Documentation

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.

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

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

    Equations
    Instances For
      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.

      Equations
      • One or more equations did not get rendered due to their size.
      Instances For
        @[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.

        Equations
        Instances For

          The explicit finite coproduct cocone is a colimit cocone.

          Equations
          • One or more equations did not get rendered due to their size.
          Instances For
            noncomputable def Stonean.coproductIsoCoproduct {α : Type} [Finite α] (X : αStonean) :

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

            Equations
            Instances For

              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.

              Equations
              Instances For
                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.

                Equations
                Instances For
                  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.

                  Equations
                  Instances For
                    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.

                    Equations
                    • Stonean.pullback.lift f hi a b w = { toFun := fun (z : ((fun (x : Stonean) => x.compHaus) W).toTop) => a z, , continuous_toFun := }
                    Instances For
                      @[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.

                      Equations
                      Instances For

                        The explicit pullback cone is a limit cone.

                        Equations
                        Instances For
                          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.

                          Equations
                          • One or more equations did not get rendered due to their size.
                          Instances For

                            The forgetful from Stonean to TopCat creates pullbacks along open embeddings

                            Equations
                            • One or more equations did not get rendered due to their size.
                            Instances For