English
For a CondensedSet X, seven conditions are mutually equivalent: X is discrete, counit is iso for the discrete underlying adjunction, EssImage of discrete; the LocallyConstant EssImage, adjunction counit iso, coherence-isConstant across Profinite, and a colimit-isColimit condition for all S.
Русский
Для CondensedSet X существует семь взаимно эквивалентных условий: X дискретна, контур единичного преобразователя дискретизации изоля, EssImage дискретизированного; LocallyConstant EssImage, изотропия переводной, согласованность по Profinite и условие существования IsColimit для всех S.
LaTeX
$$$\\mathrm{TFAE}\\left[\\mathrm{IsDiscrete}(X), \\mathrm{IsIso}(\\mathrm{counit}_{\\text{discrete}}(X)), \\mathrm{essImage}(\\mathrm{discrete},X), \\mathrm{LocallyConstant.essImage}(X), \\mathrm{IsIso}(\\mathrm{counit}_{\\mathrm{LocallyConstant}}(X)), \\mathrm{IsConstant}(\\mathrm{coherentTopology Profinite}, (\\mathrm{equivalence}.inverse.obj X)), \\forall S,\\; \\text{IsColimit}( (profiniteToCompHaus.op.comp X.val).mapCocone S.asLimitCone.op)\\right]$$$
Lean4
theorem isDiscrete_tfae (X : CondensedSet.{u}) :
TFAE
[X.IsDiscrete, IsIso ((Condensed.discreteUnderlyingAdj _).counit.app X), (Condensed.discrete _).essImage X,
CondensedSet.LocallyConstant.functor.essImage X, IsIso (CondensedSet.LocallyConstant.adjunction.counit.app X),
Sheaf.IsConstant (coherentTopology Profinite) ((Condensed.ProfiniteCompHaus.equivalence _).inverse.obj X),
∀ S : Profinite.{u}, Nonempty (IsColimit <| (profiniteToCompHaus.op ⋙ X.val).mapCocone S.asLimitCone.op)] :=
by
tfae_have 1 ↔ 2 := Sheaf.isConstant_iff_isIso_counit_app _ _ _
tfae_have 1 ↔ 3 := ⟨fun ⟨h⟩ ↦ h, fun h ↦ ⟨h⟩⟩
tfae_have 1 ↔ 4 := Sheaf.isConstant_iff_mem_essImage _ CompHaus.isTerminalPUnit adjunction _
tfae_have 1 ↔ 5 :=
have : functor.Faithful := inferInstance
have : functor.Full := inferInstance
Sheaf.isConstant_iff_isIso_counit_app' _ CompHaus.isTerminalPUnit adjunction _
tfae_have 1 ↔ 6 :=
(Sheaf.isConstant_iff_of_equivalence (coherentTopology Profinite) (coherentTopology CompHaus) profiniteToCompHaus
Profinite.isTerminalPUnit CompHaus.isTerminalPUnit _).symm
tfae_have 7 → 4 := fun h ↦ mem_locallyConstant_essImage_of_isColimit_mapCocone X (fun S ↦ (h S).some)
tfae_have 4 → 7 := fun ⟨Y, ⟨i⟩⟩ S ↦
⟨IsColimit.mapCoconeEquiv (isoWhiskerLeft profiniteToCompHaus.op ((sheafToPresheaf _ _).mapIso i))
(Condensed.isColimitLocallyConstantPresheafDiagram Y S)⟩
tfae_finish
