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map {F G : J ⥤ C} (s : cone F) {t : cone G} (P : is_limit t)
(α : F ⟶ G) : s.X ⟶ t.X | P.lift ((cones.postcompose α).obj s) | def | category_theory.limits.is_limit.map | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | Given a natural transformation `α : F ⟶ G`, we give a morphism from the cone point
of any cone over `F` to the cone point of a limit cone over `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_π {F G : J ⥤ C} (c : cone F) {d : cone G} (hd : is_limit d)
(α : F ⟶ G) (j : J) : hd.map c α ≫ d.π.app j = c.π.app j ≫ α.app j | fac _ _ _ | lemma | category_theory.limits.is_limit.map_π | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_self {c : cone F} (t : is_limit c) : t.lift c = 𝟙 c.X | (t.uniq _ _ (λ j, id_comp _)).symm | lemma | category_theory.limits.is_limit.lift_self | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_cone_morphism {t : cone F} (h : is_limit t) (s : cone F) : s ⟶ t | { hom := h.lift s } | def | category_theory.limits.is_limit.lift_cone_morphism | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | The universal morphism from any other cone to a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniq_cone_morphism {s t : cone F} (h : is_limit t) {f f' : s ⟶ t} :
f = f' | have ∀ {g : s ⟶ t}, g = h.lift_cone_morphism s, by intro g; ext; exact h.uniq _ _ g.w,
this.trans this.symm | lemma | category_theory.limits.is_limit.uniq_cone_morphism | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_unique {t : cone F} (h : is_limit t) (s : cone F) :
∃! (l : s.X ⟶ t.X), ∀ j, l ≫ t.π.app j = s.π.app j | ⟨h.lift s, h.fac s, h.uniq s⟩ | lemma | category_theory.limits.is_limit.exists_unique | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | Restating the definition of a limit cone in terms of the ∃! operator. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_exists_unique {t : cone F}
(ht : ∀ s : cone F, ∃! l : s.X ⟶ t.X, ∀ j, l ≫ t.π.app j = s.π.app j) : is_limit t | by { choose s hs hs' using ht, exact ⟨s, hs, hs'⟩ } | def | category_theory.limits.is_limit.of_exists_unique | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | Noncomputably make a colimit cocone from the existence of unique factorizations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_cone_morphism {t : cone F}
(lift : Π (s : cone F), s ⟶ t)
(uniq' : ∀ (s : cone F) (m : s ⟶ t), m = lift s) : is_limit t | { lift := λ s, (lift s).hom,
uniq' := λ s m w,
have cone_morphism.mk m w = lift s, by apply uniq',
congr_arg cone_morphism.hom this } | def | category_theory.limits.is_limit.mk_cone_morphism | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [
"lift"
] | Alternative constructor for `is_limit`,
providing a morphism of cones rather than a morphism between the cone points
and separately the factorisation condition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_up_to_iso {s t : cone F} (P : is_limit s) (Q : is_limit t) : s ≅ t | { hom := Q.lift_cone_morphism s,
inv := P.lift_cone_morphism t,
hom_inv_id' := P.uniq_cone_morphism,
inv_hom_id' := Q.uniq_cone_morphism } | def | category_theory.limits.is_limit.unique_up_to_iso | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | Limit cones on `F` are unique up to isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_is_iso {s t : cone F} (P : is_limit s) (Q : is_limit t) (f : s ⟶ t) : is_iso f | ⟨⟨P.lift_cone_morphism t, ⟨P.uniq_cone_morphism, Q.uniq_cone_morphism⟩⟩⟩ | lemma | category_theory.limits.is_limit.hom_is_iso | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | Any cone morphism between limit cones is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_point_unique_up_to_iso {s t : cone F} (P : is_limit s) (Q : is_limit t) : s.X ≅ t.X | (cones.forget F).map_iso (unique_up_to_iso P Q) | def | category_theory.limits.is_limit.cone_point_unique_up_to_iso | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | Limits of `F` are unique up to isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_point_unique_up_to_iso_hom_comp {s t : cone F} (P : is_limit s)
(Q : is_limit t) (j : J) : (cone_point_unique_up_to_iso P Q).hom ≫ t.π.app j = s.π.app j | (unique_up_to_iso P Q).hom.w _ | lemma | category_theory.limits.is_limit.cone_point_unique_up_to_iso_hom_comp | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cone_point_unique_up_to_iso_inv_comp {s t : cone F} (P : is_limit s)
(Q : is_limit t) (j : J) : (cone_point_unique_up_to_iso P Q).inv ≫ s.π.app j = t.π.app j | (unique_up_to_iso P Q).inv.w _ | lemma | category_theory.limits.is_limit.cone_point_unique_up_to_iso_inv_comp | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_comp_cone_point_unique_up_to_iso_hom {r s t : cone F}
(P : is_limit s) (Q : is_limit t) :
P.lift r ≫ (cone_point_unique_up_to_iso P Q).hom = Q.lift r | Q.uniq _ _ (by simp) | lemma | category_theory.limits.is_limit.lift_comp_cone_point_unique_up_to_iso_hom | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_comp_cone_point_unique_up_to_iso_inv {r s t : cone F}
(P : is_limit s) (Q : is_limit t) :
