CategoryTheory.Functor.comp
comp_hasColimit
CategoryTheory.Limits.colimit.pre
comp_hasColimit
CategoryTheory.Limits.colimit.pre_eq
CategoryTheory.Limits.IsColimit.desc_self
CategoryTheory.IsIso.comp_isIso
CategoryTheory.Iso.isIso_hom
CategoryTheory.IsIso.id
CategoryTheory.Iso.isIso_inv
comp_hasColimit 📖mathematicalCategoryTheory.Limits.HasColimit
CategoryTheory.Functor.comp
comp_preservesColimit 📖mathematicalCategoryTheory.Limits.PreservesColimit
CategoryTheory.Functor.comp
CategoryTheory.Functor.map_comp
CategoryTheory.Category.comp_id
comp_reflectsColimit 📖mathematicalCategoryTheory.Limits.ReflectsColimit
CategoryTheory.Functor.comp
CategoryTheory.Category.comp_id
CategoryTheory.Functor.map_comp
extendCocone_map_hom 📖mathematicalCategoryTheory.Limits.CoconeMorphism.hom
CategoryTheory.Limits.Cocone.pt
CategoryTheory.Functor.comp
CategoryTheory.Functor.obj
CategoryTheory.Functor
CategoryTheory.Functor.category
CategoryTheory.Functor.const
CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
lift
CategoryTheory.Functor.map
homToLift
CategoryTheory.NatTrans.app
CategoryTheory.Limits.Cocone.ι
CategoryTheory.Limits.Cocone
CategoryTheory.Limits.Cocone.category
extendCocone
extendCocone_obj_pt 📖mathematicalCategoryTheory.Limits.Cocone.pt
CategoryTheory.Functor.obj
CategoryTheory.Limits.Cocone
CategoryTheory.Functor.comp
CategoryTheory.Limits.Cocone.category
extendCocone
extendCocone_obj_ι_app 📖mathematicalCategoryTheory.NatTrans.app
CategoryTheory.Functor.obj
CategoryTheory.Functor
CategoryTheory.Functor.category
CategoryTheory.Functor.const
CategoryTheory.Limits.Cocone.pt
CategoryTheory.Functor.comp
CategoryTheory.Limits.Cocone.ι
CategoryTheory.Limits.Cocone
CategoryTheory.Limits.Cocone.category
extendCocone
CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
lift
CategoryTheory.Functor.map
homToLift
extendCocone_obj_ι_app' 📖mathematicalCategoryTheory.NatTrans.app
CategoryTheory.Functor.obj
CategoryTheory.Functor
CategoryTheory.Functor.category
CategoryTheory.Functor.const
CategoryTheory.Limits.Cocone.pt
CategoryTheory.Limits.Cocone
CategoryTheory.Functor.comp
CategoryTheory.Limits.Cocone.category
extendCocone
CategoryTheory.Limits.Cocone.ι
CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
CategoryTheory.Functor.map
CategoryTheory.Functor.map_comp
CategoryTheory.Category.assoc
CategoryTheory.NatTrans.naturality
CategoryTheory.Category.comp_id
hasColimit_comp_iff 📖mathematicalCategoryTheory.Limits.HasColimit
CategoryTheory.Functor.comp
hasColimit_of_comp
comp_hasColimit
hasColimit_of_comp 📖mathematicalCategoryTheory.Limits.HasColimit
hasColimitsOfShape_of_final 📖mathematicalCategoryTheory.Limits.HasColimitsOfShapehasColimit_of_comp
CategoryTheory.Limits.hasColimitOfHasColimitsOfShape
instNonemptyStructuredArrow 📖mathematicalCategoryTheory.StructuredArrowCategoryTheory.IsConnected.is_nonempty
out
out 📖mathematicalCategoryTheory.IsConnected
CategoryTheory.StructuredArrow
CategoryTheory.instCategoryStructuredArrow
preservesColimit_of_comp 📖mathematicalCategoryTheory.Limits.PreservesColimitCategoryTheory.Category.comp_id
preservesColimitsOfShape_of_final 📖mathematicalCategoryTheory.Limits.PreservesColimitsOfShapepreservesColimit_of_comp
CategoryTheory.Limits.PreservesColimitsOfShape.preservesColimit
reflectsColimit_of_comp 📖mathematicalCategoryTheory.Limits.ReflectsColimitCategoryTheory.Category.comp_id
reflectsColimitsOfShape_of_final 📖mathematicalCategoryTheory.Limits.ReflectsColimitsOfShapereflectsColimit_of_comp
CategoryTheory.Limits.reflectsColimit_of_reflectsColimitsOfShape
zigzag_of_eqvGen_colimitTypeRel 📖mathematicalRelation.EqvGen
CategoryTheory.Functor.obj
CategoryTheory.types
CategoryTheory.Functor.comp
Opposite
CategoryTheory.Category.opposite
CategoryTheory.Functor
CategoryTheory.Functor.category
CategoryTheory.coyoneda
Opposite.op
CategoryTheory.Functor.ColimitTypeRel
CategoryTheory.Zigzag
CategoryTheory.StructuredArrow
CategoryTheory.instCategoryStructuredArrow
Quiver.Hom
CategoryTheory.CategoryStruct.toQuiver
CategoryTheory.Category.toCategoryStruct
CategoryTheory.zigzag_symmetric
Relation.ReflTransGen.trans
ι_colimitIso_hom 📖mathematicalCategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
CategoryTheory.Functor.obj
CategoryTheory.Functor.comp
CategoryTheory.Limits.colimit
comp_hasColimit
CategoryTheory.Limits.colimit.ι
CategoryTheory.Iso.hom
colimitIso
comp_hasColimit
CategoryTheory.Limits.colimit.ι_pre
ι_colimitIso_hom_assoc 📖mathematicalCategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
CategoryTheory.Functor.obj
CategoryTheory.Limits.colimit
CategoryTheory.Functor.comp
comp_hasColimit
CategoryTheory.Limits.colimit.ι
CategoryTheory.Iso.hom
colimitIso
comp_hasColimit
CategoryTheory.Category.assoc
Mathlib.Tactic.Reassoc.eq_whisker'
ι_colimitIso_hom
ι_colimitIso_inv 📖mathematicalCategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
CategoryTheory.Functor.obj
CategoryTheory.Limits.colimit
CategoryTheory.Functor.comp
comp_hasColimit
CategoryTheory.Limits.colimit.ι
CategoryTheory.Iso.inv
colimitIso
comp_hasColimit
CategoryTheory.Limits.colimit.ι_inv_pre
colimit_pre_isIso
ι_colimitIso_inv_assoc 📖mathematicalCategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
CategoryTheory.Functor.obj
CategoryTheory.Limits.colimit
CategoryTheory.Limits.colimit.ι
CategoryTheory.Functor.comp
comp_hasColimit
CategoryTheory.Iso.inv
colimitIso
comp_hasColimit
CategoryTheory.Category.assoc
Mathlib.Tactic.Reassoc.eq_whisker'
ι_colimitIso_inv

