ProCGroups.ProC.InverseLimits.Predicates
This module sets up the finite-stage and inverse-limit description of the construction. It records the stage maps, projections, and comparison lemmas used to pass back to the completed object.
import
def IsPronilpotentGroup
(G : Type u) [Group G] [TopologicalSpace G] : Prop :=
IsProfiniteGroup G ∧ ∀ U : OpenNormalSubgroup G, Group.IsNilpotent (G ⧸ (U : Subgroup G))def IsProsolvableGroup
(G : Type u) [Group G] [TopologicalSpace G] : Prop :=
IsProfiniteGroup G ∧ ∀ U : OpenNormalSubgroup G, IsSolvable (G ⧸ (U : Subgroup G))theorem isProfiniteGroup (hG : IsPronilpotentGroup G) : IsProfiniteGroup GThe underlying profiniteness of a pronilpotent group.
Show proof
hG.1Proof. Work with open normal subgroups and finite quotients of the profinite or pro-\(C\) group. Finite quotients separate points, and continuous homomorphisms are determined by their compatible quotient-level maps. For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. Continuity is checked by composing with all finite quotient projections; each composite is a continuous map between finite or profinite quotient spaces. Compatibility under refinement then assembles the finite-quotient data and proves the required profinite or pro-\(C\) statement. The quotient-level assertions are stable under passing to smaller open normal subgroups, so they define compatible data in the inverse system of finite quotients. For pro-\(C\) permanence, the construction uses only operations for which the chosen finite group class is closed. Hence the finite verification assembles to the desired profinite map, subgroup, or quotient statement. For categorical constructions such as products, pullbacks, and inverse limits, the universal property is checked by composing with the coordinate projections. The uniqueness part follows because any two candidate maps with the same finite quotient composites have the same value on every separating finite quotient. Existence is obtained by assembling the compatible coordinate maps supplied by the finite quotient construction. The topological assertion is checked by the initial topology of the inverse limit. After composing with each finite-stage projection, the relevant map is a continuous finite-stage homomorphism or an operation on a finite product; compactness, Hausdorffness, total disconnectedness, and profiniteness are then inherited from the finite stages by the standard inverse-limit argument.
□theorem isProfiniteGroup (hG : IsProsolvableGroup G) : IsProfiniteGroup GThe underlying profiniteness of a prosolvable group.
Show proof
hG.1Proof. Work with open normal subgroups and finite quotients of the profinite or pro-\(C\) group. Finite quotients separate points, and continuous homomorphisms are determined by their compatible quotient-level maps. For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. Continuity is checked by composing with all finite quotient projections; each composite is a continuous map between finite or profinite quotient spaces. Compatibility under refinement then assembles the finite-quotient data and proves the required profinite or pro-\(C\) statement. The quotient-level assertions are stable under passing to smaller open normal subgroups, so they define compatible data in the inverse system of finite quotients. For pro-\(C\) permanence, the construction uses only operations for which the chosen finite group class is closed. Hence the finite verification assembles to the desired profinite map, subgroup, or quotient statement. For categorical constructions such as products, pullbacks, and inverse limits, the universal property is checked by composing with the coordinate projections. The uniqueness part follows because any two candidate maps with the same finite quotient composites have the same value on every separating finite quotient. Existence is obtained by assembling the compatible coordinate maps supplied by the finite quotient construction. The topological assertion is checked by the initial topology of the inverse limit. After composing with each finite-stage projection, the relevant map is a continuous finite-stage homomorphism or an operation on a finite product; compactness, Hausdorffness, total disconnectedness, and profiniteness are then inherited from the finite stages by the standard inverse-limit argument.
□theorem quotient_isSolvable (hG : IsProsolvableGroup G) (U : OpenNormalSubgroup G) :
IsSolvable (G ⧸ (U : Subgroup G))Every quotient by an open normal subgroup of a prosolvable group is solvable.
