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is_initial.to_self {X : C} (t : is_initial X) : t.to X = 𝟙 X | t.hom_ext _ _ | lemma | category_theory.limits.is_initial.to_self | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_terminal.is_split_mono_from {X Y : C} (t : is_terminal X) (f : X ⟶ Y) :
is_split_mono f | is_split_mono.mk' ⟨t.from _, t.hom_ext _ _⟩ | lemma | category_theory.limits.is_terminal.is_split_mono_from | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | Any morphism from a terminal object is split mono. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_initial.is_split_epi_to {X Y : C} (t : is_initial X) (f : Y ⟶ X) :
is_split_epi f | is_split_epi.mk' ⟨t.to _, t.hom_ext _ _⟩ | lemma | category_theory.limits.is_initial.is_split_epi_to | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | Any morphism to an initial object is split epi. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_terminal.mono_from {X Y : C} (t : is_terminal X) (f : X ⟶ Y) : mono f | by haveI := t.is_split_mono_from f; apply_instance | lemma | category_theory.limits.is_terminal.mono_from | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | Any morphism from a terminal object is mono. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_initial.epi_to {X Y : C} (t : is_initial X) (f : Y ⟶ X) : epi f | by haveI := t.is_split_epi_to f; apply_instance | lemma | category_theory.limits.is_initial.epi_to | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | Any morphism to an initial object is epi. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_terminal.unique_up_to_iso {T T' : C} (hT : is_terminal T) (hT' : is_terminal T') : T ≅ T' | { hom := hT'.from _,
inv := hT.from _ } | def | category_theory.limits.is_terminal.unique_up_to_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | If `T` and `T'` are terminal, they are isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_initial.unique_up_to_iso {I I' : C} (hI : is_initial I) (hI' : is_initial I') : I ≅ I' | { hom := hI.to _,
inv := hI'.to _ } | def | category_theory.limits.is_initial.unique_up_to_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | If `I` and `I'` are initial, they are isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_terminal | has_limits_of_shape (discrete.{0} pempty) C | abbreviation | category_theory.limits.has_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"pempty"
] | A category has a terminal object if it has a limit over the empty diagram.
Use `has_terminal_of_unique` to construct instances. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_initial | has_colimits_of_shape (discrete.{0} pempty) C | abbreviation | category_theory.limits.has_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"pempty"
] | A category has an initial object if it has a colimit over the empty diagram.
Use `has_initial_of_unique` to construct instances. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_change_empty_cone {c₁ : cone F₁} (hl : is_limit c₁)
(c₂ : cone F₂) (hi : c₁.X ≅ c₂.X) : is_limit c₂ | { lift := λ c, hl.lift ⟨c.X, by tidy⟩ ≫ hi.hom,
fac' := λ _ j, j.as.elim,
uniq' := λ c f _, by { erw ← hl.uniq ⟨c.X, by tidy⟩ (f ≫ hi.inv) (λ j, j.as.elim), simp } } | def | category_theory.limits.is_limit_change_empty_cone | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"lift"
] | Being terminal is independent of the empty diagram, its universe, and the cone over it,
as long as the cone points are isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_empty_cone_equiv (c₁ : cone F₁) (c₂ : cone F₂) (h : c₁.X ≅ c₂.X) :
is_limit c₁ ≃ is_limit c₂ | { to_fun := λ hl, is_limit_change_empty_cone C hl c₂ h,
inv_fun := λ hl, is_limit_change_empty_cone C hl c₁ h.symm,
left_inv := by tidy,
right_inv := by tidy } | def | category_theory.limits.is_limit_empty_cone_equiv | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"inv_fun"
] | Replacing an empty cone in `is_limit` by another with the same cone point
is an equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_terminal_change_diagram (h : has_limit F₁) : has_limit F₂ | ⟨⟨⟨⟨limit F₁, by tidy⟩, is_limit_change_empty_cone C (limit.is_limit F₁) _ (eq_to_iso rfl)⟩⟩⟩ | lemma | category_theory.limits.has_terminal_change_diagram | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_terminal_change_universe [h : has_limits_of_shape (discrete.{w} pempty) C] :
has_limits_of_shape (discrete.{w'} pempty) C | { has_limit := λ J, has_terminal_change_diagram C (let f := h.1 in f (functor.empty C)) } | lemma | category_theory.limits.has_terminal_change_universe | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"pempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_colimit_change_empty_cocone {c₁ : cocone F₁} (hl : is_colimit c₁)
(c₂ : cocone F₂) (hi : c₁.X ≅ c₂.X) : is_colimit c₂ | { desc := λ c, hi.inv ≫ hl.desc ⟨c.X, by tidy⟩,
fac' := λ _ j, j.as.elim,
uniq' := λ c f _, by { erw ← hl.uniq ⟨c.X, by tidy⟩ (hi.hom ≫ f) (λ j, j.as.elim), simp } } | def | category_theory.limits.is_colimit_change_empty_cocone | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | Being initial is independent of the empty diagram, its universe, and the cocone over it,