CategoryTheory.Functor.Initial

Definitions

NameCategoryTheorems
compCreatesLimit 📖CompOp
conesEquiv 📖CompOp
4 mathmath: conesEquiv_counitIso, conesEquiv_unitIso, conesEquiv_inverse, conesEquiv_functor
createsLimitOfComp 📖CompOp
createsLimitsOfShapeOfInitial 📖CompOp
extendCone 📖CompOp
9 mathmath: extendCone_obj_pt, conesEquiv_counitIso, extendCone_obj_π_app', conesEquiv_unitIso, extendCone_map_hom, limitConeOfComp_cone, extendCone_obj_π_app, conesEquiv_functor, limitConeOfComp_isLimit
homToLift 📖CompOp
3 mathmath: extendCone_map_hom, limit_cone_comp_aux, extendCone_obj_π_app
induction 📖CompOp
isLimitExtendConeEquiv 📖CompOp
1 mathmath: limitConeOfComp_isLimit
isLimitWhiskerEquiv 📖CompOp
1 mathmath: limitConeComp_isLimit
limIso 📖CompOp
limitConeComp 📖CompOp
2 mathmath: limitConeComp_isLimit, limitConeComp_cone
limitConeOfComp 📖CompOp
2 mathmath: limitConeOfComp_cone, limitConeOfComp_isLimit
limitIso 📖CompOp
2 mathmath: limitIso_hom, limitIso_inv