Show proof
hG.2 UProof. Work with open normal subgroups and finite quotients of the profinite or pro-\(C\) group. Finite quotients separate points, and continuous homomorphisms are determined by their compatible quotient-level maps. For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. Compatibility under refinement then assembles the finite-quotient data and proves the required profinite or pro-\(C\) statement. The quotient-level assertions are stable under passing to smaller open normal subgroups, so they define compatible data in the inverse system of finite quotients. For pro-\(C\) permanence, the construction uses only operations for which the chosen finite group class is closed. Hence the finite verification assembles to the desired profinite map, subgroup, or quotient statement. For categorical constructions such as products, pullbacks, and inverse limits, the universal property is checked by composing with the coordinate projections. The uniqueness part follows because any two candidate maps with the same finite quotient composites have the same value on every separating finite quotient. Existence is obtained by assembling the compatible coordinate maps supplied by the finite quotient construction. For quotient and subgroup claims, representatives are chosen in the ambient group and the induced map is checked to send the class of an element to the class of its image. Normality, openness, and membership in the finite quotient class are preserved by the subgroup, quotient, intersection, or inverse-image closure property being invoked. For equivalence and homeomorphism statements, the two comparison maps are composed in both orders and evaluated on the coordinates that determine the source. Each composite reduces to the identity transition or to the chosen representative identity on finite stages, so the algebraic inverse laws and the topological inverse laws follow simultaneously.
□def IsProPGroup (p : ℕ)
(G : Type u) [Group G] [TopologicalSpace G] : Prop :=
IsProCGroup (FiniteGroupClass.pGroup p) GPro-\(p\)-groups as pro-\(C\) groups for the finite \(p\)-group class.
def IsProSigmaGroup (sigma : Set ℕ)
(G : Type u) [Group G] [TopologicalSpace G] : Prop :=
IsProCGroup (FiniteGroupClass.sigmaGroup sigma) GPro-\(\Sigma\) groups as pro-\(C\) groups for the finite \(\Sigma\)-group class.
theorem isPronilpotentGroup_of_isProC_nilpotent
(hG : IsProCGroup FiniteGroupClass.nilpotent G) : IsPronilpotentGroup GA pro-nilpotent group is pronilpotent.
Show proof
by
letI : IsTopologicalGroup G := hG.isTopologicalGroup
refine ⟨hG.isProfinite, ?_⟩
intro U
exact
(IsProCGroup.hasAllOpenNormalQuotientsInClass_of_basis_of_quotientClosed
FiniteGroupClass.nilpotent_isomClosed FiniteGroupClass.nilpotent_quotientClosed hG U).2Proof. Work with open normal subgroups and finite quotients of the profinite or pro-\(C\) group. Finite quotients separate points, and continuous homomorphisms are determined by their compatible quotient-level maps. For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. Compatibility under refinement then assembles the finite-quotient data and proves the required profinite or pro-\(C\) statement. The quotient-level assertions are stable under passing to smaller open normal subgroups, so they define compatible data in the inverse system of finite quotients. For pro-\(C\) permanence, the construction uses only operations for which the chosen finite group class is closed. Hence the finite verification assembles to the desired profinite map, subgroup, or quotient statement. For categorical constructions such as products, pullbacks, and inverse limits, the universal property is checked by composing with the coordinate projections. The uniqueness part follows because any two candidate maps with the same finite quotient composites have the same value on every separating finite quotient. Existence is obtained by assembling the compatible coordinate maps supplied by the finite quotient construction. For quotient and subgroup claims, representatives are chosen in the ambient group and the induced map is checked to send the class of an element to the class of its image. Normality, openness, and membership in the finite quotient class are preserved by the subgroup, quotient, intersection, or inverse-image closure property being invoked. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps.
□theorem isProsolvableGroup_of_isProC_solvable
(hG : IsProCGroup FiniteGroupClass.solvable G) : IsProsolvableGroup GA pro-solvable group is prosolvable.