Q.lift r ≫ (cone_point_unique_up_to_iso P Q).inv = P.lift r | P.uniq _ _ (by simp) | lemma | category_theory.limits.is_limit.lift_comp_cone_point_unique_up_to_iso_inv | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_iso_limit {r t : cone F} (P : is_limit r) (i : r ≅ t) : is_limit t | is_limit.mk_cone_morphism
(λ s, P.lift_cone_morphism s ≫ i.hom)
(λ s m, by rw ←i.comp_inv_eq; apply P.uniq_cone_morphism) | def | category_theory.limits.is_limit.of_iso_limit | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | Transport evidence that a cone is a limit cone across an isomorphism of cones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_iso_limit_lift {r t : cone F} (P : is_limit r) (i : r ≅ t) (s) :
(P.of_iso_limit i).lift s = P.lift s ≫ i.hom.hom | rfl | lemma | category_theory.limits.is_limit.of_iso_limit_lift | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_iso_limit {r t : cone F} (i : r ≅ t) : is_limit r ≃ is_limit t | { to_fun := λ h, h.of_iso_limit i,
inv_fun := λ h, h.of_iso_limit i.symm,
left_inv := by tidy,
right_inv := by tidy } | def | category_theory.limits.is_limit.equiv_iso_limit | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [
"inv_fun"
] | Isomorphism of cones preserves whether or not they are limiting cones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_iso_limit_apply {r t : cone F} (i : r ≅ t) (P : is_limit r) :
equiv_iso_limit i P = P.of_iso_limit i | rfl | lemma | category_theory.limits.is_limit.equiv_iso_limit_apply | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_iso_limit_symm_apply {r t : cone F} (i : r ≅ t) (P : is_limit t) :
(equiv_iso_limit i).symm P = P.of_iso_limit i.symm | rfl | lemma | category_theory.limits.is_limit.equiv_iso_limit_symm_apply | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_point_iso {r t : cone F} (P : is_limit r) [i : is_iso (P.lift t)] : is_limit t | of_iso_limit P
begin
haveI : is_iso (P.lift_cone_morphism t).hom := i,
haveI : is_iso (P.lift_cone_morphism t) := cones.cone_iso_of_hom_iso _,
symmetry,
apply as_iso (P.lift_cone_morphism t),
end | def | category_theory.limits.is_limit.of_point_iso | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | If the canonical morphism from a cone point to a limiting cone point is an iso, then the
first cone was limiting also. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_lift (h : is_limit t) {W : C} (m : W ⟶ t.X) :
m = h.lift { X := W, π := { app := λ b, m ≫ t.π.app b } } | h.uniq { X := W, π := { app := λ b, m ≫ t.π.app b } } m (λ b, rfl) | lemma | category_theory.limits.is_limit.hom_lift | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_ext (h : is_limit t) {W : C} {f f' : W ⟶ t.X}
(w : ∀ j, f ≫ t.π.app j = f' ≫ t.π.app j) : f = f' | by rw [h.hom_lift f, h.hom_lift f']; congr; exact funext w | lemma | category_theory.limits.is_limit.hom_ext | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [
"hom_ext"
] | Two morphisms into a limit are equal if their compositions with
each cone morphism are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_right_adjoint {D : Type u₄} [category.{v₄} D] {G : K ⥤ D}
(h : cone G ⥤ cone F) [is_right_adjoint h] {c : cone G} (t : is_limit c) :
is_limit (h.obj c) | mk_cone_morphism
(λ s, (adjunction.of_right_adjoint h).hom_equiv s c (t.lift_cone_morphism _))
(λ s m, (adjunction.eq_hom_equiv_apply _ _ _).2 t.uniq_cone_morphism) | def | category_theory.limits.is_limit.of_right_adjoint | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | Given a right adjoint functor between categories of cones,
the image of a limit cone is a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_cone_equiv {D : Type u₄} [category.{v₄} D] {G : K ⥤ D}
(h : cone G ≌ cone F) {c : cone G} :
is_limit (h.functor.obj c) ≃ is_limit c | { to_fun := λ P, of_iso_limit (of_right_adjoint h.inverse P) (h.unit_iso.symm.app c),
inv_fun := of_right_adjoint h.functor,
left_inv := by tidy,
right_inv := by tidy, } | def | category_theory.limits.is_limit.of_cone_equiv | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [
"inv_fun"
] | Given two functors which have equivalent categories of cones, we can transport a limiting cone
across the equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_cone_equiv_apply_desc {D : Type u₄} [category.{v₄} D] {G : K ⥤ D}
(h : cone G ≌ cone F) {c : cone G} (P : is_limit (h.functor.obj c)) (s) :
(of_cone_equiv h P).lift s =
((h.unit_iso.hom.app s).hom ≫
(h.functor.inv.map (P.lift_cone_morphism (h.functor.obj s))).hom) ≫
(h.unit_iso.inv.app c).hom | rfl | lemma | category_theory.limits.is_limit.of_cone_equiv_apply_desc | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_cone_equiv_symm_apply_desc {D : Type u₄} [category.{v₄} D] {G : K ⥤ D}
(h : cone G ≌ cone F) {c : cone G} (P : is_limit c) (s) :
((of_cone_equiv h).symm P).lift s =
(h.counit_iso.inv.app s).hom ≫ (h.functor.map (P.lift_cone_morphism (h.inverse.obj s))).hom | rfl | lemma | category_theory.limits.is_limit.of_cone_equiv_symm_apply_desc | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
postcompose_hom_equiv {F G : J ⥤ C} (α : F ≅ G) (c : cone F) :
is_limit ((cones.postcompose α.hom).obj c) ≃ is_limit c | of_cone_equiv (cones.postcompose_equivalence α) | def | category_theory.limits.is_limit.postcompose_hom_equiv | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | A cone postcomposed with a natural isomorphism is a limit cone if and only if the original cone is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
