statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
is_limit_map_cone_empty_cone_equiv :
is_limit (G.map_cone (as_empty_cone X)) ≃ is_terminal (G.obj X) | is_limit_empty_cone_equiv D _ _ (eq_to_iso rfl) | def | category_theory.limits.is_limit_map_cone_empty_cone_equiv | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/terminal.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.preserves.basic"
] | [] | The map of an empty cone is a limit iff the mapped object is terminal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_terminal.is_terminal_obj [preserves_limit (functor.empty.{0} C) G]
(l : is_terminal X) : is_terminal (G.obj X) | is_limit_map_cone_empty_cone_equiv G X (preserves_limit.preserves l) | def | category_theory.limits.is_terminal.is_terminal_obj | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/terminal.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.preserves.basic"
] | [] | The property of preserving terminal objects expressed in terms of `is_terminal`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_terminal.is_terminal_of_obj [reflects_limit (functor.empty.{0} C) G]
(l : is_terminal (G.obj X)) : is_terminal X | reflects_limit.reflects ((is_limit_map_cone_empty_cone_equiv G X).symm l) | def | category_theory.limits.is_terminal.is_terminal_of_obj | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/terminal.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.preserves.basic"
] | [] | The property of reflecting terminal objects expressed in terms of `is_terminal`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_of_shape_pempty_of_preserves_terminal
[preserves_limit (functor.empty.{0} C) G] : preserves_limits_of_shape (discrete pempty) G | { preserves_limit := λ K,
preserves_limit_of_iso_diagram G (functor.empty_ext (functor.empty.{0} C) _) } | def | category_theory.limits.preserves_limits_of_shape_pempty_of_preserves_terminal | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/terminal.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.preserves.basic"
] | [
"pempty"
] | Preserving the terminal object implies preserving all limits of the empty diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_of_has_terminal_of_preserves_limit [preserves_limit (functor.empty.{0} C) G] :
is_terminal (G.obj (⊤_ C)) | terminal_is_terminal.is_terminal_obj G (⊤_ C) | def | category_theory.limits.is_limit_of_has_terminal_of_preserves_limit | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/terminal.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.preserves.basic"
] | [] | If `G` preserves the terminal object and `C` has a terminal object, then the image of the terminal
object is terminal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_terminal_of_has_terminal_of_preserves_limit [preserves_limit (functor.empty.{0} C) G] :
has_terminal D | ⟨λ F,
begin
haveI := has_limit.mk ⟨_, is_limit_of_has_terminal_of_preserves_limit G⟩,
apply has_limit_of_iso F.unique_from_empty.symm,
end⟩ | lemma | category_theory.limits.has_terminal_of_has_terminal_of_preserves_limit | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/terminal.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.preserves.basic"
] | [] | If `C` has a terminal object and `G` preserves terminal objects, then `D` has a terminal object
also.
Note this property is somewhat unique to (co)limits of the empty diagram: for general `J`, if `C`
has limits of shape `J` and `G` preserves them, then `D` does not necessarily have limits of shape
`J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_terminal.of_iso_comparison
[i : is_iso (terminal_comparison G)] : preserves_limit (functor.empty C) G | begin
apply preserves_limit_of_preserves_limit_cone terminal_is_terminal,
apply (is_limit_map_cone_empty_cone_equiv _ _).symm _,
apply is_limit.of_point_iso (limit.is_limit (functor.empty.{0} D)),
apply i,
end | def | category_theory.limits.preserves_terminal.of_iso_comparison | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/terminal.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.preserves.basic"
] | [] | If the terminal comparison map for `G` is an isomorphism, then `G` preserves terminal objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_terminal_of_is_iso
(f : G.obj (⊤_ C) ⟶ ⊤_ D) [i : is_iso f] : preserves_limit (functor.empty C) G | begin
rw subsingleton.elim f (terminal_comparison G) at i,
exactI preserves_terminal.of_iso_comparison G,
end | def | category_theory.limits.preserves_terminal_of_is_iso | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/terminal.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.preserves.basic"
] | [] | If there is any isomorphism `G.obj ⊤ ⟶ ⊤`, then `G` preserves terminal objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_terminal_of_iso
(f : G.obj (⊤_ C) ≅ ⊤_ D) : preserves_limit (functor.empty C) G | preserves_terminal_of_is_iso G f.hom | def | category_theory.limits.preserves_terminal_of_iso | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/terminal.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.preserves.basic"
