File: Mathlib/Condensed/TopComparison.lean

full_name: factorsThrough_of_pullbackCondition
code: /--
An auxiliary lemma to that allows us to use `QuotientMap.lift` in the proof of
`equalizerCondition_yonedaPresheaf`.
-/
theorem factorsThrough_of_pullbackCondition {Z B : C} {π : Z ⟶ B} [HasPullback π π]
    [PreservesLimit (cospan π π) G]
    {a : C(G.obj Z, X)}
    (ha : a ∘ (G.map (pullback.fst _ _)) = a ∘ (G.map (pullback.snd π π))) :
    Function.FactorsThrough a (G.map π) 
goal: Function.FactorsThrough ⇑a ⇑(G.map π)

full_name: equalizerCondition_yonedaPresheaf
code: /--
If `G` preserves the relevant pullbacks and every effective epi in `C` is a quotient map (which is
the case when `C` is `CompHaus` or `Profinite`), then `yonedaPresheaf` satisfies the equalizer
condition which is required to be a sheaf for the regular topology.
-/
theorem equalizerCondition_yonedaPresheaf
    [∀ (Z B : C) (π : Z ⟶ B) [EffectiveEpi π], PreservesLimit (cospan π π) G]
    (hq : ∀ (Z B : C) (π : Z ⟶ B) [EffectiveEpi π], QuotientMap (G.map π)) :
      EqualizerCondition (yonedaPresheaf G X) 
goal: CategoryTheory.regularTopology.EqualizerCondition (ContinuousMap.yonedaPresheaf G X)

full_name: instPreservesFiniteProductsOppositeYonedaPresheafOfPreservesFiniteCoproductsTopCat
code: /--
If `G` preserves finite coproducts (which is the case when `C` is `CompHaus`, `Profinite` or
`Stonean`), then `yonedaPresheaf` preserves finite products, which is required to be a sheaf for
the extensive topology.
-/
noncomputable instance [PreservesFiniteCoproducts G] :
    PreservesFiniteProducts (yonedaPresheaf G X) 
goal: CategoryTheory.Limits.PreservesFiniteProducts (ContinuousMap.yonedaPresheaf G X)

full_name: TopCat.toCondensedSet_val_map
code: /--
Associate to a `(u+1)`-small topological space the corresponding condensed set, given by
`yonedaPresheaf`.
-/
@[simps! val_obj val_map]
noncomputable def TopCat.toCondensedSet (X : TopCat.{u+1}) : CondensedSet.{u} 
goal: ∀ {X_1 Y : CompHausᵒᵖ} (f : X_1 ⟶ Y) (g : C(↑(compHausToTop.obj (Opposite.unop X_1)), ↑X)),
  X.toCondensedSet.val.map f g = g.comp f.unop

full_name: TopCat.toCondensedSet_val_obj
code: /--
Associate to a `(u+1)`-small topological space the corresponding condensed set, given by
`yonedaPresheaf`.
-/
@[simps! val_obj val_map]
noncomputable def TopCat.toCondensedSet (X : TopCat.{u+1}) : CondensedSet.{u} 
goal: ∀ (X_1 : CompHausᵒᵖ), X.toCondensedSet.val.obj X_1 = C(↑(Opposite.unop X_1).toTop, ↑X)

full_name: TopCat.toCondensedSet
code: /--
Associate to a `(u+1)`-small topological space the corresponding condensed set, given by
`yonedaPresheaf`.
-/
@[simps! val_obj val_map]
noncomputable def TopCat.toCondensedSet (X : TopCat.{u+1}) : CondensedSet.{u} 
goal: CondensedSet

full_name: topCatToCondensedSet
code: /--
`TopCat.toCondensedSet` yields a functor from `TopCat.{u+1}` to `CondensedSet.{u}`.
-/
@[simps]
noncomputable def topCatToCondensedSet : TopCat.{u+1} ⥤ CondensedSet.{u} 
goal: TopCat ⥤ CondensedSet

full_name: topCatToCondensedSet_obj
code: /--
`TopCat.toCondensedSet` yields a functor from `TopCat.{u+1}` to `CondensedSet.{u}`.
-/
@[simps]
noncomputable def topCatToCondensedSet : TopCat.{u+1} ⥤ CondensedSet.{u} 
goal: topCatToCondensedSet.obj X = X.toCondensedSet

full_name: topCatToCondensedSet_map_val_app
code: /--
`TopCat.toCondensedSet` yields a functor from `TopCat.{u+1}` to `CondensedSet.{u}`.
-/
@[simps]
noncomputable def topCatToCondensedSet : TopCat.{u+1} ⥤ CondensedSet.{u} 
goal: ∀ {X Y : TopCat} (f : X ⟶ Y) (x : CompHausᵒᵖ) (g : ((fun X => X.toCondensedSet) X).val.obj x),
  (topCatToCondensedSet.map f).val.app x g = ContinuousMap.comp f g