Theorem: AlgebraicGeometry.PresheafedSpace.stalkMap.stalkSpecializes_stalkMap
File: Mathlib/AlgebraicGeometry/Stalks.lean

tactic: dsimp [stalkSpecializes, stalkMap, stalkFunctor, stalkPushforward]
state_before: C : Type u
inst✝¹ : Category C
inst✝ : HasColimits C
X Y : PresheafedSpace C
f : X ⟶ Y
x y : ↑↑X
h : x ⤳ y
⊢ stalkSpecializes Y.presheaf (_ : ↑f.base x ⤳ ↑f.base y) ≫ stalkMap f x = stalkMap f y ≫ stalkSpecializes X.presheaf h

tactic: refine colimit.hom_ext fun j => ?_
state_before: C : Type u
inst✝¹ : Category C
inst✝ : HasColimits C
X Y : PresheafedSpace C
f : X ⟶ Y
x y : ↑↑X
h : x ⤳ y
⊢ colimit.desc (Functor.op (OpenNhds.inclusion (↑f.base y)) ⋙ Y.presheaf)
        { pt := colimit (Functor.op (OpenNhds.inclusion (↑f.base x)) ⋙ Y.presheaf),
          ι :=
            NatTrans.mk fun U =>
              colimit.ι (Functor.op (OpenNhds.inclusion (↑f.base x)) ⋙ Y.presheaf)
                { obj := U.unop.obj, property := (_ : ↑f.base x ∈ U.unop.obj.carrier) }.op } ≫
      colimMap (whiskerLeft (Functor.op (OpenNhds.inclusion ((CategoryTheory.forget TopCat).map f.base x))) f.c) ≫
        colimMap (whiskerRight (𝟙 (Functor.op (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x))) X.presheaf) ≫
          colimit.pre (Functor.op (OpenNhds.inclusion x) ⋙ X.presheaf) (Functor.op (OpenNhds.map f.base x)) =
    (colimMap (whiskerLeft (Functor.op (OpenNhds.inclusion ((CategoryTheory.forget TopCat).map f.base y))) f.c) ≫
        colimMap (whiskerRight (𝟙 (Functor.op (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y))) X.presheaf) ≫
          colimit.pre (Functor.op (OpenNhds.inclusion y) ⋙ X.presheaf) (Functor.op (OpenNhds.map f.base y))) ≫
      colimit.desc (Functor.op (OpenNhds.inclusion y) ⋙ X.presheaf)
        { pt := colimit (Functor.op (OpenNhds.inclusion x) ⋙ X.presheaf),
          ι :=
            NatTrans.mk fun U =>
              colimit.ι (Functor.op (OpenNhds.inclusion x) ⋙ X.presheaf)
                { obj := U.unop.obj, property := (_ : x ∈ U.unop.obj.carrier) }.op }