] | [] | If there is any isomorphism `G.obj ⊤ ≅ ⊤`, then `G` preserves terminal objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_terminal.iso : G.obj (⊤_ C) ≅ ⊤_ D | (is_limit_of_has_terminal_of_preserves_limit G).cone_point_unique_up_to_iso (limit.is_limit _) | def | category_theory.limits.preserves_terminal.iso | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/terminal.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.preserves.basic"
] | [] | If `G` preserves terminal objects, then the terminal comparison map for `G` is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_terminal.iso_hom : (preserves_terminal.iso G).hom = terminal_comparison G | rfl | lemma | category_theory.limits.preserves_terminal.iso_hom | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/terminal.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_colimit_map_cocone_empty_cocone_equiv :
is_colimit (G.map_cocone (as_empty_cocone.{v₁} X)) ≃ is_initial (G.obj X) | is_colimit_empty_cocone_equiv D _ _ (eq_to_iso rfl) | def | category_theory.limits.is_colimit_map_cocone_empty_cocone_equiv | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/terminal.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.preserves.basic"
] | [] | The map of an empty cocone is a colimit iff the mapped object is initial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_initial.is_initial_obj [preserves_colimit (functor.empty.{0} C) G]
(l : is_initial X) : is_initial (G.obj X) | is_colimit_map_cocone_empty_cocone_equiv G X (preserves_colimit.preserves l) | def | category_theory.limits.is_initial.is_initial_obj | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/terminal.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.preserves.basic"
] | [] | The property of preserving initial objects expressed in terms of `is_initial`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_initial.is_initial_of_obj [reflects_colimit (functor.empty.{0} C) G]
(l : is_initial (G.obj X)) : is_initial X | reflects_colimit.reflects ((is_colimit_map_cocone_empty_cocone_equiv G X).symm l) | def | category_theory.limits.is_initial.is_initial_of_obj | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/terminal.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.preserves.basic"
] | [] | The property of reflecting initial objects expressed in terms of `is_initial`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits_of_shape_pempty_of_preserves_initial
[preserves_colimit (functor.empty.{0} C) G] : preserves_colimits_of_shape (discrete pempty) G | { preserves_colimit := λ K,
preserves_colimit_of_iso_diagram G (functor.empty_ext (functor.empty.{0} C) _) } | def | category_theory.limits.preserves_colimits_of_shape_pempty_of_preserves_initial | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/terminal.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.preserves.basic"
] | [
"pempty"
] | Preserving the initial object implies preserving all colimits of the empty diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_of_has_initial_of_preserves_colimit [preserves_colimit (functor.empty.{0} C) G] :
is_initial (G.obj (⊥_ C)) | initial_is_initial.is_initial_obj G (⊥_ C) | def | category_theory.limits.is_colimit_of_has_initial_of_preserves_colimit | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/terminal.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.preserves.basic"
] | [] | If `G` preserves the initial object and `C` has a initial object, then the image of the initial
object is initial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_initial_of_has_initial_of_preserves_colimit [preserves_colimit (functor.empty.{0} C) G] :
has_initial D | ⟨λ F,
begin
haveI := has_colimit.mk ⟨_, is_colimit_of_has_initial_of_preserves_colimit G⟩,
apply has_colimit_of_iso F.unique_from_empty,
end⟩ | lemma | category_theory.limits.has_initial_of_has_initial_of_preserves_colimit | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/terminal.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.preserves.basic"
] | [] | If `C` has a initial object and `G` preserves initial objects, then `D` has a initial object
also.
Note this property is somewhat unique to colimits of the empty diagram: for general `J`, if `C`
has colimits of shape `J` and `G` preserves them, then `D` does not necessarily have colimits of
shape `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_initial.of_iso_comparison
[i : is_iso (initial_comparison G)] : preserves_colimit (functor.empty C) G | begin
apply preserves_colimit_of_preserves_colimit_cocone initial_is_initial,
apply (is_colimit_map_cocone_empty_cocone_equiv _ _).symm _,
apply is_colimit.of_point_iso (colimit.is_colimit (functor.empty.{0} D)),
apply i,
end | def | category_theory.limits.preserves_initial.of_iso_comparison | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/terminal.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.preserves.basic"
] | [] | If the initial comparison map for `G` is an isomorphism, then `G` preserves initial objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_initial_of_is_iso
(f : ⊥_ D ⟶ G.obj (⊥_ C)) [i : is_iso f] : preserves_colimit (functor.empty C) G | begin
rw subsingleton.elim f (initial_comparison G) at i,
exactI preserves_initial.of_iso_comparison G,