as long as the cocone points are isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_empty_cocone_equiv (c₁ : cocone F₁) (c₂ : cocone F₂) (h : c₁.X ≅ c₂.X) :
is_colimit c₁ ≃ is_colimit c₂ | { to_fun := λ hl, is_colimit_change_empty_cocone C hl c₂ h,
inv_fun := λ hl, is_colimit_change_empty_cocone C hl c₁ h.symm,
left_inv := by tidy,
right_inv := by tidy } | def | category_theory.limits.is_colimit_empty_cocone_equiv | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"inv_fun"
] | Replacing an empty cocone in `is_colimit` by another with the same cocone point
is an equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_initial_change_diagram (h : has_colimit F₁) : has_colimit F₂ | ⟨⟨⟨⟨colimit F₁, by tidy⟩,
is_colimit_change_empty_cocone C (colimit.is_colimit F₁) _ (eq_to_iso rfl)⟩⟩⟩ | lemma | category_theory.limits.has_initial_change_diagram | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_initial_change_universe [h : has_colimits_of_shape (discrete.{w} pempty) C] :
has_colimits_of_shape (discrete.{w'} pempty) C | { has_colimit := λ J, has_initial_change_diagram C (let f := h.1 in f (functor.empty C)) } | lemma | category_theory.limits.has_initial_change_universe | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"pempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
terminal [has_terminal C] : C | limit (functor.empty.{0} C) | abbreviation | category_theory.limits.terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | An arbitrary choice of terminal object, if one exists.
You can use the notation `⊤_ C`.
This object is characterized by having a unique morphism from any object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
initial [has_initial C] : C | colimit (functor.empty.{0} C) | abbreviation | category_theory.limits.initial | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | An arbitrary choice of initial object, if one exists.
You can use the notation `⊥_ C`.
This object is characterized by having a unique morphism to any object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_terminal_of_unique (X : C) [h : Π Y : C, unique (Y ⟶ X)] : has_terminal C | { has_limit := λ F, has_limit.mk ⟨_, (is_terminal_equiv_unique F X).inv_fun h⟩ } | lemma | category_theory.limits.has_terminal_of_unique | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"inv_fun",
"unique"
] | We can more explicitly show that a category has a terminal object by specifying the object,
and showing there is a unique morphism to it from any other object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_terminal.has_terminal {X : C} (h : is_terminal X) : has_terminal C | { has_limit := λ F, has_limit.mk ⟨⟨X, by tidy⟩, is_limit_change_empty_cone _ h _ (iso.refl _)⟩ } | lemma | category_theory.limits.is_terminal.has_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_initial_of_unique (X : C) [h : Π Y : C, unique (X ⟶ Y)] : has_initial C | { has_colimit := λ F, has_colimit.mk ⟨_, (is_initial_equiv_unique F X).inv_fun h⟩ } | lemma | category_theory.limits.has_initial_of_unique | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"inv_fun",
"unique"
] | We can more explicitly show that a category has an initial object by specifying the object,
and showing there is a unique morphism from it to any other object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_initial.has_initial {X : C} (h : is_initial X) : has_initial C | { has_colimit := λ F, has_colimit.mk
⟨⟨X, by tidy⟩, is_colimit_change_empty_cocone _ h _ (iso.refl _)⟩ } | lemma | category_theory.limits.is_initial.has_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
terminal.from [has_terminal C] (P : C) : P ⟶ ⊤_ C | limit.lift (functor.empty C) (as_empty_cone P) | abbreviation | category_theory.limits.terminal.from | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | The map from an object to the terminal object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
initial.to [has_initial C] (P : C) : ⊥_ C ⟶ P | colimit.desc (functor.empty C) (as_empty_cocone P) | abbreviation | category_theory.limits.initial.to | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | The map to an object from the initial object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
terminal_is_terminal [has_terminal C] : is_terminal (⊤_ C) | { lift := λ s, terminal.from _ } | def | category_theory.limits.terminal_is_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"lift"
] | A terminal object is terminal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
initial_is_initial [has_initial C] : is_initial (⊥_ C) | { desc := λ s, initial.to _ } | def | category_theory.limits.initial_is_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | An initial object is initial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_to_terminal [has_terminal C] (P : C) : unique (P ⟶ ⊤_ C) | is_terminal_equiv_unique _ (⊤_ C) terminal_is_terminal P | instance | category_theory.limits.unique_to_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_from_initial [has_initial C] (P : C) : unique (⊥_ C ⟶ P) | is_initial_equiv_unique _ (⊥_ C) initial_is_initial P | instance | category_theory.limits.unique_from_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