tactic: induction j with | h j => ?_
state_before: C : Type u
inst✝¹ : Category C
inst✝ : HasColimits C
X Y : PresheafedSpace C
f : X ⟶ Y
x y : ↑↑X
h : x ⤳ y
j : (OpenNhds (↑f.base y))ᵒᵖ
⊢ colimit.ι
        (((whiskeringLeft (OpenNhds (↑f.base y))ᵒᵖ (Opens ↑↑Y)ᵒᵖ C).obj
              (Functor.op (OpenNhds.inclusion (↑f.base y)))).obj
          Y.presheaf)
        j ≫
      colimit.desc (Functor.op (OpenNhds.inclusion (↑f.base y)) ⋙ Y.presheaf)
          { pt := colimit (Functor.op (OpenNhds.inclusion (↑f.base x)) ⋙ Y.presheaf),
            ι :=
              NatTrans.mk fun U =>
                colimit.ι (Functor.op (OpenNhds.inclusion (↑f.base x)) ⋙ Y.presheaf)
                  { obj := U.unop.obj, property := (_ : ↑f.base x ∈ U.unop.obj.carrier) }.op } ≫
        colimMap (whiskerLeft (Functor.op (OpenNhds.inclusion ((CategoryTheory.forget TopCat).map f.base x))) f.c) ≫
          colimMap (whiskerRight (𝟙 (Functor.op (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x))) X.presheaf) ≫
            colimit.pre (Functor.op (OpenNhds.inclusion x) ⋙ X.presheaf) (Functor.op (OpenNhds.map f.base x)) =
    colimit.ι
        (((whiskeringLeft (OpenNhds (↑f.base y))ᵒᵖ (Opens ↑↑Y)ᵒᵖ C).obj
              (Functor.op (OpenNhds.inclusion (↑f.base y)))).obj
          Y.presheaf)
        j ≫
      (colimMap (whiskerLeft (Functor.op (OpenNhds.inclusion ((CategoryTheory.forget TopCat).map f.base y))) f.c) ≫
          colimMap (whiskerRight (𝟙 (Functor.op (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y))) X.presheaf) ≫
            colimit.pre (Functor.op (OpenNhds.inclusion y) ⋙ X.presheaf) (Functor.op (OpenNhds.map f.base y))) ≫
        colimit.desc (Functor.op (OpenNhds.inclusion y) ⋙ X.presheaf)
          { pt := colimit (Functor.op (OpenNhds.inclusion x) ⋙ X.presheaf),
            ι :=
              NatTrans.mk fun U =>
                colimit.ι (Functor.op (OpenNhds.inclusion x) ⋙ X.presheaf)
                  { obj := U.unop.obj, property := (_ : x ∈ U.unop.obj.carrier) }.op }

tactic: dsimp
state_before: case h
C : Type u
inst✝¹ : Category C
inst✝ : HasColimits C
X Y : PresheafedSpace C
f : X ⟶ Y
x y : ↑↑X
h : x ⤳ y
j : OpenNhds (↑f.base y)
⊢ colimit.ι
        (((whiskeringLeft (OpenNhds (↑f.base y))ᵒᵖ (Opens ↑↑Y)ᵒᵖ C).obj
              (Functor.op (OpenNhds.inclusion (↑f.base y)))).obj
          Y.presheaf)
        j.op ≫
      colimit.desc (Functor.op (OpenNhds.inclusion (↑f.base y)) ⋙ Y.presheaf)
          { pt := colimit (Functor.op (OpenNhds.inclusion (↑f.base x)) ⋙ Y.presheaf),
            ι :=
              NatTrans.mk fun U =>
                colimit.ι (Functor.op (OpenNhds.inclusion (↑f.base x)) ⋙ Y.presheaf)
                  { obj := U.unop.obj, property := (_ : ↑f.base x ∈ U.unop.obj.carrier) }.op } ≫
        colimMap (whiskerLeft (Functor.op (OpenNhds.inclusion ((CategoryTheory.forget TopCat).map f.base x))) f.c) ≫
          colimMap (whiskerRight (𝟙 (Functor.op (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x))) X.presheaf) ≫
            colimit.pre (Functor.op (OpenNhds.inclusion x) ⋙ X.presheaf) (Functor.op (OpenNhds.map f.base x)) =
    colimit.ι
        (((whiskeringLeft (OpenNhds (↑f.base y))ᵒᵖ (Opens ↑↑Y)ᵒᵖ C).obj
              (Functor.op (OpenNhds.inclusion (↑f.base y)))).obj
          Y.presheaf)
        j.op ≫
      (colimMap (whiskerLeft (Functor.op (OpenNhds.inclusion ((CategoryTheory.forget TopCat).map f.base y))) f.c) ≫
          colimMap (whiskerRight (𝟙 (Functor.op (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y))) X.presheaf) ≫
            colimit.pre (Functor.op (OpenNhds.inclusion y) ⋙ X.presheaf) (Functor.op (OpenNhds.map f.base y))) ≫
        colimit.desc (Functor.op (OpenNhds.inclusion y) ⋙ X.presheaf)
          { pt := colimit (Functor.op (OpenNhds.inclusion x) ⋙ X.presheaf),
            ι :=
              NatTrans.mk fun U =>
                colimit.ι (Functor.op (OpenNhds.inclusion x) ⋙ X.presheaf)
                  { obj := U.unop.obj, property := (_ : x ∈ U.unop.obj.carrier) }.op }