end | def | category_theory.limits.preserves_initial_of_is_iso | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/terminal.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.preserves.basic"
] | [] | If there is any isomorphism `⊥ ⟶ G.obj ⊥`, then `G` preserves initial objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_initial_of_iso
(f : ⊥_ D ≅ G.obj (⊥_ C)) : preserves_colimit (functor.empty C) G | preserves_initial_of_is_iso G f.hom | def | category_theory.limits.preserves_initial_of_iso | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/terminal.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.preserves.basic"
] | [] | If there is any isomorphism `⊥ ≅ G.obj ⊥ `, then `G` preserves initial objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_initial.iso : G.obj (⊥_ C) ≅ ⊥_ D | (is_colimit_of_has_initial_of_preserves_colimit G).cocone_point_unique_up_to_iso
(colimit.is_colimit _) | def | category_theory.limits.preserves_initial.iso | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/terminal.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.preserves.basic"
] | [] | If `G` preserves initial objects, then the initial comparison map for `G` is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_initial.iso_hom : (preserves_initial.iso G).inv = initial_comparison G | rfl | lemma | category_theory.limits.preserves_initial.iso_hom | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/terminal.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_zero_morphisms (F : C ⥤ D) : Prop | (map_zero' : ∀ (X Y : C), F.map (0 : X ⟶ Y) = 0 . obviously) | class | category_theory.functor.preserves_zero_morphisms | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/zero.lean | [
"category_theory.limits.preserves.shapes.terminal",
"category_theory.limits.shapes.zero_morphisms"
] | [] | A functor preserves zero morphisms if it sends zero morphisms to zero morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_zero (F : C ⥤ D) [preserves_zero_morphisms F] (X Y : C) :
F.map (0 : X ⟶ Y) = 0 | preserves_zero_morphisms.map_zero' _ _ | lemma | category_theory.functor.map_zero | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/zero.lean | [
"category_theory.limits.preserves.shapes.terminal",
"category_theory.limits.shapes.zero_morphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_of_map_zero (F : C ⥤ D) [preserves_zero_morphisms F] [faithful F] {X Y : C}
(f : X ⟶ Y) (h : F.map f = 0) : f = 0 | F.map_injective $ h.trans $ eq.symm $ F.map_zero _ _ | lemma | category_theory.functor.zero_of_map_zero | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/zero.lean | [
"category_theory.limits.preserves.shapes.terminal",
"category_theory.limits.shapes.zero_morphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_zero_iff (F : C ⥤ D) [preserves_zero_morphisms F] [faithful F] {X Y : C} {f : X ⟶ Y} :
F.map f = 0 ↔ f = 0 | ⟨F.zero_of_map_zero _, by { rintro rfl, exact F.map_zero _ _ }⟩ | lemma | category_theory.functor.map_eq_zero_iff | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/zero.lean | [
"category_theory.limits.preserves.shapes.terminal",
"category_theory.limits.shapes.zero_morphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_zero_morphisms_of_is_left_adjoint (F : C ⥤ D) [is_left_adjoint F] :
preserves_zero_morphisms F | { map_zero' := λ X Y, let adj := adjunction.of_left_adjoint F in
begin
calc F.map (0 : X ⟶ Y) = F.map 0 ≫ F.map (adj.unit.app Y) ≫ adj.counit.app (F.obj Y) : _
... = F.map 0 ≫ F.map ((right_adjoint F).map (0 : F.obj X ⟶ _)) ≫ adj.counit.app (F.obj Y) : _
... = 0 : _,
{ rw adjunction.left_triangle_comp... | instance | category_theory.functor.preserves_zero_morphisms_of_is_left_adjoint | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/zero.lean | [
"category_theory.limits.preserves.shapes.terminal",
"category_theory.limits.shapes.zero_morphisms"
] | [
"adj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_zero_morphisms_of_is_right_adjoint (G : C ⥤ D) [is_right_adjoint G] :
preserves_zero_morphisms G | { map_zero' := λ X Y, let adj := adjunction.of_right_adjoint G in
begin
calc G.map (0 : X ⟶ Y) = adj.unit.app (G.obj X) ≫ G.map (adj.counit.app X) ≫ G.map 0 : _
... = adj.unit.app (G.obj X) ≫ G.map ((left_adjoint G).map (0 : _ ⟶ G.obj X)) ≫ G.map 0 : _
... = 0 : _,
{ rw adjunction.right_triangle_compo... | instance | category_theory.functor.preserves_zero_morphisms_of_is_right_adjoint | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/zero.lean | [
"category_theory.limits.preserves.shapes.terminal",
"category_theory.limits.shapes.zero_morphisms"
] | [
"adj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_zero_morphisms_of_full (F : C ⥤ D) [full F] : preserves_zero_morphisms F | { map_zero' := λ X Y, calc
F.map (0 : X ⟶ Y) = F.map (0 ≫ (F.preimage (0 : F.obj Y ⟶ F.obj Y))) : by rw zero_comp
... = 0 : by rw [F.map_comp, F.image_preimage, comp_zero] } | instance | category_theory.functor.preserves_zero_morphisms_of_full | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/zero.lean | [
"category_theory.limits.preserves.shapes.terminal",
"category_theory.limits.shapes.zero_morphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_zero_object [preserves_zero_morphisms F] : F.obj 0 ≅ 0 | { hom := 0,
inv := 0,
