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cospan_op {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z) :
cospan f.op g.op ≅ walking_span_op_equiv.inverse ⋙ (span f g).op | nat_iso.of_components (by { rintro (_|_|_); refl, })
(by { rintros (_|_|_) (_|_|_) f; cases f; tidy, }) | def | category_theory.limits.cospan_op | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | The canonical isomorphism relating `cospan f.op g.op` and `(span f g).op` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_span {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z) :
(span f g).op ≅ walking_span_op_equiv.functor ⋙ cospan f.op g.op | calc (span f g).op ≅ 𝟭 _ ⋙ (span f g).op : by refl
... ≅ (walking_span_op_equiv.functor ⋙ walking_span_op_equiv.inverse) ⋙ (span f g).op :
iso_whisker_right walking_span_op_equiv.unit_iso _
... ≅ walking_span_op_equiv.functor ⋙ (walking_span_op_equiv.inverse ⋙ (span f g).op) :
functor.associator _ _ _
... ≅ walkin... | def | category_theory.limits.op_span | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | The canonical isomorphism relating `(span f g).op` and `cospan f.op g.op` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop {X Y Z : Cᵒᵖ} {f : X ⟶ Y} {g : X ⟶ Z} (c : pushout_cocone f g) :
pullback_cone f.unop g.unop | cocone.unop ((cocones.precompose (op_cospan f.unop g.unop).hom).obj
(cocone.whisker walking_cospan_op_equiv.functor c)) | def | category_theory.limits.pushout_cocone.unop | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | The obvious map `pushout_cocone f g → pullback_cone f.unop g.unop` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_fst {X Y Z : Cᵒᵖ} {f : X ⟶ Y} {g : X ⟶ Z} (c : pushout_cocone f g) :
c.unop.fst = c.inl.unop | by { change (_ : limits.cone _).π.app _ = _,
simp only [pushout_cocone.ι_app_left, pushout_cocone.unop_π_app], tidy } | lemma | category_theory.limits.pushout_cocone.unop_fst | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_snd {X Y Z : Cᵒᵖ} {f : X ⟶ Y} {g : X ⟶ Z} (c : pushout_cocone f g) :
c.unop.snd = c.inr.unop | by { change (_ : limits.cone _).π.app _ = _,
simp only [pushout_cocone.unop_π_app, pushout_cocone.ι_app_right], tidy, } | lemma | category_theory.limits.pushout_cocone.unop_snd | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} (c : pushout_cocone f g) :
pullback_cone f.op g.op | (cones.postcompose ((cospan_op f g).symm).hom).obj
(cone.whisker walking_span_op_equiv.inverse (cocone.op c)) | def | category_theory.limits.pushout_cocone.op | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | The obvious map `pushout_cocone f.op g.op → pullback_cone f g` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_fst {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} (c : pushout_cocone f g) :
c.op.fst = c.inl.op | by { change (_ : limits.cone _).π.app _ = _, apply category.comp_id, } | lemma | category_theory.limits.pushout_cocone.op_fst | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_snd {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} (c : pushout_cocone f g) :
c.op.snd = c.inr.op | by { change (_ : limits.cone _).π.app _ = _, apply category.comp_id, } | lemma | category_theory.limits.pushout_cocone.op_snd | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop {X Y Z : Cᵒᵖ} {f : X ⟶ Z} {g : Y ⟶ Z} (c : pullback_cone f g) :
pushout_cocone f.unop g.unop | cone.unop ((cones.postcompose (op_span f.unop g.unop).symm.hom).obj
(cone.whisker walking_span_op_equiv.functor c)) | def | category_theory.limits.pullback_cone.unop | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | The obvious map `pullback_cone f g → pushout_cocone f.unop g.unop` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_inl {X Y Z : Cᵒᵖ} {f : X ⟶ Z} {g : Y ⟶ Z} (c : pullback_cone f g) :
c.unop.inl = c.fst.unop | begin
change ((_ : limits.cocone _).ι.app _) = _,
dsimp only [unop, op_span],
simp, dsimp, simp, dsimp, simp
end | lemma | category_theory.limits.pullback_cone.unop_inl | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_inr {X Y Z : Cᵒᵖ} {f : X ⟶ Z} {g : Y ⟶ Z} (c : pullback_cone f g) :
c.unop.inr = c.snd.unop | begin
change ((_ : limits.cocone _).ι.app _) = _,
apply quiver.hom.op_inj,
simp [unop_ι_app], dsimp, simp,
apply category.comp_id,
end | lemma | category_theory.limits.pullback_cone.unop_inr | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [
"quiver.hom.op_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op {X Y Z : C} {f : X ⟶ Z} {g : Y ⟶ Z} (c : pullback_cone f g) :
pushout_cocone f.op g.op | (cocones.precompose (span_op f g).hom).obj
(cocone.whisker walking_cospan_op_equiv.inverse (cone.op c)) | def | category_theory.limits.pullback_cone.op | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | The obvious map `pullback_cone f g → pushout_cocone f.op g.op` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_inl {X Y Z : C} {f : X ⟶ Z} {g : Y ⟶ Z} (c : pullback_cone f g) :
c.op.inl = c.fst.op | by { change (_ : limits.cocone _).ι.app _ = _, apply category.id_comp, } | lemma | category_theory.limits.pullback_cone.op_inl | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_inr {X Y Z : C} {f : X ⟶ Z} {g : Y ⟶ Z} (c : pullback_cone f g) :
c.op.inr = c.snd.op | by { change (_ : limits.cocone _).ι.app _ = _, apply category.id_comp, } | lemma | category_theory.limits.pullback_cone.op_inr | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_unop {X Y Z : C} {f : X ⟶ Z} {g : Y ⟶ Z} (c : pullback_cone f g) : c.op.unop ≅ c | pullback_cone.ext (iso.refl _) (by simp) (by simp) | def | category_theory.limits.pullback_cone.op_unop | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | If `c` is a pullback cone, then `c.op.unop` is isomorphic to `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_op {X Y Z : Cᵒᵖ} {f : X ⟶ Z} {g : Y ⟶ Z} (c : pullback_cone f g) : c.unop.op ≅ c | pullback_cone.ext (iso.refl _) (by simp) (by simp) | def | category_theory.limits.pullback_cone.unop_op | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | If `c` is a pullback cone in `Cᵒᵖ`, then `c.unop.op` is isomorphic to `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_unop {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} (c : pushout_cocone f g) : c.op.unop ≅ c | pushout_cocone.ext (iso.refl _) (by simp) (by simp) | def | category_theory.limits.pushout_cocone.op_unop | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | If `c` is a pushout cocone, then `c.op.unop` is isomorphic to `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_op {X Y Z : Cᵒᵖ} {f : X ⟶ Y} {g : X ⟶ Z} (c : pushout_cocone f g) : c.unop.op ≅ c | pushout_cocone.ext (iso.refl _) (by simp) (by simp) | def | category_theory.limits.pushout_cocone.unop_op | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | If `c` is a pushout cocone in `Cᵒᵖ`, then `c.unop.op` is isomorphic to `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_equiv_is_limit_op {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} (c : pushout_cocone f g) :