Theorems

NameKindAssumesProvesValidatesDepends On
comp_hasLimit 📖mathematicalCategoryTheory.Limits.HasLimit
CategoryTheory.Functor.comp
comp_preservesLimit 📖mathematicalCategoryTheory.Limits.PreservesLimit
CategoryTheory.Functor.comp
CategoryTheory.Functor.map_comp
CategoryTheory.Category.id_comp
comp_reflectsLimit 📖mathematicalCategoryTheory.Limits.ReflectsLimit
CategoryTheory.Functor.comp
CategoryTheory.Functor.map_comp
CategoryTheory.Category.id_comp
conesEquiv_counitIso 📖mathematicalCategoryTheory.Equivalence.counitIso
CategoryTheory.Limits.Cone
CategoryTheory.Functor.comp
CategoryTheory.Limits.Cone.category
conesEquiv
CategoryTheory.NatIso.ofComponents
CategoryTheory.Limits.Cones.whiskering
extendCone
CategoryTheory.Functor.id
CategoryTheory.Limits.Cones.ext
CategoryTheory.Functor.obj
CategoryTheory.Iso.refl
CategoryTheory.Limits.Cone.pt
conesEquiv_functor 📖mathematicalCategoryTheory.Equivalence.functor
CategoryTheory.Limits.Cone
CategoryTheory.Functor.comp
CategoryTheory.Limits.Cone.category
conesEquiv
extendCone
conesEquiv_inverse 📖mathematicalCategoryTheory.Equivalence.inverse
CategoryTheory.Limits.Cone
CategoryTheory.Functor.comp
CategoryTheory.Limits.Cone.category
conesEquiv
CategoryTheory.Limits.Cones.whiskering
conesEquiv_unitIso 📖mathematicalCategoryTheory.Equivalence.unitIso
CategoryTheory.Limits.Cone
CategoryTheory.Functor.comp
CategoryTheory.Limits.Cone.category
conesEquiv
CategoryTheory.NatIso.ofComponents
CategoryTheory.Functor.id
extendCone
CategoryTheory.Limits.Cones.whiskering
CategoryTheory.Limits.Cones.ext
CategoryTheory.Functor.obj
CategoryTheory.Iso.refl
CategoryTheory.Limits.Cone.pt
extendCone_map_hom 📖mathematicalCategoryTheory.Limits.ConeMorphism.hom
CategoryTheory.Limits.Cone.pt
CategoryTheory.Functor.comp
CategoryTheory.Functor.obj
CategoryTheory.Functor
CategoryTheory.Functor.category
CategoryTheory.Functor.const
CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
lift
CategoryTheory.NatTrans.app
CategoryTheory.Limits.Cone.π
CategoryTheory.Functor.map
homToLift
CategoryTheory.Limits.Cone
CategoryTheory.Limits.Cone.category
extendCone
extendCone_obj_pt 📖mathematicalCategoryTheory.Limits.Cone.pt
CategoryTheory.Functor.obj
CategoryTheory.Limits.Cone
CategoryTheory.Functor.comp
CategoryTheory.Limits.Cone.category
extendCone
extendCone_obj_π_app 📖mathematicalCategoryTheory.NatTrans.app
CategoryTheory.Functor.obj
CategoryTheory.Functor
CategoryTheory.Functor.category
CategoryTheory.Functor.const
CategoryTheory.Limits.Cone.pt
CategoryTheory.Functor.comp
CategoryTheory.Limits.Cone.π
CategoryTheory.Limits.Cone
CategoryTheory.Limits.Cone.category
extendCone
CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
lift
CategoryTheory.Functor.map
homToLift
extendCone_obj_π_app' 📖mathematicalCategoryTheory.NatTrans.app
CategoryTheory.Functor.obj
CategoryTheory.Functor
CategoryTheory.Functor.category
CategoryTheory.Functor.const
CategoryTheory.Limits.Cone.pt
CategoryTheory.Limits.Cone
CategoryTheory.Functor.comp
CategoryTheory.Limits.Cone.category
extendCone
CategoryTheory.Limits.Cone.π
CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
CategoryTheory.Functor.map
CategoryTheory.Functor.map_comp
CategoryTheory.Limits.Cone.w_assoc
hasLimit_comp_iff 📖mathematicalCategoryTheory.Limits.HasLimit
CategoryTheory.Functor.comp