terminal.comp_from [has_terminal C] {P Q : C} (f : P ⟶ Q) :
f ≫ terminal.from Q = terminal.from P | by tidy | lemma | category_theory.limits.terminal.comp_from | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
initial.to_comp [has_initial C] {P Q : C} (f : P ⟶ Q) :
initial.to P ≫ f = initial.to Q | by tidy | lemma | category_theory.limits.initial.to_comp | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
initial_iso_is_initial [has_initial C] {P : C} (t : is_initial P) : ⊥_ C ≅ P | initial_is_initial.unique_up_to_iso t | def | category_theory.limits.initial_iso_is_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | The (unique) isomorphism between the chosen initial object and any other initial object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
terminal_iso_is_terminal [has_terminal C] {P : C} (t : is_terminal P) : ⊤_ C ≅ P | terminal_is_terminal.unique_up_to_iso t | def | category_theory.limits.terminal_iso_is_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | The (unique) isomorphism between the chosen terminal object and any other terminal object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
terminal.is_split_mono_from {Y : C} [has_terminal C] (f : ⊤_ C ⟶ Y) : is_split_mono f | is_terminal.is_split_mono_from terminal_is_terminal _ | instance | category_theory.limits.terminal.is_split_mono_from | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | Any morphism from a terminal object is split mono. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
initial.is_split_epi_to {Y : C} [has_initial C] (f : Y ⟶ ⊥_ C) : is_split_epi f | is_initial.is_split_epi_to initial_is_initial _ | instance | category_theory.limits.initial.is_split_epi_to | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | Any morphism to an initial object is split epi. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
terminal_op_of_initial {X : C} (t : is_initial X) : is_terminal (opposite.op X) | { lift := λ s, (t.to s.X.unop).op,
uniq' := λ s m w, quiver.hom.unop_inj (t.hom_ext _ _) } | def | category_theory.limits.terminal_op_of_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"lift",
"opposite.op",
"quiver.hom.unop_inj"
] | An initial object is terminal in the opposite category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
terminal_unop_of_initial {X : Cᵒᵖ} (t : is_initial X) : is_terminal X.unop | { lift := λ s, (t.to (opposite.op s.X)).unop,
uniq' := λ s m w, quiver.hom.op_inj (t.hom_ext _ _) } | def | category_theory.limits.terminal_unop_of_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"lift",
"opposite.op",
"quiver.hom.op_inj"
] | An initial object in the opposite category is terminal in the original category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
initial_op_of_terminal {X : C} (t : is_terminal X) : is_initial (opposite.op X) | { desc := λ s, (t.from s.X.unop).op,
uniq' := λ s m w, quiver.hom.unop_inj (t.hom_ext _ _) } | def | category_theory.limits.initial_op_of_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"opposite.op",
"quiver.hom.unop_inj"
] | A terminal object is initial in the opposite category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
initial_unop_of_terminal {X : Cᵒᵖ} (t : is_terminal X) : is_initial X.unop | { desc := λ s, (t.from (opposite.op s.X)).unop,
uniq' := λ s m w, quiver.hom.op_inj (t.hom_ext _ _) } | def | category_theory.limits.initial_unop_of_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"opposite.op",
"quiver.hom.op_inj"
] | A terminal object in the opposite category is initial in the original category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_initial_op_of_has_terminal [has_terminal C] : has_initial Cᵒᵖ | (initial_op_of_terminal terminal_is_terminal).has_initial | instance | category_theory.limits.has_initial_op_of_has_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_terminal_op_of_has_initial [has_initial C] : has_terminal Cᵒᵖ | (terminal_op_of_initial initial_is_initial).has_terminal | instance | category_theory.limits.has_terminal_op_of_has_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_terminal_of_has_initial_op [has_initial Cᵒᵖ] : has_terminal C | (terminal_unop_of_initial initial_is_initial).has_terminal | lemma | category_theory.limits.has_terminal_of_has_initial_op | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_initial_of_has_terminal_op [has_terminal Cᵒᵖ] : has_initial C | (initial_unop_of_terminal terminal_is_terminal).has_initial | lemma | category_theory.limits.has_initial_of_has_terminal_op | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_const_terminal {J : Type*} [category J] {C : Type*} [category C] [has_terminal C] :
limit ((category_theory.functor.const J).obj (⊤_ C)) ≅ ⊤_ C | { hom := terminal.from _,
inv := limit.lift ((category_theory.functor.const J).obj (⊤_ C))