is_colimit c ≃ is_limit c.op | begin
apply equiv_of_subsingleton_of_subsingleton,
{ intro h,
equiv_rw is_limit.postcompose_hom_equiv _ _,
equiv_rw (is_limit.whisker_equivalence_equiv walking_span_op_equiv.symm).symm,
exact is_limit_cocone_op _ h, },
{ intro h,
equiv_rw is_colimit.equiv_iso_colimit c.op_unop.symm,
apply is_c... | def | category_theory.limits.pushout_cocone.is_colimit_equiv_is_limit_op | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [
"equiv_of_subsingleton_of_subsingleton"
] | A pushout cone is a colimit cocone if and only if the corresponding pullback cone
in the opposite category is a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_equiv_is_limit_unop {X Y Z : Cᵒᵖ} {f : X ⟶ Y} {g : X ⟶ Z}
(c : pushout_cocone f g) : is_colimit c ≃ is_limit c.unop | begin
apply equiv_of_subsingleton_of_subsingleton,
{ intro h,
apply is_limit_cocone_unop,
equiv_rw is_colimit.precompose_hom_equiv _ _,
equiv_rw (is_colimit.whisker_equivalence_equiv _).symm,
exact h, },
{ intro h,
equiv_rw is_colimit.equiv_iso_colimit c.unop_op.symm,
equiv_rw is_colimit.p... | def | category_theory.limits.pushout_cocone.is_colimit_equiv_is_limit_unop | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [
"equiv_of_subsingleton_of_subsingleton"
] | A pushout cone is a colimit cocone in `Cᵒᵖ` if and only if the corresponding pullback cone
in `C` is a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_equiv_is_colimit_op {X Y Z : C} {f : X ⟶ Z} {g : Y ⟶ Z}
(c : pullback_cone f g) : is_limit c ≃ is_colimit c.op | (is_limit.equiv_iso_limit c.op_unop).symm.trans c.op.is_colimit_equiv_is_limit_unop.symm | def | category_theory.limits.pullback_cone.is_limit_equiv_is_colimit_op | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | A pullback cone is a limit cone if and only if the corresponding pushout cocone
in the opposite category is a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_equiv_is_colimit_unop {X Y Z : Cᵒᵖ} {f : X ⟶ Z} {g : Y ⟶ Z}
(c : pullback_cone f g) : is_limit c ≃ is_colimit c.unop | (is_limit.equiv_iso_limit c.unop_op).symm.trans c.unop.is_colimit_equiv_is_limit_op.symm | def | category_theory.limits.pullback_cone.is_limit_equiv_is_colimit_unop | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | A pullback cone is a limit cone in `Cᵒᵖ` if and only if the corresponding pushout cocone
in `C` is a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_iso_unop_pushout {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z)
[has_pullback f g] [has_pushout f.op g.op] : pullback f g ≅ unop (pushout f.op g.op) | is_limit.cone_point_unique_up_to_iso (limit.is_limit _)
((pushout_cocone.is_colimit_equiv_is_limit_unop _) (colimit.is_colimit (span f.op g.op))) | def | category_theory.limits.pullback_iso_unop_pushout | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | The pullback of `f` and `g` in `C` is isomorphic to the pushout of
`f.op` and `g.op` in `Cᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_iso_unop_pushout_inv_fst {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z)
[has_pullback f g] [has_pushout f.op g.op] :
(pullback_iso_unop_pushout f g).inv ≫ pullback.fst =
(pushout.inl : _ ⟶ pushout f.op g.op).unop | (is_limit.cone_point_unique_up_to_iso_inv_comp _ _ _).trans (by simp) | lemma | category_theory.limits.pullback_iso_unop_pushout_inv_fst | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_iso_unop_pushout_inv_snd {X Y Z : C} (f : X ⟶ Z)
(g : Y ⟶ Z) [has_pullback f g] [has_pushout f.op g.op] :
(pullback_iso_unop_pushout f g).inv ≫ pullback.snd =
(pushout.inr : _ ⟶ pushout f.op g.op).unop | (is_limit.cone_point_unique_up_to_iso_inv_comp _ _ _).trans (by simp) | lemma | category_theory.limits.pullback_iso_unop_pushout_inv_snd | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_iso_unop_pushout_hom_inl {X Y Z : C} (f : X ⟶ Z)
(g : Y ⟶ Z) [has_pullback f g] [has_pushout f.op g.op] :
pushout.inl ≫ (pullback_iso_unop_pushout f g).hom.op = pullback.fst.op | begin
apply quiver.hom.unop_inj,
dsimp,
rw [← pullback_iso_unop_pushout_inv_fst, iso.hom_inv_id_assoc],
end | lemma | category_theory.limits.pullback_iso_unop_pushout_hom_inl | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [
"quiver.hom.unop_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_iso_unop_pushout_hom_inr {X Y Z : C} (f : X ⟶ Z)
(g : Y ⟶ Z) [has_pullback f g] [has_pushout f.op g.op] :
pushout.inr ≫ (pullback_iso_unop_pushout f g).hom.op = pullback.snd.op | begin
apply quiver.hom.unop_inj,
dsimp,
rw [← pullback_iso_unop_pushout_inv_snd, iso.hom_inv_id_assoc],
end | lemma | category_theory.limits.pullback_iso_unop_pushout_hom_inr | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [
"quiver.hom.unop_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushout_iso_unop_pullback {X Y Z : C} (f : X ⟶ Z) (g : X ⟶ Y)
[has_pushout f g] [has_pullback f.op g.op] : pushout f g ≅ unop (pullback f.op g.op) | is_colimit.cocone_point_unique_up_to_iso (colimit.is_colimit _)
((pullback_cone.is_limit_equiv_is_colimit_unop _) (limit.is_limit (cospan f.op g.op))) | def | category_theory.limits.pushout_iso_unop_pullback | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | The pushout of `f` and `g` in `C` is isomorphic to the pullback of
`f.op` and `g.op` in `Cᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout_iso_unop_pullback_inl_hom {X Y Z : C} (f : X ⟶ Z) (g : X ⟶ Y)
[has_pushout f g] [has_pullback f.op g.op] :