tactic: simp only [colimit.ι_desc_assoc, comp_obj, op_obj, unop_op, ι_colimMap_assoc, colimit.map_desc,
  OpenNhds.inclusion_obj, pushforwardObj_obj, whiskerLeft_app, OpenNhds.map_obj, whiskerRight_app,
  NatTrans.id_app, map_id, colimit.ι_pre, id_comp, assoc, colimit.pre_desc]
state_before: case h
C : Type u
inst✝¹ : Category C
inst✝ : HasColimits C
X Y : PresheafedSpace C
f : X ⟶ Y
x y : ↑↑X
h : x ⤳ y
j : OpenNhds (↑f.base y)
⊢ colimit.ι (Functor.op (OpenNhds.inclusion (↑f.base y)) ⋙ Y.presheaf) j.op ≫
      colimit.desc (Functor.op (OpenNhds.inclusion (↑f.base y)) ⋙ Y.presheaf)
          { pt := colimit (Functor.op (OpenNhds.inclusion (↑f.base x)) ⋙ Y.presheaf),
            ι :=
              NatTrans.mk fun U =>
                colimit.ι (Functor.op (OpenNhds.inclusion (↑f.base x)) ⋙ Y.presheaf)
                  { obj := U.unop.obj, property := (_ : ↑f.base x ∈ U.unop.obj.carrier) }.op } ≫
        colimMap (whiskerLeft (Functor.op (OpenNhds.inclusion ((CategoryTheory.forget TopCat).map f.base x))) f.c) ≫
          colimMap (whiskerRight (𝟙 (Functor.op (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x))) X.presheaf) ≫
            colimit.pre (Functor.op (OpenNhds.inclusion x) ⋙ X.presheaf) (Functor.op (OpenNhds.map f.base x)) =
    colimit.ι (Functor.op (OpenNhds.inclusion (↑f.base y)) ⋙ Y.presheaf) j.op ≫
      (colimMap (whiskerLeft (Functor.op (OpenNhds.inclusion ((CategoryTheory.forget TopCat).map f.base y))) f.c) ≫
          colimMap (whiskerRight (𝟙 (Functor.op (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y))) X.presheaf) ≫
            colimit.pre (Functor.op (OpenNhds.inclusion y) ⋙ X.presheaf) (Functor.op (OpenNhds.map f.base y))) ≫
        colimit.desc (Functor.op (OpenNhds.inclusion y) ⋙ X.presheaf)
          { pt := colimit (Functor.op (OpenNhds.inclusion x) ⋙ X.presheaf),
            ι :=
              NatTrans.mk fun U =>
                colimit.ι (Functor.op (OpenNhds.inclusion x) ⋙ X.presheaf)
                  { obj := U.unop.obj, property := (_ : x ∈ U.unop.obj.carrier) }.op }

tactic: erw [colimit.ι_desc]
state_before: case h
C : Type u
inst✝¹ : Category C
inst✝ : HasColimits C
X Y : PresheafedSpace C
f : X ⟶ Y
x y : ↑↑X
h : x ⤳ y
j : OpenNhds (↑f.base y)
⊢ f.c.app j.obj.op ≫
      colimit.ι (Functor.op (OpenNhds.inclusion x) ⋙ X.presheaf)
        { obj := (Opens.map f.base).obj j.obj, property := (_ : ↑f.base x ∈ j.op.unop.obj.carrier) }.op =
    colimit.ι (Functor.op (OpenNhds.inclusion (↑f.base y)) ⋙ Y.presheaf) j.op ≫
      colimit.desc (Functor.op (OpenNhds.inclusion ((CategoryTheory.forget TopCat).map f.base y)) ⋙ Y.presheaf)
        ((Cocones.precompose
              (whiskerLeft (Functor.op (OpenNhds.inclusion ((CategoryTheory.forget TopCat).map f.base y))) f.c)).obj
          ((Cocones.precompose
                (whiskerRight (𝟙 (Functor.op (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y))) X.presheaf)).obj
            (Cocone.whisker (Functor.op (OpenNhds.map f.base y))
              { pt := colimit (Functor.op (OpenNhds.inclusion x) ⋙ X.presheaf),
                ι :=
                  NatTrans.mk fun U =>
                    colimit.ι (Functor.op (OpenNhds.inclusion x) ⋙ X.presheaf)
                      { obj := U.unop.obj, property := (_ : x ∈ U.unop.obj.carrier) }.op })))