hom_inv_id' := by rw [← F.map_id, id_zero, F.map_zero, zero_comp],
inv_hom_id' := by rw [id_zero, comp_zero] } | def | category_theory.functor.map_zero_object | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/zero.lean | [
"category_theory.limits.preserves.shapes.terminal",
"category_theory.limits.shapes.zero_morphisms"
] | [] | A functor that preserves zero morphisms also preserves the zero object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_zero_morphisms_of_map_zero_object (i : F.obj 0 ≅ 0) : preserves_zero_morphisms F | { map_zero' := λ X Y, calc
F.map (0 : X ⟶ Y) = F.map (0 : X ⟶ 0) ≫ F.map 0 : by rw [← functor.map_comp, comp_zero]
... = F.map 0 ≫ (i.hom ≫ i.inv) ≫ F.map 0
: by rw [iso.hom_inv_id, category.id_comp]
... = 0 : by simp only [zero_of_to_zero i.hom, zero_comp, comp... | lemma | category_theory.functor.preserves_zero_morphisms_of_map_zero_object | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/zero.lean | [
"category_theory.limits.preserves.shapes.terminal",
"category_theory.limits.shapes.zero_morphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_zero_morphisms_of_preserves_initial_object
[preserves_colimit (functor.empty.{0} C) F] : preserves_zero_morphisms F | preserves_zero_morphisms_of_map_zero_object $
F.map_iso has_zero_object.zero_iso_initial
≪≫ preserves_initial.iso F ≪≫ has_zero_object.zero_iso_initial.symm | instance | category_theory.functor.preserves_zero_morphisms_of_preserves_initial_object | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/zero.lean | [
"category_theory.limits.preserves.shapes.terminal",
"category_theory.limits.shapes.zero_morphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_zero_morphisms_of_preserves_terminal_object
[preserves_limit (functor.empty.{0} C) F] : preserves_zero_morphisms F | preserves_zero_morphisms_of_map_zero_object $
F.map_iso has_zero_object.zero_iso_terminal
≪≫ preserves_terminal.iso F ≪≫ has_zero_object.zero_iso_terminal.symm | instance | category_theory.functor.preserves_zero_morphisms_of_preserves_terminal_object | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/zero.lean | [
"category_theory.limits.preserves.shapes.terminal",
"category_theory.limits.shapes.zero_morphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_terminal_object_of_preserves_zero_morphisms
[preserves_zero_morphisms F] : preserves_limit (functor.empty C) F | preserves_terminal_of_iso F $
F.map_iso has_zero_object.zero_iso_terminal.symm
≪≫ map_zero_object F ≪≫ has_zero_object.zero_iso_terminal | def | category_theory.functor.preserves_terminal_object_of_preserves_zero_morphisms | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/zero.lean | [
"category_theory.limits.preserves.shapes.terminal",
"category_theory.limits.shapes.zero_morphisms"
] | [] | Preserving zero morphisms implies preserving terminal objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_initial_object_of_preserves_zero_morphisms
[preserves_zero_morphisms F] : preserves_colimit (functor.empty C) F | preserves_initial_of_iso F $
has_zero_object.zero_iso_initial.symm ≪≫ (map_zero_object F).symm
≪≫ (F.map_iso has_zero_object.zero_iso_initial.symm).symm | def | category_theory.functor.preserves_initial_object_of_preserves_zero_morphisms | category_theory.limits.preserves.shapes | src/category_theory/limits/preserves/shapes/zero.lean | [
"category_theory.limits.preserves.shapes.terminal",
"category_theory.limits.shapes.zero_morphisms"
] | [] | Preserving zero morphisms implies preserving terminal objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
walking_pair : Type
| left | right | inductive | category_theory.limits.walking_pair | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | The type of objects for the diagram indexing a binary (co)product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_pair.swap : walking_pair ≃ walking_pair | { to_fun := λ j, walking_pair.rec_on j right left,
inv_fun := λ j, walking_pair.rec_on j right left,
left_inv := λ j, by { cases j; refl, },
right_inv := λ j, by { cases j; refl, }, } | def | category_theory.limits.walking_pair.swap | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [
"inv_fun"
] | The equivalence swapping left and right. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
walking_pair.swap_apply_left : walking_pair.swap left = right | rfl | lemma | category_theory.limits.walking_pair.swap_apply_left | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_pair.swap_apply_right : walking_pair.swap right = left | rfl | lemma | category_theory.limits.walking_pair.swap_apply_right | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_pair.swap_symm_apply_tt : walking_pair.swap.symm left = right | rfl | lemma | category_theory.limits.walking_pair.swap_symm_apply_tt | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_pair.swap_symm_apply_ff : walking_pair.swap.symm right = left | rfl | lemma | category_theory.limits.walking_pair.swap_symm_apply_ff | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_pair.equiv_bool : walking_pair ≃ bool | { to_fun := λ j, walking_pair.rec_on j tt ff, -- to match equiv.sum_equiv_sigma_bool
inv_fun := λ b, bool.rec_on b right left,
left_inv := λ j, by { cases j; refl, },