pushout.inl ≫ (pushout_iso_unop_pullback f g).hom =
(pullback.fst : pullback f.op g.op ⟶ _).unop | (is_colimit.comp_cocone_point_unique_up_to_iso_hom _ _ _).trans (by simp) | lemma | category_theory.limits.pushout_iso_unop_pullback_inl_hom | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushout_iso_unop_pullback_inr_hom {X Y Z : C} (f : X ⟶ Z) (g : X ⟶ Y)
[has_pushout f g] [has_pullback f.op g.op] :
pushout.inr ≫ (pushout_iso_unop_pullback f g).hom =
(pullback.snd : pullback f.op g.op ⟶ _).unop | (is_colimit.comp_cocone_point_unique_up_to_iso_hom _ _ _).trans (by simp) | lemma | category_theory.limits.pushout_iso_unop_pullback_inr_hom | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushout_iso_unop_pullback_inv_fst {X Y Z : C} (f : X ⟶ Z) (g : X ⟶ Y)
[has_pushout f g] [has_pullback f.op g.op] :
(pushout_iso_unop_pullback f g).inv.op ≫ pullback.fst = pushout.inl.op | begin
apply quiver.hom.unop_inj,
dsimp,
rw [← pushout_iso_unop_pullback_inl_hom, category.assoc, iso.hom_inv_id, category.comp_id],
end | lemma | category_theory.limits.pushout_iso_unop_pullback_inv_fst | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [
"quiver.hom.unop_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushout_iso_unop_pullback_inv_snd {X Y Z : C} (f : X ⟶ Z) (g : X ⟶ Y)
[has_pushout f g] [has_pullback f.op g.op] :
(pushout_iso_unop_pullback f g).inv.op ≫ pullback.snd = pushout.inr.op | begin
apply quiver.hom.unop_inj,
dsimp,
rw [← pushout_iso_unop_pullback_inr_hom, category.assoc, iso.hom_inv_id, category.comp_id],
end | lemma | category_theory.limits.pushout_iso_unop_pullback_inv_snd | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [
"quiver.hom.unop_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimit_of_has_colimit_comp_forget
(F : J ⥤ over X) [i : has_colimit (F ⋙ forget X)] : has_colimit F | @@costructured_arrow.has_colimit _ _ _ _ i _ | instance | category_theory.over.has_colimit_of_has_colimit_comp_forget | category_theory.limits | src/category_theory/limits/over.lean | [
"category_theory.over",
"category_theory.adjunction.opposites",
"category_theory.limits.preserves.basic",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.creates",
"category_theory.limits.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
creates_colimits : creates_colimits (forget X) | costructured_arrow.creates_colimits | instance | category_theory.over.creates_colimits | category_theory.limits | src/category_theory/limits/over.lean | [
"category_theory.over",
"category_theory.adjunction.opposites",
"category_theory.limits.preserves.basic",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.creates",
"category_theory.limits.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_left_of_epi [has_pushouts C] {f g : over X} (h : f ⟶ g) [epi h] : epi h.left | costructured_arrow.epi_left_of_epi _ | lemma | category_theory.over.epi_left_of_epi | category_theory.limits | src/category_theory/limits/over.lean | [
"category_theory.over",
"category_theory.adjunction.opposites",
"category_theory.limits.preserves.basic",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.creates",
"category_theory.limits.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_iff_epi_left [has_pushouts C] {f g : over X} (h : f ⟶ g) : epi h ↔ epi h.left | costructured_arrow.epi_iff_epi_left _ | lemma | category_theory.over.epi_iff_epi_left | category_theory.limits | src/category_theory/limits/over.lean | [
"category_theory.over",
"category_theory.adjunction.opposites",
"category_theory.limits.preserves.basic",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.creates",
"category_theory.limits.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback {X Y : C} (f : X ⟶ Y) : over Y ⥤ over X | { obj := λ g, over.mk (pullback.snd : pullback g.hom f ⟶ X),
map := λ g h k,
over.hom_mk
(pullback.lift (pullback.fst ≫ k.left) pullback.snd (by simp [pullback.condition]))
(by tidy) } | def | category_theory.over.pullback | category_theory.limits | src/category_theory/limits/over.lean | [
"category_theory.over",
"category_theory.adjunction.opposites",
"category_theory.limits.preserves.basic",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.creates",
"category_theory.limits.comma"
] | [] | When `C` has pullbacks, a morphism `f : X ⟶ Y` induces a functor `over Y ⥤ over X`,
by pulling back a morphism along `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_pullback_adj {A B : C} (f : A ⟶ B) :
over.map f ⊣ pullback f | adjunction.mk_of_hom_equiv
{ hom_equiv := λ g h,
{ to_fun := λ X, over.hom_mk (pullback.lift X.left g.hom (over.w X)) (pullback.lift_snd _ _ _),
inv_fun := λ Y,
begin
refine over.hom_mk _ _,
refine Y.left ≫ pullback.fst,
dsimp,
rw [← over.w Y, category.assoc, pullback.condition, catego... | def | category_theory.over.map_pullback_adj | category_theory.limits | src/category_theory/limits/over.lean | [
"category_theory.over",
"category_theory.adjunction.opposites",
"category_theory.limits.preserves.basic",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.creates",
"category_theory.limits.comma"
] | [
"inv_fun"
] | `over.map f` is left adjoint to `over.pullback f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_id {A : C} : pullback (𝟙 A) ≅ 𝟭 _ | adjunction.right_adjoint_uniq
(map_pullback_adj _)
(adjunction.id.of_nat_iso_left over.map_id.symm) | def | category_theory.over.pullback_id | category_theory.limits | src/category_theory/limits/over.lean | [
"category_theory.over",
"category_theory.adjunction.opposites",
"category_theory.limits.preserves.basic",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.creates",
"category_theory.limits.comma"
] | [] | pullback (𝟙 A) : over A ⥤ over A is the identity functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_comp {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) :
pullback (f ≫ g) ≅ pullback g ⋙ pullback f | adjunction.right_adjoint_uniq
(map_pullback_adj _)
(((map_pullback_adj _).comp (map_pullback_adj _)).of_nat_iso_left
(over.map_comp _ _).symm) | def | category_theory.over.pullback_comp | category_theory.limits | src/category_theory/limits/over.lean | [
"category_theory.over",
"category_theory.adjunction.opposites",