tactic: dsimp
state_before: case h
C : Type u
inst✝¹ : Category C
inst✝ : HasColimits C
X Y : PresheafedSpace C
f : X ⟶ Y
x y : ↑↑X
h : x ⤳ y
j : OpenNhds (↑f.base y)
⊢ f.c.app j.obj.op ≫
      colimit.ι (Functor.op (OpenNhds.inclusion x) ⋙ X.presheaf)
        { obj := (Opens.map f.base).obj j.obj, property := (_ : ↑f.base x ∈ j.op.unop.obj.carrier) }.op =
    ((Cocones.precompose
                (whiskerLeft (Functor.op (OpenNhds.inclusion ((CategoryTheory.forget TopCat).map f.base y))) f.c)).obj
            ((Cocones.precompose
                  (whiskerRight (𝟙 (Functor.op (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y))) X.presheaf)).obj
              (Cocone.whisker (Functor.op (OpenNhds.map f.base y))
                { pt := colimit (Functor.op (OpenNhds.inclusion x) ⋙ X.presheaf),
                  ι :=
                    NatTrans.mk fun U =>
                      colimit.ι (Functor.op (OpenNhds.inclusion x) ⋙ X.presheaf)
                        { obj := U.unop.obj, property := (_ : x ∈ U.unop.obj.carrier) }.op }))).ι.app
      j.op

tactic: erw [X.presheaf.map_id, id_comp]
state_before: case h
C : Type u
inst✝¹ : Category C
inst✝ : HasColimits C
X Y : PresheafedSpace C
f : X ⟶ Y
x y : ↑↑X
h : x ⤳ y
j : OpenNhds (↑f.base y)
⊢ f.c.app j.obj.op ≫
      colimit.ι (Functor.op (OpenNhds.inclusion x) ⋙ X.presheaf)
        { obj := (Opens.map f.base).obj j.obj, property := (_ : ↑f.base x ∈ j.op.unop.obj.carrier) }.op =
    f.c.app ((OpenNhds.inclusion ((CategoryTheory.forget TopCat).map f.base y)).obj j).op ≫
      X.presheaf.map (𝟙 ((OpenNhds.inclusion y).obj ((OpenNhds.map f.base y).obj j)).op) ≫
        colimit.ι (Functor.op (OpenNhds.inclusion x) ⋙ X.presheaf)
          { obj := ((OpenNhds.map f.base y).obj j).obj,
              property := (_ : x ∈ ((OpenNhds.map f.base y).obj j).op.unop.obj.carrier) }.op

tactic: rfl
state_before: case h
C : Type u
inst✝¹ : Category C
inst✝ : HasColimits C
X Y : PresheafedSpace C
f : X ⟶ Y
x y : ↑↑X
h : x ⤳ y
j : OpenNhds (↑f.base y)
⊢ f.c.app j.obj.op ≫
      colimit.ι (Functor.op (OpenNhds.inclusion x) ⋙ X.presheaf)
        { obj := (Opens.map f.base).obj j.obj, property := (_ : ↑f.base x ∈ j.op.unop.obj.carrier) }.op =
    f.c.app ((OpenNhds.inclusion ((CategoryTheory.forget TopCat).map f.base y)).obj j).op ≫
      colimit.ι (Functor.op (OpenNhds.inclusion x) ⋙ X.presheaf)
        { obj := ((OpenNhds.map f.base y).obj j).obj,
            property := (_ : x ∈ ((OpenNhds.map f.base y).obj j).op.unop.obj.carrier) }.op