postcompose_inv_equiv {F G : J ⥤ C} (α : F ≅ G) (c : cone G) :
is_limit ((cones.postcompose α.inv).obj c) ≃ is_limit c | postcompose_hom_equiv α.symm c | def | category_theory.limits.is_limit.postcompose_inv_equiv | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | A cone postcomposed with the inverse of a natural isomorphism is a limit cone if and only if
the original cone is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_of_nat_iso_of_iso {F G : J ⥤ C} (α : F ≅ G) (c : cone F) (d : cone G)
(w : (cones.postcompose α.hom).obj c ≅ d) :
is_limit c ≃ is_limit d | (postcompose_hom_equiv α _).symm.trans (equiv_iso_limit w) | def | category_theory.limits.is_limit.equiv_of_nat_iso_of_iso | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | Constructing an equivalence `is_limit c ≃ is_limit d` from a natural isomorphism
between the underlying functors, and then an isomorphism between `c` transported along this and `d`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_points_iso_of_nat_iso {F G : J ⥤ C} {s : cone F} {t : cone G}
(P : is_limit s) (Q : is_limit t) (w : F ≅ G) : s.X ≅ t.X | { hom := Q.map s w.hom,
inv := P.map t w.inv,
hom_inv_id' := P.hom_ext (by tidy),
inv_hom_id' := Q.hom_ext (by tidy), } | def | category_theory.limits.is_limit.cone_points_iso_of_nat_iso | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | The cone points of two limit cones for naturally isomorphic functors
are themselves isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_points_iso_of_nat_iso_hom_comp {F G : J ⥤ C} {s : cone F} {t : cone G}
(P : is_limit s) (Q : is_limit t) (w : F ≅ G) (j : J) :
(cone_points_iso_of_nat_iso P Q w).hom ≫ t.π.app j = s.π.app j ≫ w.hom.app j | by simp | lemma | category_theory.limits.is_limit.cone_points_iso_of_nat_iso_hom_comp | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cone_points_iso_of_nat_iso_inv_comp {F G : J ⥤ C} {s : cone F} {t : cone G}
(P : is_limit s) (Q : is_limit t) (w : F ≅ G) (j : J) :
(cone_points_iso_of_nat_iso P Q w).inv ≫ s.π.app j = t.π.app j ≫ w.inv.app j | by simp | lemma | category_theory.limits.is_limit.cone_points_iso_of_nat_iso_inv_comp | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_comp_cone_points_iso_of_nat_iso_hom {F G : J ⥤ C} {r s : cone F} {t : cone G}
(P : is_limit s) (Q : is_limit t) (w : F ≅ G) :
P.lift r ≫ (cone_points_iso_of_nat_iso P Q w).hom = Q.map r w.hom | Q.hom_ext (by simp) | lemma | category_theory.limits.is_limit.lift_comp_cone_points_iso_of_nat_iso_hom | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_comp_cone_points_iso_of_nat_iso_inv {F G : J ⥤ C} {r s : cone G} {t : cone F}
(P : is_limit t) (Q : is_limit s) (w : F ≅ G) :
Q.lift r ≫ (cone_points_iso_of_nat_iso P Q w).inv = P.map r w.inv | P.hom_ext (by simp) | lemma | category_theory.limits.is_limit.lift_comp_cone_points_iso_of_nat_iso_inv | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_equivalence {s : cone F} (P : is_limit s) (e : K ≌ J) :
is_limit (s.whisker e.functor) | of_right_adjoint (cones.whiskering_equivalence e).functor P | def | category_theory.limits.is_limit.whisker_equivalence | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | If `s : cone F` is a limit cone, so is `s` whiskered by an equivalence `e`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_whisker_equivalence {s : cone F} (e : K ≌ J) (P : is_limit (s.whisker e.functor)) :
is_limit s | equiv_iso_limit ((cones.whiskering_equivalence e).unit_iso.app s).symm
(of_right_adjoint (cones.whiskering_equivalence e).inverse P : _) | def | category_theory.limits.is_limit.of_whisker_equivalence | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | If `s : cone F` whiskered by an equivalence `e` is a limit cone, so is `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whisker_equivalence_equiv {s : cone F} (e : K ≌ J) :
is_limit s ≃ is_limit (s.whisker e.functor) | ⟨λ h, h.whisker_equivalence e, of_whisker_equivalence e, by tidy, by tidy⟩ | def | category_theory.limits.is_limit.whisker_equivalence_equiv | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | Given an equivalence of diagrams `e`, `s` is a limit cone iff `s.whisker e.functor` is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_points_iso_of_equivalence {F : J ⥤ C} {s : cone F} {G : K ⥤ C} {t : cone G}
(P : is_limit s) (Q : is_limit t) (e : J ≌ K) (w : e.functor ⋙ G ≅ F) : s.X ≅ t.X | let w' : e.inverse ⋙ F ≅ G := (iso_whisker_left e.inverse w).symm ≪≫ inv_fun_id_assoc e G in
{ hom := Q.lift ((cones.equivalence_of_reindexing e.symm w').functor.obj s),
inv := P.lift ((cones.equivalence_of_reindexing e w).functor.obj t),
hom_inv_id' :=
begin
apply hom_ext P, intros j,
dsimp,
simp onl... | def | category_theory.limits.is_limit.cone_points_iso_of_equivalence | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [
"hom_ext"
] | We can prove two cone points `(s : cone F).X` and `(t.cone G).X` are isomorphic if
* both cones are limit cones
* their indexing categories are equivalent via some `e : J ≌ K`,
* the triangle of functors commutes up to a natural isomorphism: `e.functor ⋙ G ≅ F`.