right_inv := λ b, by { cases b; refl, }, } | def | category_theory.limits.walking_pair.equiv_bool | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [
"inv_fun"
] | An equivalence from `walking_pair` to `bool`, sometimes useful when reindexing limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
walking_pair.equiv_bool_apply_left : walking_pair.equiv_bool left = tt | rfl | lemma | category_theory.limits.walking_pair.equiv_bool_apply_left | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_pair.equiv_bool_apply_right : walking_pair.equiv_bool right = ff | rfl | lemma | category_theory.limits.walking_pair.equiv_bool_apply_right | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_pair.equiv_bool_symm_apply_tt : walking_pair.equiv_bool.symm tt = left | rfl | lemma | category_theory.limits.walking_pair.equiv_bool_symm_apply_tt | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_pair.equiv_bool_symm_apply_ff : walking_pair.equiv_bool.symm ff = right | rfl | lemma | category_theory.limits.walking_pair.equiv_bool_symm_apply_ff | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pair_function (X Y : C) : walking_pair → C | λ j, walking_pair.cases_on j X Y | def | category_theory.limits.pair_function | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | The function on the walking pair, sending the two points to `X` and `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pair_function_left (X Y : C) : pair_function X Y left = X | rfl | lemma | category_theory.limits.pair_function_left | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pair_function_right (X Y : C) : pair_function X Y right = Y | rfl | lemma | category_theory.limits.pair_function_right | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pair (X Y : C) : discrete walking_pair ⥤ C | discrete.functor (λ j, walking_pair.cases_on j X Y) | def | category_theory.limits.pair | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | The diagram on the walking pair, sending the two points to `X` and `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pair_obj_left (X Y : C) : (pair X Y).obj ⟨left⟩ = X | rfl | lemma | category_theory.limits.pair_obj_left | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pair_obj_right (X Y : C) : (pair X Y).obj ⟨right⟩ = Y | rfl | lemma | category_theory.limits.pair_obj_right | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_pair : F ⟶ G | { app := λ j, discrete.rec_on j (λ j, walking_pair.cases_on j f g) } | def | category_theory.limits.map_pair | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | The natural transformation between two functors out of the
walking pair, specified by its
components. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_pair_left : (map_pair f g).app ⟨left⟩ = f | rfl | lemma | category_theory.limits.map_pair_left | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_pair_right : (map_pair f g).app ⟨right⟩ = g | rfl | lemma | category_theory.limits.map_pair_right | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_pair_iso (f : F.obj ⟨left⟩ ≅ G.obj ⟨left⟩) (g : F.obj ⟨right⟩ ≅ G.obj ⟨right⟩) : F ≅ G | nat_iso.of_components (λ j, discrete.rec_on j (λ j, walking_pair.cases_on j f g)) (by tidy) | def | category_theory.limits.map_pair_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | The natural isomorphism between two functors out of the walking pair, specified by its
components. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagram_iso_pair (F : discrete walking_pair ⥤ C) :
F ≅ pair (F.obj ⟨walking_pair.left⟩) (F.obj ⟨walking_pair.right⟩) | map_pair_iso (iso.refl _) (iso.refl _) | def | category_theory.limits.diagram_iso_pair | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | Every functor out of the walking pair is naturally isomorphic (actually, equal) to a `pair` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pair_comp (X Y : C) (F : C ⥤ D) : pair X Y ⋙ F ≅ pair (F.obj X) (F.obj Y) | diagram_iso_pair _ | def | category_theory.limits.pair_comp | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | The natural isomorphism between `pair X Y ⋙ F` and `pair (F.obj X) (F.obj Y)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_fan (X Y : C) | cone (pair X Y) | abbreviation | category_theory.limits.binary_fan | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | A binary fan is just a cone on a diagram indexing a product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_fan.fst {X Y : C} (s : binary_fan X Y) | s.π.app ⟨walking_pair.left⟩ | abbreviation | category_theory.limits.binary_fan.fst | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | The first projection of a binary fan. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_fan.snd {X Y : C} (s : binary_fan X Y) | s.π.app ⟨walking_pair.right⟩ | abbreviation | category_theory.limits.binary_fan.snd | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | The second projection of a binary fan. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_fan.π_app_left {X Y : C} (s : binary_fan X Y) :
s.π.app ⟨walking_pair.left⟩ = s.fst | rfl | lemma | category_theory.limits.binary_fan.π_app_left | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_fan.π_app_right {X Y : C} (s : binary_fan X Y) :