"category_theory.limits.preserves.basic",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.creates",
"category_theory.limits.comma"
] | [] | pullback commutes with composition (up to natural isomorphism). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_is_right_adjoint {A B : C} (f : A ⟶ B) :
is_right_adjoint (pullback f) | ⟨_, map_pullback_adj f⟩ | instance | category_theory.over.pullback_is_right_adjoint | category_theory.limits | src/category_theory/limits/over.lean | [
"category_theory.over",
"category_theory.adjunction.opposites",
"category_theory.limits.preserves.basic",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.creates",
"category_theory.limits.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limit_of_has_limit_comp_forget
(F : J ⥤ under X) [i : has_limit (F ⋙ forget X)] : has_limit F | @@structured_arrow.has_limit _ _ _ _ i _ | instance | category_theory.under.has_limit_of_has_limit_comp_forget | category_theory.limits | src/category_theory/limits/over.lean | [
"category_theory.over",
"category_theory.adjunction.opposites",
"category_theory.limits.preserves.basic",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.creates",
"category_theory.limits.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_right_of_mono [has_pullbacks C] {f g : under X} (h : f ⟶ g) [mono h] : mono h.right | structured_arrow.mono_right_of_mono _ | lemma | category_theory.under.mono_right_of_mono | category_theory.limits | src/category_theory/limits/over.lean | [
"category_theory.over",
"category_theory.adjunction.opposites",
"category_theory.limits.preserves.basic",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.creates",
"category_theory.limits.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_iff_mono_right [has_pullbacks C] {f g : under X} (h : f ⟶ g) : mono h ↔ mono h.right | structured_arrow.mono_iff_mono_right _ | lemma | category_theory.under.mono_iff_mono_right | category_theory.limits | src/category_theory/limits/over.lean | [
"category_theory.over",
"category_theory.adjunction.opposites",
"category_theory.limits.preserves.basic",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.creates",
"category_theory.limits.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
creates_limits : creates_limits (forget X) | structured_arrow.creates_limits | instance | category_theory.under.creates_limits | category_theory.limits | src/category_theory/limits/over.lean | [
"category_theory.over",
"category_theory.adjunction.opposites",
"category_theory.limits.preserves.basic",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.creates",
"category_theory.limits.comma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushout {X Y : C} (f : X ⟶ Y) : under X ⥤ under Y | { obj := λ g, under.mk (pushout.inr : Y ⟶ pushout g.hom f),
map := λ g h k,
under.hom_mk
(pushout.desc (k.right ≫ pushout.inl) pushout.inr (by { simp [←pushout.condition], }))
(by tidy) } | def | category_theory.under.pushout | category_theory.limits | src/category_theory/limits/over.lean | [
"category_theory.over",
"category_theory.adjunction.opposites",
"category_theory.limits.preserves.basic",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.creates",
"category_theory.limits.comma"
] | [] | When `C` has pushouts, a morphism `f : X ⟶ Y` induces a functor `under X ⥤ under Y`,
by pushing a morphism forward along `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_comp_eval (c : cone F) (i : I) : cone (F ⋙ pi.eval C i) | { X := c.X i,
π :=
{ app := λ j, c.π.app j i,
naturality' := λ j j' f, congr_fun (c.π.naturality f) i, } } | def | category_theory.pi.cone_comp_eval | category_theory.limits | src/category_theory/limits/pi.lean | [
"category_theory.pi.basic",
"category_theory.limits.has_limits"
] | [] | A cone over `F : J ⥤ Π i, C i` has as its components cones over each of the `F ⋙ pi.eval C i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_comp_eval (c : cocone F) (i : I) : cocone (F ⋙ pi.eval C i) | { X := c.X i,
ι :=
{ app := λ j, c.ι.app j i,
naturality' := λ j j' f, congr_fun (c.ι.naturality f) i, } } | def | category_theory.pi.cocone_comp_eval | category_theory.limits | src/category_theory/limits/pi.lean | [
"category_theory.pi.basic",
"category_theory.limits.has_limits"
] | [] | A cocone over `F : J ⥤ Π i, C i` has as its components cocones over each of the `F ⋙ pi.eval C i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_of_cone_comp_eval (c : Π i, cone (F ⋙ pi.eval C i)) : cone F | { X := λ i, (c i).X,
π :=
{ app := λ j i, (c i).π.app j,
naturality' := λ j j' f, by { ext i, exact (c i).π.naturality f, } } } | def | category_theory.pi.cone_of_cone_comp_eval | category_theory.limits | src/category_theory/limits/pi.lean | [
"category_theory.pi.basic",
"category_theory.limits.has_limits"
] | [] | Given a family of cones over the `F ⋙ pi.eval C i`, we can assemble these together as a `cone F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_of_cocone_comp_eval (c : Π i, cocone (F ⋙ pi.eval C i)) : cocone F | { X := λ i, (c i).X,
ι :=
{ app := λ j i, (c i).ι.app j,
naturality' := λ j j' f, by { ext i, exact (c i).ι.naturality f, } } } | def | category_theory.pi.cocone_of_cocone_comp_eval | category_theory.limits | src/category_theory/limits/pi.lean | [
"category_theory.pi.basic",
"category_theory.limits.has_limits"
] | [] | Given a family of cocones over the `F ⋙ pi.eval C i`,
we can assemble these together as a `cocone F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_of_cone_eval_is_limit {c : Π i, cone (F ⋙ pi.eval C i)} (P : Π i, is_limit (c i)) :
is_limit (cone_of_cone_comp_eval c) | { lift := λ s i, (P i).lift (cone_comp_eval s i),
fac' := λ s j,
begin
ext i,
exact (P i).fac (cone_comp_eval s i) j,
end,
uniq' := λ s m w,
begin
ext i,
exact (P i).uniq (cone_comp_eval s i) (m i) (λ j, congr_fun (w j) i)
end } | def | category_theory.pi.cone_of_cone_eval_is_limit | category_theory.limits | src/category_theory/limits/pi.lean | [
"category_theory.pi.basic",
"category_theory.limits.has_limits"
] | [
"lift"
] | Given a family of limit cones over the `F ⋙ pi.eval C i`,
assembling them together as a `cone F` produces a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_of_cocone_eval_is_colimit