{ X := ⊤_ C, π := { app := λ j, terminal.from _, }}, } | def | category_theory.limits.limit_const_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"category_theory.functor.const"
] | The limit of the constant `⊤_ C` functor is `⊤_ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_const_terminal_inv_π
{J : Type*} [category J] {C : Type*} [category C] [has_terminal C] {j : J} :
limit_const_terminal.inv ≫ limit.π ((category_theory.functor.const J).obj (⊤_ C)) j =
terminal.from _ | by ext ⟨⟨⟩⟩ | lemma | category_theory.limits.limit_const_terminal_inv_π | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"category_theory.functor.const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_const_initial {J : Type*} [category J] {C : Type*} [category C] [has_initial C] :
colimit ((category_theory.functor.const J).obj (⊥_ C)) ≅ ⊥_ C | { hom := colimit.desc ((category_theory.functor.const J).obj (⊥_ C))
{ X := ⊥_ C, ι := { app := λ j, initial.to _, }, },
inv := initial.to _, } | def | category_theory.limits.colimit_const_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"category_theory.functor.const"
] | The colimit of the constant `⊥_ C` functor is `⊥_ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_colimit_const_initial_hom
{J : Type*} [category J] {C : Type*} [category C] [has_initial C] {j : J} :
colimit.ι ((category_theory.functor.const J).obj (⊥_ C)) j ≫ colimit_const_initial.hom =
initial.to _ | by ext ⟨⟨⟩⟩ | lemma | category_theory.limits.ι_colimit_const_initial_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"category_theory.functor.const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
initial_mono_class (C : Type u₁) [category.{v₁} C] : Prop | (is_initial_mono_from : ∀ {I} (X : C) (hI : is_initial I), mono (hI.to X)) | class | category_theory.limits.initial_mono_class | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | A category is a `initial_mono_class` if the canonical morphism of an initial object is a
monomorphism. In practice, this is most useful when given an arbitrary morphism out of the chosen
initial object, see `initial.mono_from`.
Given a terminal object, this is equivalent to the assumption that the unique morphism from... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_initial.mono_from [initial_mono_class C] {I} {X : C} (hI : is_initial I) (f : I ⟶ X) :
mono f | begin
rw hI.hom_ext f (hI.to X),
apply initial_mono_class.is_initial_mono_from,
end | lemma | category_theory.limits.is_initial.mono_from | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
initial.mono_from [has_initial C] [initial_mono_class C] (X : C) (f : ⊥_ C ⟶ X) :
mono f | initial_is_initial.mono_from f | instance | category_theory.limits.initial.mono_from | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
initial_mono_class.of_is_initial {I : C} (hI : is_initial I) (h : ∀ X, mono (hI.to X)) :
initial_mono_class C | { is_initial_mono_from := λ I' X hI',
begin
rw hI'.hom_ext (hI'.to X) ((hI'.unique_up_to_iso hI).hom ≫ hI.to X),
apply mono_comp,
end } | lemma | category_theory.limits.initial_mono_class.of_is_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | To show a category is a `initial_mono_class` it suffices to give an initial object such that
every morphism out of it is a monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
initial_mono_class.of_initial [has_initial C] (h : ∀ X : C, mono (initial.to X)) :
initial_mono_class C | initial_mono_class.of_is_initial initial_is_initial h | lemma | category_theory.limits.initial_mono_class.of_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | To show a category is a `initial_mono_class` it suffices to show every morphism out of the
initial object is a monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
initial_mono_class.of_is_terminal {I T : C} (hI : is_initial I) (hT : is_terminal T)
(f : mono (hI.to T)) :
initial_mono_class C | initial_mono_class.of_is_initial hI (λ X, mono_of_mono_fac (hI.hom_ext (_ ≫ hT.from X) (hI.to T))) | lemma | category_theory.limits.initial_mono_class.of_is_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | To show a category is a `initial_mono_class` it suffices to show the unique morphism from an
initial object to a terminal object is a monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
initial_mono_class.of_terminal [has_initial C] [has_terminal C]
(h : mono (initial.to (⊤_ C))) :
initial_mono_class C | initial_mono_class.of_is_terminal initial_is_initial terminal_is_terminal h | lemma | category_theory.limits.initial_mono_class.of_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | To show a category is a `initial_mono_class` it suffices to show the unique morphism from the
initial object to a terminal object is a monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
terminal_comparison [has_terminal C] [has_terminal D] :
G.obj (⊤_ C) ⟶ ⊤_ D | terminal.from _ | def | category_theory.limits.terminal_comparison | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | The comparison morphism from the image of a terminal object to the terminal object in the target
category.