Show proof
by
letI : IsTopologicalGroup G := hG.isTopologicalGroup
refine ⟨hG.isProfinite, ?_⟩
intro U
exact
(IsProCGroup.hasAllOpenNormalQuotientsInClass_of_basis_of_quotientClosed
FiniteGroupClass.solvable_isomClosed FiniteGroupClass.solvable_quotientClosed hG U).2Proof. Work with open normal subgroups and finite quotients of the profinite or pro-\(C\) group. Finite quotients separate points, and continuous homomorphisms are determined by their compatible quotient-level maps. For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. Compatibility under refinement then assembles the finite-quotient data and proves the required profinite or pro-\(C\) statement. The quotient-level assertions are stable under passing to smaller open normal subgroups, so they define compatible data in the inverse system of finite quotients. For pro-\(C\) permanence, the construction uses only operations for which the chosen finite group class is closed. Hence the finite verification assembles to the desired profinite map, subgroup, or quotient statement. For categorical constructions such as products, pullbacks, and inverse limits, the universal property is checked by composing with the coordinate projections. The uniqueness part follows because any two candidate maps with the same finite quotient composites have the same value on every separating finite quotient. Existence is obtained by assembling the compatible coordinate maps supplied by the finite quotient construction. For quotient and subgroup claims, representatives are chosen in the ambient group and the induced map is checked to send the class of an element to the class of its image. Normality, openness, and membership in the finite quotient class are preserved by the subgroup, quotient, intersection, or inverse-image closure property being invoked. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps.
□theorem isPronilpotentGroup (hG : IsProcyclicGroup G) : IsPronilpotentGroup GA procyclic group is pronilpotent.
Show proof
by
letI : IsTopologicalGroup G := hG.isTopologicalGroup
refine ⟨hG.isProfinite, ?_⟩
intro U
exact
(FiniteGroupClass.cyclic_to_nilpotent
(IsProCGroup.hasAllOpenNormalQuotientsInClass_of_basis_of_quotientClosed
FiniteGroupClass.cyclic_isomClosed FiniteGroupClass.cyclic_quotientClosed hG U)).2Proof. Work with open normal subgroups and finite quotients of the profinite or pro-\(C\) group. Finite quotients separate points, and continuous homomorphisms are determined by their compatible quotient-level maps. For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. Continuity is checked by composing with all finite quotient projections; each composite is a continuous map between finite or profinite quotient spaces. Compatibility under refinement then assembles the finite-quotient data and proves the required profinite or pro-\(C\) statement. The quotient-level assertions are stable under passing to smaller open normal subgroups, so they define compatible data in the inverse system of finite quotients. For pro-\(C\) permanence, the construction uses only operations for which the chosen finite group class is closed. Hence the finite verification assembles to the desired profinite map, subgroup, or quotient statement. For categorical constructions such as products, pullbacks, and inverse limits, the universal property is checked by composing with the coordinate projections. The uniqueness part follows because any two candidate maps with the same finite quotient composites have the same value on every separating finite quotient. Existence is obtained by assembling the compatible coordinate maps supplied by the finite quotient construction. For quotient and subgroup claims, representatives are chosen in the ambient group and the induced map is checked to send the class of an element to the class of its image. Normality, openness, and membership in the finite quotient class are preserved by the subgroup, quotient, intersection, or inverse-image closure property being invoked.
□theorem isProsolvableGroup (hG : IsProcyclicGroup G) : IsProsolvableGroup GA procyclic group is prosolvable.
Show proof
by
letI : IsTopologicalGroup G := hG.isTopologicalGroup
refine ⟨hG.isProfinite, ?_⟩
intro U
exact
(FiniteGroupClass.cyclic_to_solvable
(IsProCGroup.hasAllOpenNormalQuotientsInClass_of_basis_of_quotientClosed
FiniteGroupClass.cyclic_isomClosed FiniteGroupClass.cyclic_quotientClosed hG U)).2Proof. Work with open normal subgroups and finite quotients of the profinite or pro-\(C\) group. Finite quotients separate points, and continuous homomorphisms are determined by their compatible quotient-level maps. For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. Continuity is checked by composing with all finite quotient projections; each composite is a continuous map between finite or profinite quotient spaces. Compatibility under refinement then assembles the finite-quotient data and proves the required profinite or pro-\(C\) statement. The quotient-level assertions are stable under passing to smaller open normal subgroups, so they define compatible data in the inverse system of finite quotients. For pro-\(C\) permanence, the construction uses only operations for which the chosen finite group class is closed. Hence the finite verification assembles to the desired profinite map, subgroup, or quotient statement. For categorical constructions such as products, pullbacks, and inverse limits, the universal property is checked by composing with the coordinate projections. The uniqueness part follows because any two candidate maps with the same finite quotient composites have the same value on every separating finite quotient. Existence is obtained by assembling the compatible coordinate maps supplied by the finite quotient construction. For quotient and subgroup claims, representatives are chosen in the ambient group and the induced map is checked to send the class of an element to the class of its image. Normality, openness, and membership in the finite quotient class are preserved by the subgroup, quotient, intersection, or inverse-image closure property being invoked.