hasLimit_of_comp
comp_hasLimit
hasLimit_of_comp 📖mathematicalCategoryTheory.Limits.HasLimit
hasLimitsOfShape_of_initial 📖mathematicalCategoryTheory.Limits.HasLimitsOfShapehasLimit_of_comp
CategoryTheory.Limits.hasLimitOfHasLimitsOfShape
instNonemptyCostructuredArrow 📖mathematicalCategoryTheory.CostructuredArrowCategoryTheory.IsConnected.is_nonempty
out
limitConeComp_cone 📖mathematicalCategoryTheory.Limits.LimitCone.cone
CategoryTheory.Functor.comp
limitConeComp
CategoryTheory.Limits.Cone.whisker
limitConeComp_isLimit 📖mathematicalCategoryTheory.Limits.LimitCone.isLimit
CategoryTheory.Functor.comp
limitConeComp
DFunLike.coe
Equiv
CategoryTheory.Limits.IsLimit
CategoryTheory.Limits.LimitCone.cone
CategoryTheory.Limits.Cone.whisker
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
isLimitWhiskerEquiv
limitConeOfComp_cone 📖mathematicalCategoryTheory.Limits.LimitCone.cone
limitConeOfComp
CategoryTheory.Functor.obj
CategoryTheory.Limits.Cone
CategoryTheory.Functor.comp
CategoryTheory.Limits.Cone.category
extendCone
limitConeOfComp_isLimit 📖mathematicalCategoryTheory.Limits.LimitCone.isLimit
limitConeOfComp
DFunLike.coe
Equiv
CategoryTheory.Limits.IsLimit
CategoryTheory.Functor.comp
CategoryTheory.Limits.LimitCone.cone
CategoryTheory.Functor.obj
CategoryTheory.Limits.Cone
CategoryTheory.Limits.Cone.category
extendCone
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
isLimitExtendConeEquiv
limitIso_hom 📖mathematicalCategoryTheory.Iso.hom
CategoryTheory.Limits.limit
CategoryTheory.Functor.comp
comp_hasLimit
limitIso
CategoryTheory.inv
CategoryTheory.Limits.limit.pre
limit_pre_isIso
comp_hasLimit
limitIso_inv 📖mathematicalCategoryTheory.Iso.inv
CategoryTheory.Limits.limit
CategoryTheory.Functor.comp
comp_hasLimit
limitIso
CategoryTheory.Limits.limit.pre
comp_hasLimit
limit_cone_comp_aux 📖mathematicalCategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
CategoryTheory.Functor.obj
CategoryTheory.Functor
CategoryTheory.Functor.category
CategoryTheory.Functor.const
CategoryTheory.Limits.Cone.pt
CategoryTheory.Functor.comp
lift
CategoryTheory.NatTrans.app
CategoryTheory.Limits.Cone.π
CategoryTheory.Functor.map
homToLift
CategoryTheory.Limits.Cone.w
CategoryTheory.Category.assoc
CategoryTheory.Functor.map_comp
limit_pre_isIso 📖mathematicalCategoryTheory.IsIso
CategoryTheory.Limits.limit
CategoryTheory.Functor.comp
comp_hasLimit
CategoryTheory.Limits.limit.pre
comp_hasLimit
CategoryTheory.Limits.limit.pre_eq
CategoryTheory.Limits.IsLimit.lift_self
CategoryTheory.IsIso.comp_isIso
CategoryTheory.Iso.isIso_hom
CategoryTheory.IsIso.id
CategoryTheory.Iso.isIso_inv
out 📖mathematicalCategoryTheory.IsConnected
CategoryTheory.CostructuredArrow
CategoryTheory.instCategoryCostructuredArrow
preservesLimit_of_comp 📖mathematicalCategoryTheory.Limits.PreservesLimitCategoryTheory.Category.id_comp
preservesLimitsOfShape_of_initial 📖mathematicalCategoryTheory.Limits.PreservesLimitsOfShapepreservesLimit_of_comp
CategoryTheory.Limits.PreservesLimitsOfShape.preservesLimit
reflectsLimit_of_comp 📖mathematicalCategoryTheory.Limits.ReflectsLimitCategoryTheory.Category.id_comp
reflectsLimitsOfShape_of_initial 📖mathematicalCategoryTheory.Limits.ReflectsLimitsOfShapereflectsLimit_of_comp
CategoryTheory.Limits.reflectsLimit_of_reflectsLimitsOfShape