This is an isomorphism iff `G` preserves terminal objects, see
`category_theory.limits.preserves_terminal.of_iso_comparison`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
initial_comparison [has_initial C] [has_initial D] :
⊥_ D ⟶ G.obj (⊥_ C) | initial.to _ | def | category_theory.limits.initial_comparison | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cone_of_diagram_initial
{X : J} (tX : is_initial X) (F : J ⥤ C) : cone F | { X := F.obj X,
π :=
{ app := λ j, F.map (tX.to j),
naturality' := λ j j' k,
begin
dsimp,
rw [← F.map_comp, category.id_comp, tX.hom_ext (tX.to j ≫ k) (tX.to j')],
end } } | def | category_theory.limits.cone_of_diagram_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | From a functor `F : J ⥤ C`, given an initial object of `J`, construct a cone for `J`.
In `limit_of_diagram_initial` we show it is a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_of_diagram_initial
{X : J} (tX : is_initial X) (F : J ⥤ C) :
is_limit (cone_of_diagram_initial tX F) | { lift := λ s, s.π.app X,
uniq' := λ s m w,
begin
rw [← w X, cone_of_diagram_initial_π_app, tX.hom_ext (tX.to X) (𝟙 _)],
dsimp, simp -- See note [dsimp, simp]
end} | def | category_theory.limits.limit_of_diagram_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"lift"
] | From a functor `F : J ⥤ C`, given an initial object of `J`, show the cone
`cone_of_diagram_initial` is a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_of_initial (F : J ⥤ C)
[has_initial J] [has_limit F] :
limit F ≅ F.obj (⊥_ J) | is_limit.cone_point_unique_up_to_iso
(limit.is_limit _)
(limit_of_diagram_initial initial_is_initial F) | def | category_theory.limits.limit_of_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | For a functor `F : J ⥤ C`, if `J` has an initial object then the image of it is isomorphic
to the limit of `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_of_diagram_terminal {X : J} (hX : is_terminal X)
(F : J ⥤ C) [∀ (i j : J) (f : i ⟶ j), is_iso (F.map f)] : cone F | { X := F.obj X,
π :=
{ app := λ i, inv (F.map (hX.from _)),
naturality' := begin
intros i j f,
dsimp,
simp only [is_iso.eq_inv_comp, is_iso.comp_inv_eq, category.id_comp,
← F.map_comp, hX.hom_ext (hX.from i) (f ≫ hX.from j)],
end } } | def | category_theory.limits.cone_of_diagram_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | From a functor `F : J ⥤ C`, given a terminal object of `J`, construct a cone for `J`,
provided that the morphisms in the diagram are isomorphisms.
In `limit_of_diagram_terminal` we show it is a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_of_diagram_terminal {X : J} (hX : is_terminal X)
(F : J ⥤ C) [∀ (i j : J) (f : i ⟶ j), is_iso (F.map f)] :
is_limit (cone_of_diagram_terminal hX F) | { lift := λ S, S.π.app _ } | def | category_theory.limits.limit_of_diagram_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [
"lift"
] | From a functor `F : J ⥤ C`, given a terminal object of `J` and that the morphisms in the
diagram are isomorphisms, show the cone `cone_of_diagram_terminal` is a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_of_terminal (F : J ⥤ C)
[has_terminal J] [has_limit F] [∀ (i j : J) (f : i ⟶ j), is_iso (F.map f)] :
limit F ≅ F.obj (⊤_ J) | is_limit.cone_point_unique_up_to_iso
(limit.is_limit _)
(limit_of_diagram_terminal terminal_is_terminal F) | def | category_theory.limits.limit_of_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | For a functor `F : J ⥤ C`, if `J` has a terminal object and all the morphisms in the diagram
are isomorphisms, then the image of the terminal object is isomorphic to the limit of `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_of_diagram_terminal
{X : J} (tX : is_terminal X) (F : J ⥤ C) : cocone F | { X := F.obj X,
ι :=
{ app := λ j, F.map (tX.from j),
naturality' := λ j j' k,
begin
dsimp,
rw [← F.map_comp, category.comp_id, tX.hom_ext (k ≫ tX.from j') (tX.from j)],
end } } | def | category_theory.limits.cocone_of_diagram_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | From a functor `F : J ⥤ C`, given a terminal object of `J`, construct a cocone for `J`.