This is the most general form of uniqueness of cone poin... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_iso (h : is_limit t) (W : C) : ulift.{u₁} (W ⟶ t.X : Type v₃) ≅ (const J).obj W ⟶ F | { hom := λ f, (t.extend f.down).π,
inv := λ π, ⟨h.lift { X := W, π := π }⟩,
hom_inv_id' := by ext f; apply h.hom_ext; intro j; simp; dsimp; refl } | def | category_theory.limits.is_limit.hom_iso | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | The universal property of a limit cone: a map `W ⟶ X` is the same as
a cone on `F` with vertex `W`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_iso_hom (h : is_limit t) {W : C} (f : ulift.{u₁} (W ⟶ t.X)) :
(is_limit.hom_iso h W).hom f = (t.extend f.down).π | rfl | lemma | category_theory.limits.is_limit.hom_iso_hom | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_iso (h : is_limit t) : yoneda.obj t.X ⋙ ulift_functor.{u₁} ≅ F.cones | nat_iso.of_components (λ W, is_limit.hom_iso h (unop W)) (by tidy). | def | category_theory.limits.is_limit.nat_iso | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | The limit of `F` represents the functor taking `W` to
the set of cones on `F` with vertex `W`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_iso' (h : is_limit t) (W : C) :
ulift.{u₁} ((W ⟶ t.X) : Type v₃) ≅
{ p : Π j, W ⟶ F.obj j // ∀ {j j'} (f : j ⟶ j'), p j ≫ F.map f = p j' } | h.hom_iso W ≪≫
{ hom := λ π,
⟨λ j, π.app j, λ j j' f,
by convert ←(π.naturality f).symm; apply id_comp⟩,
inv := λ p,
{ app := λ j, p.1 j,
naturality' := λ j j' f, begin dsimp, rw [id_comp], exact (p.2 f).symm end } } | def | category_theory.limits.is_limit.hom_iso' | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | Another, more explicit, formulation of the universal property of a limit cone.
See also `hom_iso`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_faithful {t : cone F} {D : Type u₄} [category.{v₄} D] (G : C ⥤ D) [faithful G]
(ht : is_limit (G.map_cone t)) (lift : Π (s : cone F), s.X ⟶ t.X)
(h : ∀ s, G.map (lift s) = ht.lift (G.map_cone s)) : is_limit t | { lift := lift,
fac' := λ s j, by apply G.map_injective; rw [G.map_comp, h]; apply ht.fac,
uniq' := λ s m w, begin
apply G.map_injective, rw h,
refine ht.uniq (G.map_cone s) _ (λ j, _),
convert ←congr_arg (λ f, G.map f) (w j),
apply G.map_comp
end } | def | category_theory.limits.is_limit.of_faithful | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [
"lift"
] | If G : C → D is a faithful functor which sends t to a limit cone,
then it suffices to check that the induced maps for the image of t
can be lifted to maps of C. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_cone_equiv {D : Type u₄} [category.{v₄} D]
{K : J ⥤ C} {F G : C ⥤ D} (h : F ≅ G) {c : cone K}
(t : is_limit (F.map_cone c)) : is_limit (G.map_cone c) | begin
apply postcompose_inv_equiv (iso_whisker_left K h : _) (G.map_cone c) _,
apply t.of_iso_limit (postcompose_whisker_left_map_cone h.symm c).symm,
end | def | category_theory.limits.is_limit.map_cone_equiv | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | If `F` and `G` are naturally isomorphic, then `F.map_cone c` being a limit implies
`G.map_cone c` is also a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_unique_cone_morphism {t : cone F} :
is_limit t ≅ Π s, unique (s ⟶ t) | { hom := λ h s,
{ default := h.lift_cone_morphism s,
uniq := λ _, h.uniq_cone_morphism },
inv := λ h,
{ lift := λ s, (h s).default.hom,
uniq' := λ s f w, congr_arg cone_morphism.hom ((h s).uniq ⟨f, w⟩) } } | def | category_theory.limits.is_limit.iso_unique_cone_morphism | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [
"lift",
"unique"
] | A cone is a limit cone exactly if
there is a unique cone morphism from any other cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_of_hom {Y : C} (f : Y ⟶ X) : cone F | { X := Y, π := h.hom.app (op Y) ⟨f⟩ } | def | category_theory.limits.is_limit.of_nat_iso.cone_of_hom | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | If `F.cones` is represented by `X`, each morphism `f : Y ⟶ X` gives a cone with cone point
`Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_of_cone (s : cone F) : s.X ⟶ X | (h.inv.app (op s.X) s.π).down | def | category_theory.limits.is_limit.of_nat_iso.hom_of_cone | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | If `F.cones` is represented by `X`, each cone `s` gives a morphism `s.X ⟶ X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_of_hom_of_cone (s : cone F) : cone_of_hom h (hom_of_cone h s) = s | begin
dsimp [cone_of_hom, hom_of_cone], cases s, congr, dsimp,
convert congr_fun (congr_fun (congr_arg nat_trans.app h.inv_hom_id) (op s_X)) s_π,
exact ulift.up_down _
end | lemma | category_theory.limits.is_limit.of_nat_iso.cone_of_hom_of_cone | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_of_cone_of_hom {Y : C} (f : Y ⟶ X) : hom_of_cone h (cone_of_hom h f) = f | congr_arg ulift.down (congr_fun (congr_fun (congr_arg nat_trans.app h.hom_inv_id) (op Y)) ⟨f⟩ : _) | lemma | category_theory.limits.is_limit.of_nat_iso.hom_of_cone_of_hom | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_cone : cone F | cone_of_hom h (𝟙 X) | def | category_theory.limits.is_limit.of_nat_iso.limit_cone | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | If `F.cones` is represented by `X`, the cone corresponding to the identity morphism on `X`