{c : Π i, cocone (F ⋙ pi.eval C i)} (P : Π i, is_colimit (c i)) :
is_colimit (cocone_of_cocone_comp_eval c) | { desc := λ s i, (P i).desc (cocone_comp_eval s i),
fac' := λ s j,
begin
ext i,
exact (P i).fac (cocone_comp_eval s i) j,
end,
uniq' := λ s m w,
begin
ext i,
exact (P i).uniq (cocone_comp_eval s i) (m i) (λ j, congr_fun (w j) i)
end } | def | category_theory.pi.cocone_of_cocone_eval_is_colimit | category_theory.limits | src/category_theory/limits/pi.lean | [
"category_theory.pi.basic",
"category_theory.limits.has_limits"
] | [] | Given a family of colimit cocones over the `F ⋙ pi.eval C i`,
assembling them together as a `cocone F` produces a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limit_of_has_limit_comp_eval : has_limit F | has_limit.mk
{ cone := cone_of_cone_comp_eval (λ i, limit.cone _),
is_limit := cone_of_cone_eval_is_limit (λ i, limit.is_limit _), } | lemma | category_theory.pi.has_limit_of_has_limit_comp_eval | category_theory.limits | src/category_theory/limits/pi.lean | [
"category_theory.pi.basic",
"category_theory.limits.has_limits"
] | [] | If we have a functor `F : J ⥤ Π i, C i` into a category of indexed families,
and we have limits for each of the `F ⋙ pi.eval C i`,
then `F` has a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimit_of_has_colimit_comp_eval : has_colimit F | has_colimit.mk
{ cocone := cocone_of_cocone_comp_eval (λ i, colimit.cocone _),
is_colimit := cocone_of_cocone_eval_is_colimit (λ i, colimit.is_colimit _), } | lemma | category_theory.pi.has_colimit_of_has_colimit_comp_eval | category_theory.limits | src/category_theory/limits/pi.lean | [
"category_theory.pi.basic",
"category_theory.limits.has_limits"
] | [] | If we have a functor `F : J ⥤ Π i, C i` into a category of indexed families,
and colimits exist for each of the `F ⋙ pi.eval C i`,
there is a colimit for `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restricted_yoneda : ℰ ⥤ (Cᵒᵖ ⥤ Type u₁) | yoneda ⋙ (whiskering_left _ _ (Type u₁)).obj (functor.op A) | def | category_theory.colimit_adj.restricted_yoneda | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [] | The functor taking `(E : ℰ) (c : Cᵒᵖ)` to the homset `(A.obj C ⟶ E)`. It is shown in `L_adjunction`
that this functor has a left adjoint (provided `E` has colimits) given by taking colimits over
categories of elements.
In the case where `ℰ = Cᵒᵖ ⥤ Type u` and `A = yoneda`, this functor is isomorphic to the identity.
D... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restricted_yoneda_yoneda : restricted_yoneda (yoneda : C ⥤ Cᵒᵖ ⥤ Type u₁) ≅ 𝟭 _ | nat_iso.of_components
(λ P, nat_iso.of_components (λ X, yoneda_sections_small X.unop _)
(λ X Y f, funext $ λ x,
begin
dsimp,
rw ← functor_to_types.naturality _ _ x f (𝟙 _),
dsimp,
simp,
end))
(λ _ _ _, rfl) | def | category_theory.colimit_adj.restricted_yoneda_yoneda | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [] | The functor `restricted_yoneda` is isomorphic to the identity functor when evaluated at the yoneda
embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_yoneda_hom_equiv (P : Cᵒᵖ ⥤ Type u₁) (E : ℰ)
{c : cocone ((category_of_elements.π P).left_op ⋙ A)} (t : is_colimit c) :
(c.X ⟶ E) ≃ (P ⟶ (restricted_yoneda A).obj E) | ((ulift_trivial _).symm ≪≫ t.hom_iso' E).to_equiv.trans
{ to_fun := λ k,
{ app := λ c p, k.1 (opposite.op ⟨_, p⟩),
naturality' := λ c c' f, funext $ λ p,
(k.2 (quiver.hom.op ⟨f, rfl⟩ :
(opposite.op ⟨c', P.map f p⟩ : P.elementsᵒᵖ) ⟶ opposite.op ⟨c, p⟩)).symm },
inv_fun := λ τ,
{ val := λ p,... | def | category_theory.colimit_adj.restrict_yoneda_hom_equiv | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [
"inv_fun",
"opposite.op",
"quiver.hom.op"
] | (Implementation). The equivalence of homsets which helps construct the left adjoint to
`colimit_adj.restricted_yoneda`.
It is shown in `restrict_yoneda_hom_equiv_natural` that this is a natural bijection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_yoneda_hom_equiv_natural (P : Cᵒᵖ ⥤ Type u₁) (E₁ E₂ : ℰ) (g : E₁ ⟶ E₂)
{c : cocone _} (t : is_colimit c) (k : c.X ⟶ E₁) :
restrict_yoneda_hom_equiv A P E₂ t (k ≫ g) =
restrict_yoneda_hom_equiv A P E₁ t k ≫ (restricted_yoneda A).map g | begin
ext _ X p,
apply (assoc _ _ _).symm,
end | lemma | category_theory.colimit_adj.restrict_yoneda_hom_equiv_natural | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [] | (Implementation). Show that the bijection in `restrict_yoneda_hom_equiv` is natural (on the right). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend_along_yoneda : (Cᵒᵖ ⥤ Type u₁) ⥤ ℰ | adjunction.left_adjoint_of_equiv
(λ P E, restrict_yoneda_hom_equiv A P E (colimit.is_colimit _))
(λ P E E' g, restrict_yoneda_hom_equiv_natural A P E E' g _) | def | category_theory.colimit_adj.extend_along_yoneda | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [] | The left adjoint to the functor `restricted_yoneda` (shown in `yoneda_adjunction`). It is also an
extension of `A` along the yoneda embedding (shown in `is_extension_along_yoneda`), in particular
it is the left Kan extension of `A` through the yoneda embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend_along_yoneda_obj (P : Cᵒᵖ ⥤ Type u₁) : (extend_along_yoneda A).obj P =
colimit ((category_of_elements.π P).left_op ⋙ A) | rfl | lemma | category_theory.colimit_adj.extend_along_yoneda_obj | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_along_yoneda_map {X Y : Cᵒᵖ ⥤ Type u₁} (f : X ⟶ Y) :
(extend_along_yoneda A).map f = colimit.pre ((category_of_elements.π Y).left_op ⋙ A)
(category_of_elements.map f).op | begin
ext J,
erw colimit.ι_pre ((category_of_elements.π Y).left_op ⋙ A) (category_of_elements.map f).op,
dsimp only [extend_along_yoneda, restrict_yoneda_hom_equiv,
is_colimit.hom_iso', is_colimit.hom_iso, ulift_trivial],
simpa
end | lemma | category_theory.colimit_adj.extend_along_yoneda_map | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
yoneda_adjunction : extend_along_yoneda A ⊣ restricted_yoneda A | adjunction.adjunction_of_equiv_left _ _ | def | category_theory.colimit_adj.yoneda_adjunction | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [] | Show `extend_along_yoneda` is left adjoint to `restricted_yoneda`.