In `colimit_of_diagram_terminal` we show it is a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_of_diagram_terminal
{X : J} (tX : is_terminal X) (F : J ⥤ C) :
is_colimit (cocone_of_diagram_terminal tX F) | { desc := λ s, s.ι.app X,
uniq' := λ s m w,
by { rw [← w X, cocone_of_diagram_terminal_ι_app, tX.hom_ext (tX.from X) (𝟙 _)], simp } } | def | category_theory.limits.colimit_of_diagram_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | From a functor `F : J ⥤ C`, given a terminal object of `J`, show the cocone
`cocone_of_diagram_terminal` is a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_of_terminal (F : J ⥤ C)
[has_terminal J] [has_colimit F] :
colimit F ≅ F.obj (⊤_ J) | is_colimit.cocone_point_unique_up_to_iso
(colimit.is_colimit _)
(colimit_of_diagram_terminal terminal_is_terminal F) | def | category_theory.limits.colimit_of_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | For a functor `F : J ⥤ C`, if `J` has a terminal object then the image of it is isomorphic
to the colimit of `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_of_diagram_initial {X : J} (hX : is_initial X) (F : J ⥤ C)
[∀ (i j : J) (f : i ⟶ j), is_iso (F.map f)] : cocone F | { X := F.obj X,
ι :=
{ app := λ i, inv (F.map (hX.to _)),
naturality' := begin
intros i j f,
dsimp,
simp only [is_iso.eq_inv_comp, is_iso.comp_inv_eq, category.comp_id,
← F.map_comp, hX.hom_ext (hX.to i ≫ f) (hX.to j)],
end } } | def | category_theory.limits.cocone_of_diagram_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | From a functor `F : J ⥤ C`, given an initial object of `J`, construct a cocone for `J`,
provided that the morphisms in the diagram are isomorphisms.
In `colimit_of_diagram_initial` we show it is a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_of_diagram_initial {X : J} (hX : is_initial X) (F : J ⥤ C)
[∀ (i j : J) (f : i ⟶ j), is_iso (F.map f)] : is_colimit (cocone_of_diagram_initial hX F) | { desc := λ S, S.ι.app _ } | def | category_theory.limits.colimit_of_diagram_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | From a functor `F : J ⥤ C`, given an initial object of `J` and that the morphisms in the
diagram are isomorphisms, show the cone `cocone_of_diagram_initial` is a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_of_initial (F : J ⥤ C)
[has_initial J] [has_colimit F] [∀ (i j : J) (f : i ⟶ j), is_iso (F.map f)] :
colimit F ≅ F.obj (⊥_ J) | is_colimit.cocone_point_unique_up_to_iso
(colimit.is_colimit _)
(colimit_of_diagram_initial initial_is_initial _) | def | category_theory.limits.colimit_of_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | For a functor `F : J ⥤ C`, if `J` has an initial object and all the morphisms in the diagram
are isomorphisms, then the image of the initial object is isomorphic to the colimit of `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_π_of_is_initial {j : J} (I : is_initial j) (F : J ⥤ C) [has_limit F] :
is_iso (limit.π F j) | ⟨⟨limit.lift _ (cone_of_diagram_initial I F), ⟨by { ext, simp }, by simp⟩⟩⟩ | lemma | category_theory.limits.is_iso_π_of_is_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | If `j` is initial in the index category, then the map `limit.π F j` is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_π_initial [has_initial J] (F : J ⥤ C) [has_limit F] :
is_iso (limit.π F (⊥_ J)) | is_iso_π_of_is_initial (initial_is_initial) F | instance | category_theory.limits.is_iso_π_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_π_of_is_terminal {j : J} (I : is_terminal j) (F : J ⥤ C)
[has_limit F] [∀ (i j : J) (f : i ⟶ j), is_iso (F.map f)] : is_iso (limit.π F j) | ⟨⟨limit.lift _ (cone_of_diagram_terminal I F), by { ext, simp }, by simp ⟩⟩ | lemma | category_theory.limits.is_iso_π_of_is_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_π_terminal [has_terminal J] (F : J ⥤ C) [has_limit F]
[∀ (i j : J) (f : i ⟶ j), is_iso (F.map f)] : is_iso (limit.π F (⊤_ J)) | is_iso_π_of_is_terminal terminal_is_terminal F | instance | category_theory.limits.is_iso_π_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_ι_of_is_terminal {j : J} (I : is_terminal j) (F : J ⥤ C) [has_colimit F] :
is_iso (colimit.ι F j) | ⟨⟨colimit.desc _ (cocone_of_diagram_terminal I F), ⟨by simp, by { ext, simp }⟩⟩⟩ | lemma | category_theory.limits.is_iso_ι_of_is_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | If `j` is terminal in the index category, then the map `colimit.ι F j` is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_ι_terminal [has_terminal J] (F : J ⥤ C) [has_colimit F] :
is_iso (colimit.ι F (⊤_ J)) | is_iso_ι_of_is_terminal (terminal_is_terminal) F | instance | category_theory.limits.is_iso_ι_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_ι_of_is_initial {j : J} (I : is_initial j) (F : J ⥤ C)