□theorem isProabelianGroup (hG : IsProcyclicGroup G) : IsProabelianGroup GA procyclic group is pro-abelian.
Show proof
by
exact hG.mono (fun {Q} [Group Q] hQ => by
rcases hQ with ⟨hfin, hcyc⟩
refine ⟨hfin, ?_⟩
letI : IsCyclic Q := hcyc
letI : CommGroup Q := IsCyclic.commGroup
intro a b
exact mul_comm a b)Proof. Work with open normal subgroups and finite quotients of the profinite or pro-\(C\) group. Finite quotients separate points, and continuous homomorphisms are determined by their compatible quotient-level maps. For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. Kernel and image statements are verified after quotienting by sufficiently small open normal subgroups, where they become ordinary finite group calculations. Compatibility under refinement then assembles the finite-quotient data and proves the required profinite or pro-\(C\) statement. The quotient-level assertions are stable under passing to smaller open normal subgroups, so they define compatible data in the inverse system of finite quotients. For pro-\(C\) permanence, the construction uses only operations for which the chosen finite group class is closed. Hence the finite verification assembles to the desired profinite map, subgroup, or quotient statement. For categorical constructions such as products, pullbacks, and inverse limits, the universal property is checked by composing with the coordinate projections. The uniqueness part follows because any two candidate maps with the same finite quotient composites have the same value on every separating finite quotient. Existence is obtained by assembling the compatible coordinate maps supplied by the finite quotient construction. For quotient and subgroup claims, representatives are chosen in the ambient group and the induced map is checked to send the class of an element to the class of its image. Normality, openness, and membership in the finite quotient class are preserved by the subgroup, quotient, intersection, or inverse-image closure property being invoked.
□theorem isProcyclicGroup_of_topologicallyGenerates_singleton
[IsTopologicalGroup G] (hG : IsProfiniteGroup G) {g : G}
(hg : Generation.TopologicallyGenerates (G := G) ({g} : Set G)) :
IsProcyclicGroup GA profinite group topologically generated by one element is procyclic.
Show proof
by
letI : CompactSpace G := IsProfiniteGroup.compactSpace hG
letI : T2Space G := IsProfiniteGroup.t2Space hG
refine IsProCGroup.of_allOpenNormalQuotients (C := FiniteGroupClass.cyclic) hG ?_
intro U
let qg : G ⧸ (U : Subgroup G) := QuotientGroup.mk' (U : Subgroup G) g
have hquot :
Generation.TopologicallyGenerates (G := G ⧸ (U : Subgroup G)) ({qg} : Set _) := by
have hmap := Generation.topologicallyGenerates_quotient_image
(G := G) (N := (U : Subgroup G)) (X := ({g} : Set G)) hg
simpa [qg] using hmap
have hdense :
Dense (((Subgroup.closure ({qg} : Set (G ⧸ (U : Subgroup G))) : Subgroup
(G ⧸ (U : Subgroup G))) : Set (G ⧸ (U : Subgroup G)))) :=
(Generation.topologicallyGenerates_iff_dense
(G := G ⧸ (U : Subgroup G)) (X := ({qg} : Set _))).1 hquot
have htop : Subgroup.zpowers qg = ⊤ := by
apply SetLike.ext'
rw [Subgroup.zpowers_eq_closure]
exact dense_discrete.1 hdense
have hcyc : IsCyclic (G ⧸ (U : Subgroup G)) :=
(isCyclic_iff_exists_zpowers_eq_top).2 ⟨qg, htop⟩
letI : Finite (G ⧸ (U : Subgroup G)) := openNormalSubgroup_finiteQuotient (G := G) U