The construction of [MM92], Chapter I, Section 5, Theorem 2. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
elements.initial (A : C) : (yoneda.obj A).elements | ⟨opposite.op A, 𝟙 _⟩ | def | category_theory.colimit_adj.elements.initial | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [] | The initial object in the category of elements for a representable functor. In `is_initial` it is
shown that this is initial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_initial (A : C) : is_initial (elements.initial A) | { desc := λ s, ⟨s.X.2.op, comp_id _⟩,
uniq' := λ s m w,
begin
simp_rw ← m.2,
dsimp [elements.initial],
simp,
end,
fac' := by rintros s ⟨⟨⟩⟩, } | def | category_theory.colimit_adj.is_initial | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [] | Show that `elements.initial A` is initial in the category of elements for the `yoneda` functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_extension_along_yoneda : (yoneda : C ⥤ Cᵒᵖ ⥤ Type u₁) ⋙ extend_along_yoneda A ≅ A | nat_iso.of_components
(λ X, (colimit.is_colimit _).cocone_point_unique_up_to_iso
(colimit_of_diagram_terminal (terminal_op_of_initial (is_initial _)) _))
begin
intros X Y f,
change (colimit.desc _ ⟨_, _⟩ ≫ colimit.desc _ _) = colimit.desc _ _ ≫ _,
apply colimit.hom_ext,
intro j,
rw [colimit.ι_desc_assoc... | def | category_theory.colimit_adj.is_extension_along_yoneda | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [] | `extend_along_yoneda A` is an extension of `A` to the presheaf category along the yoneda embedding.
`unique_extension_along_yoneda` shows it is unique among functors preserving colimits with this
property (up to isomorphism).
The first part of [MM92], Chapter I, Section 5, Corollary 4.
See Property 1 of <https://ncatl... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend_along_yoneda_iso_Kan_app (X) :
(extend_along_yoneda A).obj X ≅ ((Lan yoneda : (_ ⥤ ℰ) ⥤ _).obj A).obj X | let eq := category_of_elements.costructured_arrow_yoneda_equivalence X in
{ hom := colimit.pre (Lan.diagram (yoneda : C ⥤ _ ⥤ Type u₁) A X) eq.functor,
inv := colimit.pre ((category_of_elements.π X).left_op ⋙ A) eq.inverse,
hom_inv_id' :=
begin
erw colimit.pre_pre ((category_of_elements.π X).left_op ⋙ A) eq.i... | def | category_theory.colimit_adj.extend_along_yoneda_iso_Kan_app | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [] | Show that the images of `X` after `extend_along_yoneda` and `Lan yoneda` are indeed isomorphic.
This follows from `category_theory.category_of_elements.costructured_arrow_yoneda_equivalence`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend_along_yoneda_iso_Kan : extend_along_yoneda A ≅ (Lan yoneda : (_ ⥤ ℰ) ⥤ _).obj A | nat_iso.of_components (extend_along_yoneda_iso_Kan_app A)
begin
intros X Y f, simp,
rw extend_along_yoneda_map,
erw colimit.pre_pre (Lan.diagram (yoneda : C ⥤ _ ⥤ Type u₁) A Y) (costructured_arrow.map f),
erw colimit.pre_pre (Lan.diagram (yoneda : C ⥤ _ ⥤ Type u₁) A Y)
(category_of_elements.costructured_arr... | def | category_theory.colimit_adj.extend_along_yoneda_iso_Kan | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [] | Verify that `extend_along_yoneda` is indeed the left Kan extension along the yoneda embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend_of_comp_yoneda_iso_Lan {D : Type u₁} [small_category D] (F : C ⥤ D) :
extend_along_yoneda (F ⋙ yoneda) ≅ Lan F.op | adjunction.nat_iso_of_right_adjoint_nat_iso
(yoneda_adjunction (F ⋙ yoneda))
(Lan.adjunction (Type u₁) F.op)
(iso_whisker_right curried_yoneda_lemma' ((whiskering_left Cᵒᵖ Dᵒᵖ (Type u₁)).obj F.op : _)) | def | category_theory.colimit_adj.extend_of_comp_yoneda_iso_Lan | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [] | extending `F ⋙ yoneda` along the yoneda embedding is isomorphic to `Lan F.op`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_yoneda_iso_yoneda_comp_Lan {D : Type u₁} [small_category D] (F : C ⥤ D) :
F ⋙ yoneda ≅ yoneda ⋙ Lan F.op | (is_extension_along_yoneda (F ⋙ yoneda)).symm ≪≫
iso_whisker_left yoneda (extend_of_comp_yoneda_iso_Lan F) | def | category_theory.comp_yoneda_iso_yoneda_comp_Lan | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [] | `F ⋙ yoneda` is naturally isomorphic to `yoneda ⋙ Lan F.op`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend_along_yoneda_yoneda : extend_along_yoneda (yoneda : C ⥤ _) ≅ 𝟭 _ | adjunction.nat_iso_of_right_adjoint_nat_iso
(yoneda_adjunction _)
adjunction.id
restricted_yoneda_yoneda | def | category_theory.extend_along_yoneda_yoneda | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [] | Since `extend_along_yoneda A` is adjoint to `restricted_yoneda A`, if we use `A = yoneda`
then `restricted_yoneda A` is isomorphic to the identity, and so `extend_along_yoneda A` is as well. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor_to_representables (P : Cᵒᵖ ⥤ Type u₁) :
(P.elements)ᵒᵖ ⥤ Cᵒᵖ ⥤ Type u₁ | (category_of_elements.π P).left_op ⋙ yoneda | def | category_theory.functor_to_representables | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocone_of_representable (P : Cᵒᵖ ⥤ Type u₁) :
cocone (functor_to_representables P) | cocone.extend (colimit.cocone _) (extend_along_yoneda_yoneda.hom.app P) | def | category_theory.cocone_of_representable | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [] | This is a cocone with point `P` for the functor `functor_to_representables P`. It is shown in
`colimit_of_representable P` that this cocone is a colimit: that is, we have exhibited an arbitrary
presheaf `P` as a colimit of representables.