[has_colimit F] [∀ (i j : J) (f : i ⟶ j), is_iso (F.map f)] : is_iso (colimit.ι F j) | ⟨⟨colimit.desc _ (cocone_of_diagram_initial I F), ⟨by tidy, by { ext, simp }⟩⟩⟩ | lemma | category_theory.limits.is_iso_ι_of_is_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_ι_initial [has_initial J] (F : J ⥤ C) [has_colimit F]
[∀ (i j : J) (f : i ⟶ j), is_iso (F.map f)] : is_iso (colimit.ι F (⊥_ J)) | is_iso_ι_of_is_initial initial_is_initial F | instance | category_theory.limits.is_iso_ι_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/terminal.lean | [
"category_theory.pempty",
"category_theory.limits.has_limits",
"category_theory.epi_mono",
"category_theory.category.preorder"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_lift_π_apply
{β : Type u} (f : β → Type u) {P : Type u} (s : Π b, P ⟶ f b) (b : β) (x : P) :
(pi.π f b : (∏ f) → f b) (@pi.lift β _ _ f _ P s x) = s b x | congr_fun (limit.lift_π (fan.mk P s) ⟨b⟩) x | lemma | category_theory.limits.types.pi_lift_π_apply | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | A restatement of `types.lift_π_apply` that uses `pi.π` and `pi.lift`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_map_π_apply {β : Type u} {f g : β → Type u} (α : Π j, f j ⟶ g j) (b : β) (x) :
(pi.π g b : (∏ g) → g b) (pi.map α x) = α b ((pi.π f b : (∏ f) → f b) x) | limit.map_π_apply _ _ _ | lemma | category_theory.limits.types.pi_map_π_apply | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | A restatement of `types.map_π_apply` that uses `pi.π` and `pi.map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
terminal_limit_cone : limits.limit_cone (functor.empty (Type u)) | { cone :=
{ X := punit,
π := by tidy, },
is_limit := by tidy, } | def | category_theory.limits.types.terminal_limit_cone | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | The category of types has `punit` as a terminal object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
terminal_iso : ⊤_ (Type u) ≅ punit | limit.iso_limit_cone terminal_limit_cone | def | category_theory.limits.types.terminal_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | The terminal object in `Type u` is `punit`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_terminal_punit : is_terminal (punit : Type u) | terminal_is_terminal.of_iso terminal_iso | def | category_theory.limits.types.is_terminal_punit | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | The terminal object in `Type u` is `punit`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
initial_colimit_cocone : limits.colimit_cocone (functor.empty (Type u)) | { cocone :=
{ X := pempty,
ι := by tidy, },
is_colimit := by tidy, } | def | category_theory.limits.types.initial_colimit_cocone | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [
"pempty"
] | The category of types has `pempty` as an initial object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
initial_iso : ⊥_ (Type u) ≅ pempty | colimit.iso_colimit_cocone initial_colimit_cocone | def | category_theory.limits.types.initial_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [
"pempty"
] | The initial object in `Type u` is `pempty`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_initial_punit : is_initial (pempty : Type u) | initial_is_initial.of_iso initial_iso | def | category_theory.limits.types.is_initial_punit | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [
"pempty"
] | The initial object in `Type u` is `pempty`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_product_cone (X Y : Type u) : binary_fan X Y | binary_fan.mk prod.fst prod.snd | def | category_theory.limits.types.binary_product_cone | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_product_cone_fst (X Y : Type u) :
(binary_product_cone X Y).fst = prod.fst | rfl | lemma | category_theory.limits.types.binary_product_cone_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_product_cone_snd (X Y : Type u) :
(binary_product_cone X Y).snd = prod.snd | rfl | lemma | category_theory.limits.types.binary_product_cone_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_product_limit (X Y : Type u) : is_limit (binary_product_cone X Y) | { lift := λ (s : binary_fan X Y) x, (s.fst x, s.snd x),
fac' := λ s j, discrete.rec_on j (λ j, walking_pair.cases_on j rfl rfl),
uniq' := λ s m w, funext $ λ x, prod.ext (congr_fun (w ⟨left⟩) x) (congr_fun (w ⟨right⟩) x) } | def | category_theory.limits.types.binary_product_limit | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [
"lift",
"prod.ext"