s.π.app ⟨walking_pair.right⟩ = s.snd | rfl | lemma | category_theory.limits.binary_fan.π_app_right | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_fan.is_limit.mk {X Y : C} (s : binary_fan X Y)
(lift : Π {T : C} (f : T ⟶ X) (g : T ⟶ Y), T ⟶ s.X)
(hl₁ : ∀ {T : C} (f : T ⟶ X) (g : T ⟶ Y), lift f g ≫ s.fst = f)
(hl₂ : ∀ {T : C} (f : T ⟶ X) (g : T ⟶ Y), lift f g ≫ s.snd = g)
(uniq : ∀ {T : C} (f : T ⟶ X) (g : T ⟶ Y) (m : T ⟶ s.X) (h₁ : m ≫ s.fst = f)
... | is_limit.mk
(λ t, lift (binary_fan.fst t) (binary_fan.snd t))
(by { rintros t (rfl|rfl), { exact hl₁ _ _ }, { exact hl₂ _ _ } })
(λ t m h, uniq _ _ _ (h ⟨walking_pair.left⟩) (h ⟨walking_pair.right⟩)) | def | category_theory.limits.binary_fan.is_limit.mk | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [
"lift"
] | A convenient way to show that a binary fan is a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_fan.is_limit.hom_ext {W X Y : C} {s : binary_fan X Y} (h : is_limit s)
{f g : W ⟶ s.X} (h₁ : f ≫ s.fst = g ≫ s.fst) (h₂ : f ≫ s.snd = g ≫ s.snd) : f = g | h.hom_ext $ λ j, discrete.rec_on j (λ j, walking_pair.cases_on j h₁ h₂) | lemma | category_theory.limits.binary_fan.is_limit.hom_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_cofan (X Y : C) | cocone (pair X Y) | abbreviation | category_theory.limits.binary_cofan | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | A binary cofan is just a cocone on a diagram indexing a coproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_cofan.inl {X Y : C} (s : binary_cofan X Y) | s.ι.app ⟨walking_pair.left⟩ | abbreviation | category_theory.limits.binary_cofan.inl | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | The first inclusion of a binary cofan. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_cofan.inr {X Y : C} (s : binary_cofan X Y) | s.ι.app ⟨walking_pair.right⟩ | abbreviation | category_theory.limits.binary_cofan.inr | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | The second inclusion of a binary cofan. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_cofan.ι_app_left {X Y : C} (s : binary_cofan X Y) :
s.ι.app ⟨walking_pair.left⟩ = s.inl | rfl | lemma | category_theory.limits.binary_cofan.ι_app_left | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_cofan.ι_app_right {X Y : C} (s : binary_cofan X Y) :
s.ι.app ⟨walking_pair.right⟩ = s.inr | rfl | lemma | category_theory.limits.binary_cofan.ι_app_right | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_cofan.is_colimit.mk {X Y : C} (s : binary_cofan X Y)
(desc : Π {T : C} (f : X ⟶ T) (g : Y ⟶ T), s.X ⟶ T)
(hd₁ : ∀ {T : C} (f : X ⟶ T) (g : Y ⟶ T), s.inl ≫ desc f g = f)
(hd₂ : ∀ {T : C} (f : X ⟶ T) (g : Y ⟶ T), s.inr ≫ desc f g = g)
(uniq : ∀ {T : C} (f : X ⟶ T) (g : Y ⟶ T) (m : s.X ⟶ T) (h₁ : s.inl ≫ m ... | is_colimit.mk
(λ t, desc (binary_cofan.inl t) (binary_cofan.inr t))
(by { rintros t (rfl|rfl), { exact hd₁ _ _ }, { exact hd₂ _ _ }})
(λ t m h, uniq _ _ _ (h ⟨walking_pair.left⟩) (h ⟨walking_pair.right⟩)) | def | category_theory.limits.binary_cofan.is_colimit.mk | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | A convenient way to show that a binary cofan is a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_cofan.is_colimit.hom_ext {W X Y : C} {s : binary_cofan X Y} (h : is_colimit s)
{f g : s.X ⟶ W} (h₁ : s.inl ≫ f = s.inl ≫ g) (h₂ : s.inr ≫ f = s.inr ≫ g) : f = g | h.hom_ext $ λ j, discrete.rec_on j (λ j, walking_pair.cases_on j h₁ h₂) | lemma | category_theory.limits.binary_cofan.is_colimit.hom_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_fan.mk {P : C} (π₁ : P ⟶ X) (π₂ : P ⟶ Y) : binary_fan X Y | { X := P,
π := { app := λ j, discrete.rec_on j (λ j, walking_pair.cases_on j π₁ π₂) }} | def | category_theory.limits.binary_fan.mk | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | A binary fan with vertex `P` consists of the two projections `π₁ : P ⟶ X` and `π₂ : P ⟶ Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_cofan.mk {P : C} (ι₁ : X ⟶ P) (ι₂ : Y ⟶ P) : binary_cofan X Y | { X := P,
ι := { app := λ j, discrete.rec_on j (λ j, walking_pair.cases_on j ι₁ ι₂) }} | def | category_theory.limits.binary_cofan.mk | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | A binary cofan with vertex `P` consists of the two inclusions `ι₁ : X ⟶ P` and `ι₂ : Y ⟶ P`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_fan.mk_fst {P : C} (π₁ : P ⟶ X) (π₂ : P ⟶ Y) :
(binary_fan.mk π₁ π₂).fst = π₁ | rfl | lemma | category_theory.limits.binary_fan.mk_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_fan.mk_snd {P : C} (π₁ : P ⟶ X) (π₂ : P ⟶ Y) :
(binary_fan.mk π₁ π₂).snd = π₂ | rfl | lemma | category_theory.limits.binary_fan.mk_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_cofan.mk_inl {P : C} (ι₁ : X ⟶ P) (ι₂ : Y ⟶ P) :
(binary_cofan.mk ι₁ ι₂).inl = ι₁ | rfl | lemma | category_theory.limits.binary_cofan.mk_inl | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_cofan.mk_inr {P : C} (ι₁ : X ⟶ P) (ι₂ : Y ⟶ P) :