exact ⟨inferInstance, hcyc⟩Proof. Work with open normal subgroups and finite quotients of the profinite or pro-\(C\) group. Finite quotients separate points, and continuous homomorphisms are determined by their compatible quotient-level maps. For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. Continuity is checked by composing with all finite quotient projections; each composite is a continuous map between finite or profinite quotient spaces. Compatibility under refinement then assembles the finite-quotient data and proves the required profinite or pro-\(C\) statement. The quotient-level assertions are stable under passing to smaller open normal subgroups, so they define compatible data in the inverse system of finite quotients. For pro-\(C\) permanence, the construction uses only operations for which the chosen finite group class is closed. Hence the finite verification assembles to the desired profinite map, subgroup, or quotient statement. For categorical constructions such as products, pullbacks, and inverse limits, the universal property is checked by composing with the coordinate projections. The uniqueness part follows because any two candidate maps with the same finite quotient composites have the same value on every separating finite quotient. Existence is obtained by assembling the compatible coordinate maps supplied by the finite quotient construction. The topological assertion is checked by the initial topology of the inverse limit. After composing with each finite-stage projection, the relevant map is a continuous finite-stage homomorphism or an operation on a finite product; compactness, Hausdorffness, total disconnectedness, and profiniteness are then inherited from the finite stages by the standard inverse-limit argument.
□theorem toIsProC_nilpotent (hG : IsPronilpotentGroup G) :
IsProCGroup FiniteGroupClass.nilpotent GRepackage a pronilpotent group as a pro-nilpotent group in the working IsProCGroup interface used for permanence arguments.
Show proof
by
letI : IsTopologicalGroup G := IsProfiniteGroup.isTopologicalGroup hG.1
refine IsProCGroup.of_allOpenNormalQuotients (C := FiniteGroupClass.nilpotent) hG.1 ?_
intro U
letI : CompactSpace G := IsProfiniteGroup.compactSpace hG.1
letI : T2Space G := IsProfiniteGroup.t2Space hG.1
exact ⟨openNormalSubgroup_finiteQuotient (G := G) U, hG.2 U⟩Proof. Work with open normal subgroups and finite quotients of the profinite or pro-\(C\) group. Finite quotients separate points, and continuous homomorphisms are determined by their compatible quotient-level maps. For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. Compatibility under refinement then assembles the finite-quotient data and proves the required profinite or pro-\(C\) statement. The quotient-level assertions are stable under passing to smaller open normal subgroups, so they define compatible data in the inverse system of finite quotients. For pro-\(C\) permanence, the construction uses only operations for which the chosen finite group class is closed. Hence the finite verification assembles to the desired profinite map, subgroup, or quotient statement. For categorical constructions such as products, pullbacks, and inverse limits, the universal property is checked by composing with the coordinate projections. The uniqueness part follows because any two candidate maps with the same finite quotient composites have the same value on every separating finite quotient. Existence is obtained by assembling the compatible coordinate maps supplied by the finite quotient construction. For equivalence and homeomorphism statements, the two comparison maps are composed in both orders and evaluated on the coordinates that determine the source. Each composite reduces to the identity transition or to the chosen representative identity on finite stages, so the algebraic inverse laws and the topological inverse laws follow simultaneously. For pro-\(C\) claims, the finite quotient obtained from the construction remains in \(C\) by the relevant closure axiom, such as closure under subgroups, quotients, finite products, or extensions. The universal or lifting property is then checked on the prescribed generators, and uniqueness follows because the generators are dense or topologically generating.
□theorem toIsProC_solvable (hG : IsProsolvableGroup G) :
IsProCGroup FiniteGroupClass.solvable GRepackage a prosolvable group as a pro-solvable group in the working IsProCGroup interface used for permanence arguments.