The construction of [MM92], Chapter I, Section 5, Corollary 3. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_of_representable_X (P : Cᵒᵖ ⥤ Type u₁) :
(cocone_of_representable P).X = P | rfl | lemma | category_theory.cocone_of_representable_X | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocone_of_representable_ι_app (P : Cᵒᵖ ⥤ Type u₁) (j : (P.elements)ᵒᵖ):
(cocone_of_representable P).ι.app j = (yoneda_sections_small _ _).inv j.unop.2 | colimit.ι_desc _ _ | lemma | category_theory.cocone_of_representable_ι_app | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocone_of_representable_naturality {P₁ P₂ : Cᵒᵖ ⥤ Type u₁} (α : P₁ ⟶ P₂)
(j : (P₁.elements)ᵒᵖ) :
(cocone_of_representable P₁).ι.app j ≫ α =
(cocone_of_representable P₂).ι.app ((category_of_elements.map α).op.obj j) | begin
ext T f,
simpa [cocone_of_representable_ι_app] using functor_to_types.naturality _ _ α f.op _,
end | lemma | category_theory.cocone_of_representable_naturality | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [] | The legs of the cocone `cocone_of_representable` are natural in the choice of presheaf. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_of_representable (P : Cᵒᵖ ⥤ Type u₁) : is_colimit (cocone_of_representable P) | begin
apply is_colimit.of_point_iso (colimit.is_colimit (functor_to_representables P)),
change is_iso (colimit.desc _ (cocone.extend _ _)),
rw [colimit.desc_extend, colimit.desc_cocone],
apply_instance,
end | def | category_theory.colimit_of_representable | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [] | The cocone with point `P` given by `the_cocone` is a colimit: that is, we have exhibited an
arbitrary presheaf `P` as a colimit of representables.
The result of [MM92], Chapter I, Section 5, Corollary 3. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_iso_of_nat_iso_on_representables (L₁ L₂ : (Cᵒᵖ ⥤ Type u₁) ⥤ ℰ)
[preserves_colimits L₁] [preserves_colimits L₂]
(h : yoneda ⋙ L₁ ≅ yoneda ⋙ L₂) : L₁ ≅ L₂ | begin
apply nat_iso.of_components _ _,
{ intro P,
refine (is_colimit_of_preserves L₁ (colimit_of_representable P)).cocone_points_iso_of_nat_iso
(is_colimit_of_preserves L₂ (colimit_of_representable P)) _,
apply functor.associator _ _ _ ≪≫ _,
exact iso_whisker_left (category_of_elements.π P).l... | def | category_theory.nat_iso_of_nat_iso_on_representables | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [
"hom_ext"
] | Given two functors L₁ and L₂ which preserve colimits, if they agree when restricted to the
representable presheaves then they agree everywhere. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_extension_along_yoneda (L : (Cᵒᵖ ⥤ Type u₁) ⥤ ℰ) (hL : yoneda ⋙ L ≅ A)
[preserves_colimits L] :
L ≅ extend_along_yoneda A | nat_iso_of_nat_iso_on_representables _ _ (hL ≪≫ (is_extension_along_yoneda _).symm) | def | category_theory.unique_extension_along_yoneda | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [] | Show that `extend_along_yoneda` is the unique colimit-preserving functor which extends `A` to
the presheaf category.
The second part of [MM92], Chapter I, Section 5, Corollary 4.
See Property 3 of https://ncatlab.org/nlab/show/Yoneda+extension#properties. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_left_adjoint_of_preserves_colimits_aux (L : (Cᵒᵖ ⥤ Type u₁) ⥤ ℰ) [preserves_colimits L] :
is_left_adjoint L | { right := restricted_yoneda (yoneda ⋙ L),
adj := (yoneda_adjunction _).of_nat_iso_left
((unique_extension_along_yoneda _ L (iso.refl _)).symm) } | def | category_theory.is_left_adjoint_of_preserves_colimits_aux | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [
"adj"
] | If `L` preserves colimits and `ℰ` has them, then it is a left adjoint. This is a special case of
`is_left_adjoint_of_preserves_colimits` used to prove that. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_left_adjoint_of_preserves_colimits (L : (C ⥤ Type u₁) ⥤ ℰ) [preserves_colimits L] :
is_left_adjoint L | let e : (_ ⥤ Type u₁) ≌ (_ ⥤ Type u₁) := (op_op_equivalence C).congr_left,
t := is_left_adjoint_of_preserves_colimits_aux (e.functor ⋙ L : _)
in by exactI adjunction.left_adjoint_of_nat_iso (e.inv_fun_id_assoc _) | def | category_theory.is_left_adjoint_of_preserves_colimits | category_theory.limits | src/category_theory/limits/presheaf.lean | [
"category_theory.adjunction.limits",
"category_theory.adjunction.opposites",
"category_theory.elements",
"category_theory.limits.functor_category",
"category_theory.limits.kan_extension",
"category_theory.limits.shapes.terminal",
"category_theory.limits.types"
] | [] | If `L` preserves colimits and `ℰ` has them, then it is a left adjoint. Note this is a (partial)
converse to `left_adjoint_preserves_colimits`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone (F : J ⥤ Type (max v u)) : cone F | { X := F.sections,
π := { app := λ j u, u.val j } } | def | category_theory.limits.types.limit_cone | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | (internal implementation) the limit cone of a functor,
implemented as flat sections of a pi type | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone_is_limit (F : J ⥤ Type (max v u)) : is_limit (limit_cone F) | { lift := λ s v, ⟨λ j, s.π.app j v, λ j j' f, congr_fun (cone.w s f) _⟩,
uniq' := by { intros, ext x j, exact congr_fun (w j) x } } | def | category_theory.limits.types.limit_cone_is_limit | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [
"lift"
] | (internal implementation) the fact that the proposed limit cone is the limit | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits_of_size : has_limits_of_size.{v} (Type (max v u)) | { has_limits_of_shape := λ J 𝒥, by exactI
{ has_limit := λ F, has_limit.mk
{ cone := limit_cone F, is_limit := limit_cone_is_limit F } } } | instance | category_theory.limits.types.has_limits_of_size | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | The category of types has all limits.