(binary_cofan.mk ι₁ ι₂).inr = ι₂ | rfl | lemma | category_theory.limits.binary_cofan.mk_inr | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_binary_fan_mk {X Y : C} (c : binary_fan X Y) : c ≅ binary_fan.mk c.fst c.snd | cones.ext (iso.refl _) (λ j, by discrete_cases; cases j; tidy) | def | category_theory.limits.iso_binary_fan_mk | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | Every `binary_fan` is isomorphic to an application of `binary_fan.mk`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_binary_cofan_mk {X Y : C} (c : binary_cofan X Y) : c ≅ binary_cofan.mk c.inl c.inr | cocones.ext (iso.refl _) (λ j, by discrete_cases; cases j; tidy) | def | category_theory.limits.iso_binary_cofan_mk | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | Every `binary_fan` is isomorphic to an application of `binary_fan.mk`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_fan.is_limit_mk {W : C} {fst : W ⟶ X} {snd : W ⟶ Y}
(lift : Π (s : binary_fan X Y), s.X ⟶ W)
(fac_left : ∀ (s : binary_fan X Y), lift s ≫ fst = s.fst)
(fac_right : ∀ (s : binary_fan X Y), lift s ≫ snd = s.snd)
(uniq : ∀ (s : binary_fan X Y) (m : s.X ⟶ W)
(w_fst : m ≫ fst = s.fst) (w_snd : m ≫ snd = s... | { lift := lift,
fac' := λ s j, by { rcases j with ⟨⟨⟩⟩, exacts [fac_left s, fac_right s], },
uniq' := λ s m w, uniq s m (w ⟨walking_pair.left⟩) (w ⟨walking_pair.right⟩) } | def | category_theory.limits.binary_fan.is_limit_mk | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [
"lift"
] | This is a more convenient formulation to show that a `binary_fan` constructed using
`binary_fan.mk` is a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_cofan.is_colimit_mk {W : C} {inl : X ⟶ W} {inr : Y ⟶ W}
(desc : Π (s : binary_cofan X Y), W ⟶ s.X)
(fac_left : ∀ (s : binary_cofan X Y), inl ≫ desc s = s.inl)
(fac_right : ∀ (s : binary_cofan X Y), inr ≫ desc s = s.inr)
(uniq : ∀ (s : binary_cofan X Y) (m : W ⟶ s.X)
(w_inl : inl ≫ m = s.inl) (w_inr :... | { desc := desc,
fac' := λ s j, by { rcases j with ⟨⟨⟩⟩, exacts [fac_left s, fac_right s], },
uniq' := λ s m w, uniq s m (w ⟨walking_pair.left⟩) (w ⟨walking_pair.right⟩) } | def | category_theory.limits.binary_cofan.is_colimit_mk | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | This is a more convenient formulation to show that a `binary_cofan` constructed using
`binary_cofan.mk` is a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_fan.is_limit.lift' {W X Y : C} {s : binary_fan X Y} (h : is_limit s) (f : W ⟶ X)
(g : W ⟶ Y) : {l : W ⟶ s.X // l ≫ s.fst = f ∧ l ≫ s.snd = g} | ⟨h.lift $ binary_fan.mk f g, h.fac _ _, h.fac _ _⟩ | def | category_theory.limits.binary_fan.is_limit.lift' | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | If `s` is a limit binary fan over `X` and `Y`, then every pair of morphisms `f : W ⟶ X` and
`g : W ⟶ Y` induces a morphism `l : W ⟶ s.X` satisfying `l ≫ s.fst = f` and `l ≫ s.snd = g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_cofan.is_colimit.desc' {W X Y : C} {s : binary_cofan X Y} (h : is_colimit s) (f : X ⟶ W)
(g : Y ⟶ W) : {l : s.X ⟶ W // s.inl ≫ l = f ∧ s.inr ≫ l = g} | ⟨h.desc $ binary_cofan.mk f g, h.fac _ _, h.fac _ _⟩ | def | category_theory.limits.binary_cofan.is_colimit.desc' | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | If `s` is a colimit binary cofan over `X` and `Y`,, then every pair of morphisms `f : X ⟶ W` and
`g : Y ⟶ W` induces a morphism `l : s.X ⟶ W` satisfying `s.inl ≫ l = f` and `s.inr ≫ l = g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_fan.is_limit_flip {X Y : C} {c : binary_fan X Y} (hc : is_limit c) :
is_limit (binary_fan.mk c.snd c.fst) | binary_fan.is_limit_mk (λ s, hc.lift (binary_fan.mk s.snd s.fst))
(λ s, hc.fac _ _) (λ s, hc.fac _ _)
(λ s m e₁ e₂, binary_fan.is_limit.hom_ext hc
(e₂.trans (hc.fac (binary_fan.mk s.snd s.fst) ⟨walking_pair.left⟩).symm)
(e₁.trans (hc.fac (binary_fan.mk s.snd s.fst) ⟨walking_pair.right⟩).symm)) | def | category_theory.limits.binary_fan.is_limit_flip | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | Binary products are symmetric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_fan.is_limit_iff_is_iso_fst {X Y : C} (h : is_terminal Y) (c : binary_fan X Y) :
nonempty (is_limit c) ↔ is_iso c.fst | begin
split,
{ rintro ⟨H⟩,
obtain ⟨l, hl, -⟩ := binary_fan.is_limit.lift' H (𝟙 X) (h.from X),
exact ⟨⟨l, binary_fan.is_limit.hom_ext H
(by simpa [hl, -category.comp_id] using category.comp_id _) (h.hom_ext _ _), hl⟩⟩ },
{ introI,
exact ⟨binary_fan.is_limit.mk _ (λ _ f _, f ≫ inv c.fst)
(λ... | lemma | category_theory.limits.binary_fan.is_limit_iff_is_iso_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_fan.is_limit_iff_is_iso_snd {X Y : C} (h : is_terminal X) (c : binary_fan X Y) :
nonempty (is_limit c) ↔ is_iso c.snd | begin
refine iff.trans _ (binary_fan.is_limit_iff_is_iso_fst h (binary_fan.mk c.snd c.fst)),
exact ⟨λ h, ⟨binary_fan.is_limit_flip h.some⟩,
λ h, ⟨(binary_fan.is_limit_flip h.some).of_iso_limit (iso_binary_fan_mk c).symm⟩⟩,
end | lemma | category_theory.limits.binary_fan.is_limit_iff_is_iso_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_fan.is_limit_comp_left_iso {X Y X' : C} (c : binary_fan X Y) (f : X ⟶ X')
[is_iso f] (h : is_limit c) : is_limit (binary_fan.mk (c.fst ≫ f) c.snd) | begin
fapply binary_fan.is_limit_mk,
{ exact λ s, h.lift (binary_fan.mk (s.fst ≫ inv f) s.snd) },
{ intro s, simp },
{ intro s, simp },
{ intros s m e₁ e₂, apply binary_fan.is_limit.hom_ext h; simpa }
