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Mathlib.Topology.Category.Stonean.Limits

Explicit (co)limits in the category of Stonean spaces #

This file describes some explicit (co)limits in Stonean

Overview #

We define explicit finite coproducts in Stonean as sigma types (disjoint unions) and explicit pullbacks where one of the maps is an open embedding

This section defines the finite Coproduct of a finite family of profinite spaces X : α → Stonean.{u}

Notes: The content is mainly copied from Mathlib/Topology/Category/CompHaus/Limits.lean

def Stonean.finiteCoproduct {α : Type} [Finite α] (X : αStonean) :

The coproduct of a finite family of objects in Stonean, constructed as the disjoint union with its usual topology.

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    def Stonean.finiteCoproduct.ι {α : Type} [Finite α] (X : αStonean) (a : α) :

    The inclusion of one of the factors into the explicit finite coproduct.

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      def Stonean.finiteCoproduct.desc {α : Type} [Finite α] (X : αStonean) {B : Stonean} (e : (a : α) → X a B) :

      To construct a morphism from the explicit finite coproduct, it suffices to specify a morphism from each of its factors. This is essentially the universal property of the coproduct.

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        @[simp]
        theorem Stonean.finiteCoproduct.ι_desc {α : Type} [Finite α] (X : αStonean) {B : Stonean} (e : (a : α) → X a B) (a : α) :
        @[reducible, inline]

        The coproduct cocone associated to the explicit finite coproduct.

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          The explicit finite coproduct cocone is a colimit cocone.

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            noncomputable def Stonean.coproductIsoCoproduct {α : Type} [Finite α] (X : αStonean) :

            The isomorphism from the explicit finite coproducts to the abstract coproduct.

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              The inclusion maps into the explicit finite coproduct are open embeddings.

              The inclusion maps into the abstract finite coproduct are open embeddings.

              def Stonean.pullback {X : Stonean} {Y : Stonean} {Z : Stonean} (f : X Z) {i : Y Z} (hi : OpenEmbedding i) :

              The pullback of a morphism f and an open embedding i in Stonean, constructed explicitly as the preimage under fof the image of i with the subspace topology.

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                def Stonean.pullback.fst {X : Stonean} {Y : Stonean} {Z : Stonean} (f : X Z) {i : Y Z} (hi : OpenEmbedding i) :

                The projection from the pullback to the first component.

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                  noncomputable def Stonean.pullback.snd {X : Stonean} {Y : Stonean} {Z : Stonean} (f : X Z) {i : Y Z} (hi : OpenEmbedding i) :

                  The projection from the pullback to the second component.

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                    def Stonean.pullback.lift {X : Stonean} {Y : Stonean} {Z : Stonean} {W : Stonean} (f : X Z) {i : Y Z} (hi : OpenEmbedding i) (a : W X) (b : W Y) (w : CategoryTheory.CategoryStruct.comp a f = CategoryTheory.CategoryStruct.comp b i) :

                    Construct a morphism to the explicit pullback given morphisms to the factors which are compatible with the maps to the base. This is essentially the universal property of the pullback.

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                    • Stonean.pullback.lift f hi a b w = { toFun := fun (z : ((fun (x : Stonean) => x.compHaus) W).toTop) => a z, , continuous_toFun := }
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                      @[simp]
                      theorem Stonean.pullback.cone_pt {X : Stonean} {Y : Stonean} {Z : Stonean} (f : X Z) {i : Y Z} (hi : OpenEmbedding i) :
                      noncomputable def Stonean.pullback.cone {X : Stonean} {Y : Stonean} {Z : Stonean} (f : X Z) {i : Y Z} (hi : OpenEmbedding i) :

                      The pullback cone whose cone point is the explicit pullback.

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                        The explicit pullback cone is a limit cone.

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                          @[simp]
                          theorem Stonean.pullbackIsoPullback_inv {X : Stonean} {Y : Stonean} {Z : Stonean} (f : X Z) {i : Y Z} (hi : OpenEmbedding i) :
                          (Stonean.pullbackIsoPullback f hi).inv = Stonean.pullback.lift f hi CategoryTheory.Limits.pullback.fst CategoryTheory.Limits.pullback.snd
                          noncomputable def Stonean.pullbackIsoPullback {X : Stonean} {Y : Stonean} {Z : Stonean} (f : X Z) {i : Y Z} (hi : OpenEmbedding i) :

                          The isomorphism from the explicit pullback to the abstract pullback.

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                            The forgetful from Stonean to TopCat creates pullbacks along open embeddings

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