CategoryTheory.Grothendieck

Definitions

NameCategoryTheorems
fiberwiseColimitMapCompEquivalence 📖CompOp
structuredArrowToStructuredArrowPre 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
final_map 📖mathematicalCategoryTheory.Functor.Final
CategoryTheory.Bundled.α
CategoryTheory.Category
CategoryTheory.Functor.obj
CategoryTheory.Cat
CategoryTheory.Cat.category
CategoryTheory.Cat.str
CategoryTheory.Cat.Hom.toFunctor
CategoryTheory.NatTrans.app
CategoryTheory.Grothendieck
instCategory
map
CategoryTheory.Functor.final_comp
CategoryTheory.Functor.final_of_isRightAdjoint
CategoryTheory.Functor.isRightAdjoint_of_isEquivalence
CategoryTheory.Equivalence.isEquivalence_inverse
CategoryTheory.Equivalence.isEquivalence_functor
CategoryTheory.Functor.final_of_natIso
CategoryTheory.Functor.final_iff_comp_equivalence
CategoryTheory.Functor.final_iff_equivalence_comp
final_pre 📖mathematicalCategoryTheory.Functor.Final
CategoryTheory.Grothendieck
CategoryTheory.Functor.comp
CategoryTheory.Cat
CategoryTheory.Cat.category
instCategory
pre
CategoryTheory.Functor.Final.instNonemptyStructuredArrow
CategoryTheory.zigzag_isConnected
CategoryTheory.Zigzag.trans
CategoryTheory.Zigzag.of_zag
CategoryTheory.Functor.map_id
ext
CategoryTheory.Category.comp_id
CategoryTheory.eqToHom_naturality_assoc
CategoryTheory.Category.id_comp
CategoryTheory.eqToHom_trans_assoc
CategoryTheory.zigzag_prefunctor_obj_of_zigzag
CategoryTheory.isPreconnected_zigzag
CategoryTheory.IsConnected.toIsPreconnected
CategoryTheory.Functor.Final.out

CategoryTheory.IsCofiltered

Theorems

NameKindAssumesProvesValidatesDepends On
of_initial 📖mathematicalCategoryTheory.IsCofilteredCategoryTheory.IsFiltered.of_final
CategoryTheory.Functor.final_op_of_initial
CategoryTheory.isFiltered_op_of_isCofiltered
CategoryTheory.isCofiltered_of_isFiltered_op

CategoryTheory.IsCofilteredOrEmpty

Theorems

NameKindAssumesProvesValidatesDepends On
of_initial 📖mathematicalCategoryTheory.IsCofilteredOrEmptyCategoryTheory.IsFilteredOrEmpty.of_final
CategoryTheory.Functor.final_op_of_initial
CategoryTheory.isFilteredOrEmpty_op_of_isCofilteredOrEmpty
CategoryTheory.isCofilteredOrEmpty_of_isFilteredOrEmpty_op

CategoryTheory.IsFiltered

Theorems

NameKindAssumesProvesValidatesDepends On
of_final 📖mathematicalCategoryTheory.IsFilteredCategoryTheory.IsFilteredOrEmpty.of_final
toIsFilteredOrEmpty
Nonempty.map
nonempty

CategoryTheory.IsFilteredOrEmpty

Theorems

NameKindAssumesProvesValidatesDepends On
of_final 📖mathematicalCategoryTheory.IsFilteredOrEmptyCategoryTheory.Functor.map_id
CategoryTheory.Category.comp_id
CategoryTheory.isPreconnected_induction
CategoryTheory.IsConnected.toIsPreconnected
CategoryTheory.Functor.Final.out
CategoryTheory.IsFiltered.span
CategoryTheory.Functor.map_comp
CategoryTheory.Category.assoc
Mathlib.Tactic.Reassoc.eq_whisker'
CategoryTheory.StructuredArrow.w_assoc
CategoryTheory.IsFiltered.coeq_condition

CategoryTheory.Limits.IsColimit

Definitions

NameCategoryTheorems
underPost 📖CompOp

CategoryTheory.Limits.IsLimit

Definitions

NameCategoryTheorems
overPost 📖CompOp

CategoryTheory.ObjectProperty

Theorems

NameKindAssumesProvesValidatesDepends On
initial_ι 📖mathematicalCategoryTheory.IsConnected
CategoryTheory.CostructuredArrow
FullSubcategory
FullSubcategory.category
ι
CategoryTheory.instCategoryCostructuredArrow
CategoryTheory.Functor.InitialCategoryTheory.zigzag_isConnected
CategoryTheory.Zigzag.trans
CategoryTheory.Zigzag.of_hom
CategoryTheory.Category.comp_id
CategoryTheory.Zigzag.of_inv

CategoryTheory.StructuredArrow

Theorems

NameKindAssumesProvesValidatesDepends On
final_pre 📖mathematicalCategoryTheory.Functor.Final
CategoryTheory.StructuredArrow
CategoryTheory.Functor.comp
CategoryTheory.instCategoryStructuredArrow
pre
CategoryTheory.isConnected_iff_of_equivalence
CategoryTheory.Functor.Final.out

---

← Back to Index