end | def | category_theory.limits.binary_fan.is_limit_comp_left_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | If `X' ≅ X`, then `X × Y` also is the product of `X'` and `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_fan.is_limit_comp_right_iso {X Y Y' : C} (c : binary_fan X Y) (f : Y ⟶ Y')
[is_iso f] (h : is_limit c) : is_limit (binary_fan.mk c.fst (c.snd ≫ f)) | binary_fan.is_limit_flip $ binary_fan.is_limit_comp_left_iso _ f (binary_fan.is_limit_flip h) | def | category_theory.limits.binary_fan.is_limit_comp_right_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | If `Y' ≅ Y`, then `X x Y` also is the product of `X` and `Y'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_cofan.is_colimit_flip {X Y : C} {c : binary_cofan X Y} (hc : is_colimit c) :
is_colimit (binary_cofan.mk c.inr c.inl) | binary_cofan.is_colimit_mk (λ s, hc.desc (binary_cofan.mk s.inr s.inl))
(λ s, hc.fac _ _) (λ s, hc.fac _ _)
(λ s m e₁ e₂, binary_cofan.is_colimit.hom_ext hc
(e₂.trans (hc.fac (binary_cofan.mk s.inr s.inl) ⟨walking_pair.left⟩).symm)
(e₁.trans (hc.fac (binary_cofan.mk s.inr s.inl) ⟨walking_pair.right⟩).symm)) | def | category_theory.limits.binary_cofan.is_colimit_flip | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | Binary coproducts are symmetric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_cofan.is_colimit_iff_is_iso_inl {X Y : C} (h : is_initial Y) (c : binary_cofan X Y) :
nonempty (is_colimit c) ↔ is_iso c.inl | begin
split,
{ rintro ⟨H⟩,
obtain ⟨l, hl, -⟩ := binary_cofan.is_colimit.desc' H (𝟙 X) (h.to X),
exact ⟨⟨l, hl, binary_cofan.is_colimit.hom_ext H (by simp [reassoc_of hl]) (h.hom_ext _ _)⟩⟩ },
{ introI,
exact ⟨binary_cofan.is_colimit.mk _ (λ _ f _, inv c.inl ≫ f)
(λ _ _ _, is_iso.hom_inv_id_asso... | lemma | category_theory.limits.binary_cofan.is_colimit_iff_is_iso_inl | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_cofan.is_colimit_iff_is_iso_inr {X Y : C} (h : is_initial X) (c : binary_cofan X Y) :
nonempty (is_colimit c) ↔ is_iso c.inr | begin
refine iff.trans _ (binary_cofan.is_colimit_iff_is_iso_inl h (binary_cofan.mk c.inr c.inl)),
exact ⟨λ h, ⟨binary_cofan.is_colimit_flip h.some⟩,
λ h, ⟨(binary_cofan.is_colimit_flip h.some).of_iso_colimit (iso_binary_cofan_mk c).symm⟩⟩,
end | lemma | category_theory.limits.binary_cofan.is_colimit_iff_is_iso_inr | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_cofan.is_colimit_comp_left_iso {X Y X' : C} (c : binary_cofan X Y) (f : X' ⟶ X)
[is_iso f] (h : is_colimit c) : is_colimit (binary_cofan.mk (f ≫ c.inl) c.inr) | begin
fapply binary_cofan.is_colimit_mk,
{ exact λ s, h.desc (binary_cofan.mk (inv f ≫ s.inl) s.inr) },
{ intro s, simp },
{ intro s, simp },
{ intros s m e₁ e₂,
apply binary_cofan.is_colimit.hom_ext h,
{ rw ← cancel_epi f, simpa using e₁ },
{ simpa } }
end | def | category_theory.limits.binary_cofan.is_colimit_comp_left_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | If `X' ≅ X`, then `X ⨿ Y` also is the coproduct of `X'` and `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_cofan.is_colimit_comp_right_iso {X Y Y' : C} (c : binary_cofan X Y) (f : Y' ⟶ Y)
[is_iso f] (h : is_colimit c) : is_colimit (binary_cofan.mk c.inl (f ≫ c.inr)) | binary_cofan.is_colimit_flip $
binary_cofan.is_colimit_comp_left_iso _ f (binary_cofan.is_colimit_flip h) | def | category_theory.limits.binary_cofan.is_colimit_comp_right_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | If `Y' ≅ Y`, then `X ⨿ Y` also is the coproduct of `X` and `Y'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_binary_product (X Y : C) | has_limit (pair X Y) | abbreviation | category_theory.limits.has_binary_product | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | An abbreviation for `has_limit (pair X Y)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_binary_coproduct (X Y : C) | has_colimit (pair X Y) | abbreviation | category_theory.limits.has_binary_coproduct | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | An abbreviation for `has_colimit (pair X Y)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod (X Y : C) [has_binary_product X Y] | limit (pair X Y) | abbreviation | category_theory.limits.prod | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | If we have a product of `X` and `Y`, we can access it using `prod X Y` or
`X ⨯ Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coprod (X Y : C) [has_binary_coproduct X Y] | colimit (pair X Y) | abbreviation | category_theory.limits.coprod | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | If we have a coproduct of `X` and `Y`, we can access it using `coprod X Y ` or
`X ⨿ Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod.fst {X Y : C} [has_binary_product X Y] : X ⨯ Y ⟶ X | limit.π (pair X Y) ⟨walking_pair.left⟩ | abbreviation | category_theory.limits.prod.fst | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | The projection map to the first component of the product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod.snd {X Y : C} [has_binary_product X Y] : X ⨯ Y ⟶ Y | limit.π (pair X Y) ⟨walking_pair.right⟩ | abbreviation | category_theory.limits.prod.snd | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | The projecton map to the second component of the product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.