will be a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_of_hom_fac {Y : C} (f : Y ⟶ X) :
cone_of_hom h f = (limit_cone h).extend f | begin
dsimp [cone_of_hom, limit_cone, cone.extend],
congr' with j,
have t := congr_fun (h.hom.naturality f.op) ⟨𝟙 X⟩,
dsimp at t,
simp only [comp_id] at t,
rw congr_fun (congr_arg nat_trans.app t) j,
refl,
end | lemma | category_theory.limits.is_limit.of_nat_iso.cone_of_hom_fac | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [
"extend"
] | If `F.cones` is represented by `X`, the cone corresponding to a morphism `f : Y ⟶ X` is
the limit cone extended by `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_fac (s : cone F) : (limit_cone h).extend (hom_of_cone h s) = s | begin
rw ←cone_of_hom_of_cone h s,
conv_lhs { simp only [hom_of_cone_of_hom] },
apply (cone_of_hom_fac _ _).symm,
end | lemma | category_theory.limits.is_limit.of_nat_iso.cone_fac | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [
"extend"
] | If `F.cones` is represented by `X`, any cone is the extension of the limit cone by the
corresponding morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_nat_iso {X : C} (h : yoneda.obj X ⋙ ulift_functor.{u₁} ≅ F.cones) :
is_limit (limit_cone h) | { lift := λ s, hom_of_cone h s,
fac' := λ s j,
begin
have h := cone_fac h s,
cases s,
injection h with h₁ h₂,
simp only [heq_iff_eq] at h₂,
conv_rhs { rw ← h₂ }, refl,
end,
uniq' := λ s m w,
begin
rw ←hom_of_cone_of_hom h m,
congr,
rw cone_of_hom_fac,
dsimp [cone.extend], c... | def | category_theory.limits.is_limit.of_nat_iso | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [
"heq_iff_eq",
"lift"
] | If `F.cones` is representable, then the cone corresponding to the identity morphism on
the representing object is a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit (t : cocone F) | (desc : Π (s : cocone F), t.X ⟶ s.X)
(fac' : ∀ (s : cocone F) (j : J), t.ι.app j ≫ desc s = s.ι.app j . obviously)
(uniq' : ∀ (s : cocone F) (m : t.X ⟶ s.X) (w : ∀ j : J, t.ι.app j ≫ m = s.ι.app j),
m = desc s . obviously) | structure | category_theory.limits.is_colimit | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | A cocone `t` on `F` is a colimit cocone if each cocone on `F` admits a unique
cocone morphism from `t`.
See <https://stacks.math.columbia.edu/tag/002F>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subsingleton {t : cocone F} : subsingleton (is_colimit t) | ⟨by intros P Q; cases P; cases Q; congr; ext; solve_by_elim⟩ | instance | category_theory.limits.is_colimit.subsingleton | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map {F G : J ⥤ C} {s : cocone F} (P : is_colimit s) (t : cocone G)
(α : F ⟶ G) : s.X ⟶ t.X | P.desc ((cocones.precompose α).obj t) | def | category_theory.limits.is_colimit.map | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | Given a natural transformation `α : F ⟶ G`, we give a morphism from the cocone point
of a colimit cocone over `F` to the cocone point of any cocone over `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_map {F G : J ⥤ C} {c : cocone F} (hc : is_colimit c) (d : cocone G) (α : F ⟶ G)
(j : J) : c.ι.app j ≫ is_colimit.map hc d α = α.app j ≫ d.ι.app j | fac _ _ _ | lemma | category_theory.limits.is_colimit.ι_map | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
desc_self {t : cocone F} (h : is_colimit t) : h.desc t = 𝟙 t.X | (h.uniq _ _ (λ j, comp_id _)).symm | lemma | category_theory.limits.is_colimit.desc_self | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
desc_cocone_morphism {t : cocone F} (h : is_colimit t) (s : cocone F) : t ⟶ s | { hom := h.desc s } | def | category_theory.limits.is_colimit.desc_cocone_morphism | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | The universal morphism from a colimit cocone to any other cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniq_cocone_morphism {s t : cocone F} (h : is_colimit t) {f f' : t ⟶ s} :
f = f' | have ∀ {g : t ⟶ s}, g = h.desc_cocone_morphism s, by intro g; ext; exact h.uniq _ _ g.w,
this.trans this.symm | lemma | category_theory.limits.is_colimit.uniq_cocone_morphism | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_unique {t : cocone F} (h : is_colimit t) (s : cocone F) :
∃! (d : t.X ⟶ s.X), ∀ j, t.ι.app j ≫ d = s.ι.app j | ⟨h.desc s, h.fac s, h.uniq s⟩ | lemma | category_theory.limits.is_colimit.exists_unique | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | Restating the definition of a colimit cocone in terms of the ∃! operator. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_exists_unique {t : cocone F}
(ht : ∀ s : cocone F, ∃! d : t.X ⟶ s.X, ∀ j, t.ι.app j ≫ d = s.ι.app j) : is_colimit t | by { choose s hs hs' using ht, exact ⟨s, hs, hs'⟩ } | def | category_theory.limits.is_colimit.of_exists_unique | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | Noncomputably make a colimit cocone from the existence of unique factorizations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_cocone_morphism {t : cocone F}
(desc : Π (s : cocone F), t ⟶ s)
(uniq' : ∀ (s : cocone F) (m : t ⟶ s), m = desc s) : is_colimit t | { desc := λ s, (desc s).hom,
uniq' := λ s m w,
have cocone_morphism.mk m w = desc s, by apply uniq',
congr_arg cocone_morphism.hom this } | def | category_theory.limits.is_colimit.mk_cocone_morphism | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | Alternative constructor for `is_colimit`,