See <https://stacks.math.columbia.edu/tag/002U>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_equiv_sections {F : J ⥤ Type (max v u)} {c : cone F} (t : is_limit c) :
c.X ≃ F.sections | (is_limit.cone_point_unique_up_to_iso t (limit_cone_is_limit F)).to_equiv | def | category_theory.limits.types.is_limit_equiv_sections | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | The equivalence between a limiting cone of `F` in `Type u` and the "concrete" definition as the
sections of `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_equiv_sections_apply
{F : J ⥤ Type (max v u)} {c : cone F} (t : is_limit c) (j : J) (x : c.X) :
(((is_limit_equiv_sections t) x) : Π j, F.obj j) j = c.π.app j x | rfl | lemma | category_theory.limits.types.is_limit_equiv_sections_apply | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_limit_equiv_sections_symm_apply
{F : J ⥤ Type (max v u)} {c : cone F} (t : is_limit c) (x : F.sections) (j : J) :
c.π.app j ((is_limit_equiv_sections t).symm x) = (x : Π j, F.obj j) j | begin
equiv_rw (is_limit_equiv_sections t).symm at x,
simp,
end | lemma | category_theory.limits.types.is_limit_equiv_sections_symm_apply | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_equiv_sections (F : J ⥤ Type (max v u)) : (limit F : Type (max v u)) ≃ F.sections | is_limit_equiv_sections (limit.is_limit _) | def | category_theory.limits.types.limit_equiv_sections | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | The equivalence between the abstract limit of `F` in `Type u`
and the "concrete" definition as the sections of `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_equiv_sections_apply (F : J ⥤ Type (max v u)) (x : limit F) (j : J) :
(((limit_equiv_sections F) x) : Π j, F.obj j) j = limit.π F j x | rfl | lemma | category_theory.limits.types.limit_equiv_sections_apply | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_equiv_sections_symm_apply (F : J ⥤ Type (max v u)) (x : F.sections) (j : J) :
limit.π F j ((limit_equiv_sections F).symm x) = (x : Π j, F.obj j) j | is_limit_equiv_sections_symm_apply _ _ _ | lemma | category_theory.limits.types.limit_equiv_sections_symm_apply | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_equiv_sections_symm_apply' (F : J ⥤ Type v) (x : F.sections) (j : J) :
limit.π F j ((limit_equiv_sections.{v v} F).symm x) = (x : Π j, F.obj j) j | is_limit_equiv_sections_symm_apply _ _ _ | lemma | category_theory.limits.types.limit_equiv_sections_symm_apply' | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit.mk (F : J ⥤ Type (max v u)) (x : Π j, F.obj j)
(h : ∀ (j j') (f : j ⟶ j'), F.map f (x j) = x j') : (limit F : Type (max v u)) | (limit_equiv_sections F).symm ⟨x, h⟩ | def | category_theory.limits.types.limit.mk | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | Construct a term of `limit F : Type u` from a family of terms `x : Π j, F.obj j`
which are "coherent": `∀ (j j') (f : j ⟶ j'), F.map f (x j) = x j'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit.π_mk (F : J ⥤ Type (max v u)) (x : Π j, F.obj j)
(h : ∀ (j j') (f : j ⟶ j'), F.map f (x j) = x j') (j) : limit.π F j (limit.mk F x h) = x j | by { dsimp [limit.mk], simp, } | lemma | category_theory.limits.types.limit.π_mk | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit.π_mk' (F : J ⥤ Type v) (x : Π j, F.obj j)
(h : ∀ (j j') (f : j ⟶ j'), F.map f (x j) = x j') (j) :
limit.π F j (limit.mk.{v v} F x h) = x j | by { dsimp [limit.mk], simp, } | lemma | category_theory.limits.types.limit.π_mk' | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_ext (F : J ⥤ Type (max v u)) (x y : limit F) (w : ∀ j, limit.π F j x = limit.π F j y) :
x = y | begin
apply (limit_equiv_sections F).injective,
ext j,
simp [w j],
end | lemma | category_theory.limits.types.limit_ext | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_ext' (F : J ⥤ Type v) (x y : limit F) (w : ∀ j, limit.π F j x = limit.π F j y) :
x = y | begin
apply (limit_equiv_sections.{v v} F).injective,
ext j,
simp [w j],
end | lemma | category_theory.limits.types.limit_ext' | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_ext_iff (F : J ⥤ Type (max v u)) (x y : limit F) :
x = y ↔ (∀ j, limit.π F j x = limit.π F j y) | ⟨λ t _, t ▸ rfl, limit_ext _ _ _⟩ | lemma | category_theory.limits.types.limit_ext_iff | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_ext_iff' (F : J ⥤ Type v) (x y : limit F) :
x = y ↔ (∀ j, limit.π F j x = limit.π F j y) | ⟨λ t _, t ▸ rfl, limit_ext _ _ _⟩ | lemma | category_theory.limits.types.limit_ext_iff' | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit.w_apply {F : J ⥤ Type (max v u)} {j j' : J} {x : limit F} (f : j ⟶ j') :
F.map f (limit.π F j x) = limit.π F j' x | congr_fun (limit.w F f) x | lemma | category_theory.limits.types.limit.w_apply | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit.lift_π_apply (F : J ⥤ Type (max v u)) (s : cone F) (j : J) (x : s.X) :
limit.π F j (limit.lift F s x) = s.π.app j x | congr_fun (limit.lift_π s j) x | lemma | category_theory.limits.types.limit.lift_π_apply | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit.map_π_apply {F G : J ⥤ Type (max v u)} (α : F ⟶ G) (j : J) (x) :
limit.π G j (lim_map α x) = α.app j (limit.π F j x) | congr_fun (lim_map_π α j) x | lemma | category_theory.limits.types.limit.map_π_apply | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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