] | The product type `X × Y` is a binary product for `X` and `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_product_limit_cone (X Y : Type u) : limits.limit_cone (pair X Y) | ⟨_, binary_product_limit X Y⟩ | def | category_theory.limits.types.binary_product_limit_cone | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | The category of types has `X × Y`, the usual cartesian product,
as the binary product of `X` and `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_product_iso (X Y : Type u) : limits.prod X Y ≅ X × Y | limit.iso_limit_cone (binary_product_limit_cone X Y) | def | category_theory.limits.types.binary_product_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | The categorical binary product in `Type u` is cartesian product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_product_iso_hom_comp_fst (X Y : Type u) :
(binary_product_iso X Y).hom ≫ prod.fst = limits.prod.fst | limit.iso_limit_cone_hom_π (binary_product_limit_cone X Y) ⟨walking_pair.left⟩ | lemma | category_theory.limits.types.binary_product_iso_hom_comp_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_product_iso_hom_comp_snd (X Y : Type u) :
(binary_product_iso X Y).hom ≫ prod.snd = limits.prod.snd | limit.iso_limit_cone_hom_π (binary_product_limit_cone X Y) ⟨walking_pair.right⟩ | lemma | category_theory.limits.types.binary_product_iso_hom_comp_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_product_iso_inv_comp_fst (X Y : Type u) :
(binary_product_iso X Y).inv ≫ limits.prod.fst = prod.fst | limit.iso_limit_cone_inv_π (binary_product_limit_cone X Y) ⟨walking_pair.left⟩ | lemma | category_theory.limits.types.binary_product_iso_inv_comp_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_product_iso_inv_comp_snd (X Y : Type u) :
(binary_product_iso X Y).inv ≫ limits.prod.snd = prod.snd | limit.iso_limit_cone_inv_π (binary_product_limit_cone X Y) ⟨walking_pair.right⟩ | lemma | category_theory.limits.types.binary_product_iso_inv_comp_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_product_functor : Type u ⥤ Type u ⥤ Type u | { obj := λ X,
{ obj := λ Y, X × Y,
map := λ Y₁ Y₂ f, (binary_product_limit X Y₂).lift (binary_fan.mk prod.fst (prod.snd ≫ f)) },
map := λ X₁ X₂ f,
{ app := λ Y, (binary_product_limit X₂ Y).lift (binary_fan.mk (prod.fst ≫ f) prod.snd) } } | def | category_theory.limits.types.binary_product_functor | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_product_iso_prod : binary_product_functor ≅ (prod.functor : Type u ⥤ _) | begin
apply nat_iso.of_components (λ X, _) _,
{ apply nat_iso.of_components (λ Y, _) _,
{ exact ((limit.is_limit _).cone_point_unique_up_to_iso (binary_product_limit X Y)).symm },
{ intros Y₁ Y₂ f,
ext1;
simp } },
{ intros X₁ X₂ g,
ext : 3;
simp }
end | def | category_theory.limits.types.binary_product_iso_prod | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | The product functor given by the instance `has_binary_products (Type u)` is isomorphic to the
explicit binary product functor given by the product type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_coproduct_cocone (X Y : Type u) : cocone (pair X Y) | binary_cofan.mk sum.inl sum.inr | def | category_theory.limits.types.binary_coproduct_cocone | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | The sum type `X ⊕ Y` forms a cocone for the binary coproduct of `X` and `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_coproduct_colimit (X Y : Type u) : is_colimit (binary_coproduct_cocone X Y) | { desc := λ (s : binary_cofan X Y), sum.elim s.inl s.inr,
fac' := λ s j, discrete.rec_on j (λ j, walking_pair.cases_on j rfl rfl),
uniq' := λ s m w, funext $ λ x, sum.cases_on x (congr_fun (w ⟨left⟩)) (congr_fun (w ⟨right⟩)) } | def | category_theory.limits.types.binary_coproduct_colimit | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [
"sum.elim"
] | The sum type `X ⊕ Y` is a binary coproduct for `X` and `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_coproduct_colimit_cocone (X Y : Type u) : limits.colimit_cocone (pair X Y) | ⟨_, binary_coproduct_colimit X Y⟩ | def | category_theory.limits.types.binary_coproduct_colimit_cocone | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | The category of types has `X ⊕ Y`,
as the binary coproduct of `X` and `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_coproduct_iso (X Y : Type u) : limits.coprod X Y ≅ X ⊕ Y | colimit.iso_colimit_cocone (binary_coproduct_colimit_cocone X Y) | def | category_theory.limits.types.binary_coproduct_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | The categorical binary coproduct in `Type u` is the sum `X ⊕ Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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