Show proof
by
letI : IsTopologicalGroup G := IsProfiniteGroup.isTopologicalGroup hG.1
refine IsProCGroup.of_allOpenNormalQuotients (C := FiniteGroupClass.solvable) hG.1 ?_
intro U
letI : CompactSpace G := IsProfiniteGroup.compactSpace hG.1
letI : T2Space G := IsProfiniteGroup.t2Space hG.1
exact ⟨openNormalSubgroup_finiteQuotient (G := G) U, hG.2 U⟩Proof. Work with open normal subgroups and finite quotients of the profinite or pro-\(C\) group. Finite quotients separate points, and continuous homomorphisms are determined by their compatible quotient-level maps. For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. Compatibility under refinement then assembles the finite-quotient data and proves the required profinite or pro-\(C\) statement. The quotient-level assertions are stable under passing to smaller open normal subgroups, so they define compatible data in the inverse system of finite quotients. For pro-\(C\) permanence, the construction uses only operations for which the chosen finite group class is closed. Hence the finite verification assembles to the desired profinite map, subgroup, or quotient statement. For categorical constructions such as products, pullbacks, and inverse limits, the universal property is checked by composing with the coordinate projections. The uniqueness part follows because any two candidate maps with the same finite quotient composites have the same value on every separating finite quotient. Existence is obtained by assembling the compatible coordinate maps supplied by the finite quotient construction. For equivalence and homeomorphism statements, the two comparison maps are composed in both orders and evaluated on the coordinates that determine the source. Each composite reduces to the identity transition or to the chosen representative identity on finite stages, so the algebraic inverse laws and the topological inverse laws follow simultaneously. For pro-\(C\) claims, the finite quotient obtained from the construction remains in \(C\) by the relevant closure axiom, such as closure under subgroups, quotients, finite products, or extensions. The universal or lifting property is then checked on the prescribed generators, and uniqueness follows because the generators are dense or topologically generating.
□theorem quotient_openNormalSubgroup (hG : IsProcyclicGroup G)
(U : OpenNormalSubgroup G) :
IsProcyclicGroup (G ⧸ (U : Subgroup G))A quotient of a profinite group by an open normal subgroup is profinite.
Show proof
by
letI : IsTopologicalGroup G := IsProfiniteGroup.isTopologicalGroup hG.isProfinite
letI : Finite (G ⧸ (U : Subgroup G)) := hG.finite_quotient U
letI : DiscreteTopology (G ⧸ (U : Subgroup G)) :=
QuotientGroup.discreteTopology (openNormalSubgroup_isOpen (G := G) U)
exact IsProCGroup.of_finite_discrete (C := FiniteGroupClass.cyclic)
FiniteGroupClass.cyclic_quotientClosed
(IsProCGroup.hasAllOpenNormalQuotientsInClass_of_basis_of_quotientClosed
FiniteGroupClass.cyclic_isomClosed FiniteGroupClass.cyclic_quotientClosed hG U)Proof. Work with open normal subgroups and finite quotients of the profinite or pro-\(C\) group. Finite quotients separate points, and continuous homomorphisms are determined by their compatible quotient-level maps. Closed-subgroup and subgroup-permanence claims use ambient open-normal approximation: an open normal subgroup of the closed subgroup is refined by the intersection with an ambient open normal subgroup of \(G\). For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. Compatibility under refinement then assembles the finite-quotient data and proves the required profinite or pro-\(C\) statement. The quotient-level assertions are stable under passing to smaller open normal subgroups, so they define compatible data in the inverse system of finite quotients. For pro-\(C\) permanence, the construction uses only operations for which the chosen finite group class is closed. Hence the finite verification assembles to the desired profinite map, subgroup, or quotient statement. For categorical constructions such as products, pullbacks, and inverse limits, the universal property is checked by composing with the coordinate projections. The uniqueness part follows because any two candidate maps with the same finite quotient composites have the same value on every separating finite quotient. Existence is obtained by assembling the compatible coordinate maps supplied by the finite quotient construction. The topological assertion is checked by the initial topology of the inverse limit. After composing with each finite-stage projection, the relevant map is a continuous finite-stage homomorphism or an operation on a finite product; compactness, Hausdorffness, total disconnectedness, and profiniteness are then inherited from the finite stages by the standard inverse-limit argument.
□