providing a morphism of cocones rather than a morphism between the cocone points
and separately the factorisation condition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_up_to_iso {s t : cocone F} (P : is_colimit s) (Q : is_colimit t) : s ≅ t | { hom := P.desc_cocone_morphism t,
inv := Q.desc_cocone_morphism s,
hom_inv_id' := P.uniq_cocone_morphism,
inv_hom_id' := Q.uniq_cocone_morphism } | def | category_theory.limits.is_colimit.unique_up_to_iso | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | Colimit cocones on `F` are unique up to isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_is_iso {s t : cocone F} (P : is_colimit s) (Q : is_colimit t) (f : s ⟶ t) : is_iso f | ⟨⟨Q.desc_cocone_morphism s, ⟨P.uniq_cocone_morphism, Q.uniq_cocone_morphism⟩⟩⟩ | lemma | category_theory.limits.is_colimit.hom_is_iso | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | Any cocone morphism between colimit cocones is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_point_unique_up_to_iso {s t : cocone F} (P : is_colimit s) (Q : is_colimit t) :
s.X ≅ t.X | (cocones.forget F).map_iso (unique_up_to_iso P Q) | def | category_theory.limits.is_colimit.cocone_point_unique_up_to_iso | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | Colimits of `F` are unique up to isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_cocone_point_unique_up_to_iso_hom {s t : cocone F} (P : is_colimit s)
(Q : is_colimit t) (j : J) : s.ι.app j ≫ (cocone_point_unique_up_to_iso P Q).hom = t.ι.app j | (unique_up_to_iso P Q).hom.w _ | lemma | category_theory.limits.is_colimit.comp_cocone_point_unique_up_to_iso_hom | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_cocone_point_unique_up_to_iso_inv {s t : cocone F} (P : is_colimit s)
(Q : is_colimit t) (j : J) : t.ι.app j ≫ (cocone_point_unique_up_to_iso P Q).inv = s.ι.app j | (unique_up_to_iso P Q).inv.w _ | lemma | category_theory.limits.is_colimit.comp_cocone_point_unique_up_to_iso_inv | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocone_point_unique_up_to_iso_hom_desc {r s t : cocone F} (P : is_colimit s)
(Q : is_colimit t) : (cocone_point_unique_up_to_iso P Q).hom ≫ Q.desc r = P.desc r | P.uniq _ _ (by simp) | lemma | category_theory.limits.is_colimit.cocone_point_unique_up_to_iso_hom_desc | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocone_point_unique_up_to_iso_inv_desc {r s t : cocone F} (P : is_colimit s)
(Q : is_colimit t) : (cocone_point_unique_up_to_iso P Q).inv ≫ P.desc r = Q.desc r | Q.uniq _ _ (by simp) | lemma | category_theory.limits.is_colimit.cocone_point_unique_up_to_iso_inv_desc | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_iso_colimit {r t : cocone F} (P : is_colimit r) (i : r ≅ t) : is_colimit t | is_colimit.mk_cocone_morphism
(λ s, i.inv ≫ P.desc_cocone_morphism s)
(λ s m, by rw i.eq_inv_comp; apply P.uniq_cocone_morphism) | def | category_theory.limits.is_colimit.of_iso_colimit | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | Transport evidence that a cocone is a colimit cocone across an isomorphism of cocones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_iso_colimit_desc {r t : cocone F} (P : is_colimit r) (i : r ≅ t) (s) :
(P.of_iso_colimit i).desc s = i.inv.hom ≫ P.desc s | rfl | lemma | category_theory.limits.is_colimit.of_iso_colimit_desc | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_iso_colimit {r t : cocone F} (i : r ≅ t) : is_colimit r ≃ is_colimit t | { to_fun := λ h, h.of_iso_colimit i,
inv_fun := λ h, h.of_iso_colimit i.symm,
left_inv := by tidy,
right_inv := by tidy } | def | category_theory.limits.is_colimit.equiv_iso_colimit | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [
"inv_fun"
] | Isomorphism of cocones preserves whether or not they are colimiting cocones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_iso_colimit_apply {r t : cocone F} (i : r ≅ t) (P : is_colimit r) :
equiv_iso_colimit i P = P.of_iso_colimit i | rfl | lemma | category_theory.limits.is_colimit.equiv_iso_colimit_apply | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_iso_colimit_symm_apply {r t : cocone F} (i : r ≅ t) (P : is_colimit t) :
(equiv_iso_colimit i).symm P = P.of_iso_colimit i.symm | rfl | lemma | category_theory.limits.is_colimit.equiv_iso_colimit_symm_apply | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_point_iso {r t : cocone F} (P : is_colimit r) [i : is_iso (P.desc t)] : is_colimit t | of_iso_colimit P
begin
haveI : is_iso (P.desc_cocone_morphism t).hom := i,
haveI : is_iso (P.desc_cocone_morphism t) := cocones.cocone_iso_of_hom_iso _,
apply as_iso (P.desc_cocone_morphism t),
end | def | category_theory.limits.is_colimit.of_point_iso | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | If the canonical morphism to a cocone point from a colimiting cocone point is an iso, then the
first cocone was colimiting also. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_desc (h : is_colimit t) {W : C} (m : t.X ⟶ W) :
m = h.desc { X := W, ι := { app := λ b, t.ι.app b ≫ m,
naturality' := by intros; erw [←assoc, t.ι.naturality, comp_id, comp_id] } } | h.uniq { X := W, ι := { app := λ b, t.ι.app b ≫ m, naturality' := _ } } m (λ b, rfl) | lemma | category_theory.limits.is_colimit.hom_desc | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_ext (h : is_colimit t) {W : C} {f f' : t.X ⟶ W}
(w : ∀ j, t.ι.app j ≫ f = t.ι.app j ≫ f') : f = f' | by rw [h.hom_desc f, h.hom_desc f']; congr; exact funext w | lemma | category_theory.limits.is_colimit.hom_ext | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [
"hom_ext"
] | Two morphisms out of a colimit are equal if their compositions with
each cocone morphism are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_left_adjoint {D : Type u₄} [category.{v₄} D] {G : K ⥤ D}
(h : cocone G ⥤ cocone F) [is_left_adjoint h] {c : cocone G} (t : is_colimit c) :
is_colimit (h.obj c) | mk_cocone_morphism
(λ s, ((adjunction.of_left_adjoint h).hom_equiv c s).symm (t.desc_cocone_morphism _))
(λ s m, (adjunction.hom_equiv_apply_eq _ _ _).1 t.uniq_cocone_morphism) | def | category_theory.limits.is_colimit.of_left_adjoint | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | Given a left adjoint functor between categories of cocones,
the image of a colimit cocone is a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_cocone_equiv {D : Type u₄} [category.{v₄} D] {G : K ⥤ D}
(h : cocone G ≌ cocone F) {c : cocone G} :
is_colimit (h.functor.obj c) ≃ is_colimit c | { to_fun := λ P, of_iso_colimit (of_left_adjoint h.inverse P) (h.unit_iso.symm.app c),
inv_fun := of_left_adjoint h.functor,
left_inv := by tidy,
right_inv := by tidy, } | def | category_theory.limits.is_colimit.of_cocone_equiv | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [
"inv_fun"
] | Given two functors which have equivalent categories of cocones,
we can transport a colimiting cocone across the equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_cocone_equiv_apply_desc {D : Type u₄} [category.{v₄} D] {G : K ⥤ D}
(h : cocone G ≌ cocone F) {c : cocone G} (P : is_colimit (h.functor.obj c)) (s) :
(of_cocone_equiv h P).desc s =
(h.unit.app c).hom ≫
(h.inverse.map (P.desc_cocone_morphism (h.functor.obj s))).hom ≫
(h.unit_inv.app s).hom | rfl | lemma | category_theory.limits.is_colimit.of_cocone_equiv_apply_desc | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_cocone_equiv_symm_apply_desc {D : Type u₄} [category.{v₄} D] {G : K ⥤ D}
(h : cocone G ≌ cocone F) {c : cocone G} (P : is_colimit c) (s) :
((of_cocone_equiv h).symm P).desc s =
(h.functor.map (P.desc_cocone_morphism (h.inverse.obj s))).hom ≫ (h.counit.app s).hom | rfl | lemma | category_theory.limits.is_colimit.of_cocone_equiv_symm_apply_desc | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
precompose_hom_equiv {F G : J ⥤ C} (α : F ≅ G) (c : cocone G) :
is_colimit ((cocones.precompose α.hom).obj c) ≃ is_colimit c | of_cocone_equiv (cocones.precompose_equivalence α) | def | category_theory.limits.is_colimit.precompose_hom_equiv | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | A cocone precomposed with a natural isomorphism is a colimit cocone
if and only if the original cocone is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
precompose_inv_equiv {F G : J ⥤ C} (α : F ≅ G) (c : cocone F) :
is_colimit ((cocones.precompose α.inv).obj c) ≃ is_colimit c | precompose_hom_equiv α.symm c | def | category_theory.limits.is_colimit.precompose_inv_equiv | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | A cocone precomposed with the inverse of a natural isomorphism is a colimit cocone
if and only if the original cocone is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_of_nat_iso_of_iso {F G : J ⥤ C} (α : F ≅ G) (c : cocone F) (d : cocone G)
(w : (cocones.precompose α.inv).obj c ≅ d) :
is_colimit c ≃ is_colimit d | (precompose_inv_equiv α _).symm.trans (equiv_iso_colimit w) | def | category_theory.limits.is_colimit.equiv_of_nat_iso_of_iso | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | Constructing an equivalence `is_colimit c ≃ is_colimit d` from a natural isomorphism
between the underlying functors, and then an isomorphism between `c` transported along this and `d`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_points_iso_of_nat_iso {F G : J ⥤ C} {s : cocone F} {t : cocone G}
(P : is_colimit s) (Q : is_colimit t) (w : F ≅ G) : s.X ≅ t.X | { hom := P.map t w.hom,
inv := Q.map s w.inv,
hom_inv_id' := P.hom_ext (by tidy),
inv_hom_id' := Q.hom_ext (by tidy) } | def | category_theory.limits.is_colimit.cocone_points_iso_of_nat_iso | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | The cocone points of two colimit cocones for naturally isomorphic functors
are themselves isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_cocone_points_iso_of_nat_iso_hom {F G : J ⥤ C} {s : cocone F} {t : cocone G}
(P : is_colimit s) (Q : is_colimit t) (w : F ≅ G) (j : J) :
s.ι.app j ≫ (cocone_points_iso_of_nat_iso P Q w).hom = w.hom.app j ≫ t.ι.app j | by simp | lemma | category_theory.limits.is_colimit.comp_cocone_points_iso_of_nat_iso_hom | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_cocone_points_iso_of_nat_iso_inv {F G : J ⥤ C} {s : cocone F} {t : cocone G}
(P : is_colimit s) (Q : is_colimit t) (w : F ≅ G) (j : J) :
t.ι.app j ≫ (cocone_points_iso_of_nat_iso P Q w).inv = w.inv.app j ≫ s.ι.app j | by simp | lemma | category_theory.limits.is_colimit.comp_cocone_points_iso_of_nat_iso_inv | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocone_points_iso_of_nat_iso_hom_desc {F G : J ⥤ C} {s : cocone F} {r t : cocone G}
(P : is_colimit s) (Q : is_colimit t) (w : F ≅ G) :
(cocone_points_iso_of_nat_iso P Q w).hom ≫ Q.desc r = P.map _ w.hom | P.hom_ext (by simp) | lemma | category_theory.limits.is_colimit.cocone_points_iso_of_nat_iso_hom_desc | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocone_points_iso_of_nat_iso_inv_desc {F G : J ⥤ C} {s : cocone G} {r t : cocone F}
(P : is_colimit t) (Q : is_colimit s) (w : F ≅ G) :
(cocone_points_iso_of_nat_iso P Q w).inv ≫ P.desc r = Q.map _ w.inv | Q.hom_ext (by simp) | lemma | category_theory.limits.is_colimit.cocone_points_iso_of_nat_iso_inv_desc | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_equivalence {s : cocone F} (P : is_colimit s) (e : K ≌ J) :
is_colimit (s.whisker e.functor) | of_left_adjoint (cocones.whiskering_equivalence e).functor P | def | category_theory.limits.is_colimit.whisker_equivalence | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | If `s : cocone F` is a colimit cocone, so is `s` whiskered by an equivalence `e`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_whisker_equivalence {s : cocone F} (e : K ≌ J) (P : is_colimit (s.whisker e.functor)) :
is_colimit s | equiv_iso_colimit ((cocones.whiskering_equivalence e).unit_iso.app s).symm
(of_left_adjoint (cocones.whiskering_equivalence e).inverse P : _) | def | category_theory.limits.is_colimit.of_whisker_equivalence | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | If `s : cocone F` whiskered by an equivalence `e` is a colimit cocone, so is `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whisker_equivalence_equiv {s : cocone F} (e : K ≌ J) :
is_colimit s ≃ is_colimit (s.whisker e.functor) | ⟨λ h, h.whisker_equivalence e, of_whisker_equivalence e, by tidy, by tidy⟩ | def | category_theory.limits.is_colimit.whisker_equivalence_equiv | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | Given an equivalence of diagrams `e`, `s` is a colimit cocone iff `s.whisker e.functor` is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_points_iso_of_equivalence {F : J ⥤ C} {s : cocone F} {G : K ⥤ C} {t : cocone G}
(P : is_colimit s) (Q : is_colimit t) (e : J ≌ K) (w : e.functor ⋙ G ≅ F) : s.X ≅ t.X | let w' : e.inverse ⋙ F ≅ G := (iso_whisker_left e.inverse w).symm ≪≫ inv_fun_id_assoc e G in
{ hom := P.desc ((cocones.equivalence_of_reindexing e w).functor.obj t),
inv := Q.desc ((cocones.equivalence_of_reindexing e.symm w').functor.obj s),
hom_inv_id' :=
begin
apply hom_ext P, intros j,
dsimp,
simp... | def | category_theory.limits.is_colimit.cocone_points_iso_of_equivalence | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [
"hom_ext"
] | We can prove two cocone points `(s : cocone F).X` and `(t.cocone G).X` are isomorphic if
* both cocones are colimit cocones
* their indexing categories are equivalent via some `e : J ≌ K`,
* the triangle of functors commutes up to a natural isomorphism: `e.functor ⋙ G ≅ F`.
This is the most general form of uniqueness ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_iso (h : is_colimit t) (W : C) : ulift.{u₁} (t.X ⟶ W : Type v₃) ≅ (F ⟶ (const J).obj W) | { hom := λ f, (t.extend f.down).ι,
inv := λ ι, ⟨h.desc { X := W, ι := ι }⟩,
hom_inv_id' := by ext f; apply h.hom_ext; intro j; simp; dsimp; refl } | def | category_theory.limits.is_colimit.hom_iso | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | The universal property of a colimit cocone: a map `X ⟶ W` is the same as
a cocone on `F` with vertex `W`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_iso_hom (h : is_colimit t) {W : C} (f : ulift (t.X ⟶ W)) :
(is_colimit.hom_iso h W).hom f = (t.extend f.down).ι | rfl | lemma | category_theory.limits.is_colimit.hom_iso_hom | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_iso (h : is_colimit t) : coyoneda.obj (op t.X) ⋙ ulift_functor.{u₁} ≅ F.cocones | nat_iso.of_components (is_colimit.hom_iso h) (by intros; ext; dsimp; rw ←assoc; refl) | def | category_theory.limits.is_colimit.nat_iso | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | The colimit of `F` represents the functor taking `W` to
the set of cocones on `F` with vertex `W`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_iso' (h : is_colimit t) (W : C) :
ulift.{u₁} ((t.X ⟶ W) : Type v₃) ≅
{ p : Π j, F.obj j ⟶ W // ∀ {j j' : J} (f : j ⟶ j'), F.map f ≫ p j' = p j } | h.hom_iso W ≪≫
{ hom := λ ι,
⟨λ j, ι.app j, λ j j' f,
by convert ←(ι.naturality f); apply comp_id⟩,
inv := λ p,
{ app := λ j, p.1 j,
naturality' := λ j j' f, begin dsimp, rw [comp_id], exact (p.2 f) end } } | def | category_theory.limits.is_colimit.hom_iso' | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | Another, more explicit, formulation of the universal property of a colimit cocone.
See also `hom_iso`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_faithful {t : cocone F} {D : Type u₄} [category.{v₄} D] (G : C ⥤ D) [faithful G]
(ht : is_colimit (G.map_cocone t)) (desc : Π (s : cocone F), t.X ⟶ s.X)
(h : ∀ s, G.map (desc s) = ht.desc (G.map_cocone s)) : is_colimit t | { desc := desc,
fac' := λ s j, by apply G.map_injective; rw [G.map_comp, h]; apply ht.fac,
uniq' := λ s m w, begin
apply G.map_injective, rw h,
refine ht.uniq (G.map_cocone s) _ (λ j, _),
convert ←congr_arg (λ f, G.map f) (w j),
apply G.map_comp
end } | def | category_theory.limits.is_colimit.of_faithful | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | If G : C → D is a faithful functor which sends t to a colimit cocone,
then it suffices to check that the induced maps for the image of t
can be lifted to maps of C. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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