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values | symbolic_name stringlengths 1 131 | library stringclasses 417
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types_hom {α β : Type u} : (α ⟶ β) = (α → β) | rfl | lemma | category_theory.types_hom | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
types_id (X : Type u) : 𝟙 X = id | rfl | lemma | category_theory.types_id | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
types_comp {X Y Z : Type u} (f : X ⟶ Y) (g : Y ⟶ Z) : f ≫ g = g ∘ f | rfl | lemma | category_theory.types_comp | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
types_id_apply (X : Type u) (x : X) : ((𝟙 X) : X → X) x = x | rfl | lemma | category_theory.types_id_apply | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
types_comp_apply {X Y Z : Type u} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) : (f ≫ g) x = g (f x) | rfl | lemma | category_theory.types_comp_apply | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_inv_id_apply {X Y : Type u} (f : X ≅ Y) (x : X) : f.inv (f.hom x) = x | congr_fun f.hom_inv_id x | lemma | category_theory.hom_inv_id_apply | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_hom_id_apply {X Y : Type u} (f : X ≅ Y) (y : Y) : f.hom (f.inv y) = y | congr_fun f.inv_hom_id y | lemma | category_theory.inv_hom_id_apply | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_hom {α β : Type u} (f : α → β) : α ⟶ β | f | abbreviation | category_theory.as_hom | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sections (F : J ⥤ Type w) : set (Π j, F.obj j) | { u | ∀ {j j'} (f : j ⟶ j'), F.map f (u j) = u j'} | def | category_theory.functor.sections | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | The sections of a functor `J ⥤ Type` are
the choices of a point `u j : F.obj j` for each `j`,
such that `F.map f (u j) = u j` for every morphism `f : j ⟶ j'`.
We later use these to define limits in `Type` and in many concrete categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_comp_apply (f : X ⟶ Y) (g : Y ⟶ Z) (a : F.obj X) :
(F.map (f ≫ g)) a = (F.map g) ((F.map f) a) | by simp [types_comp] | lemma | category_theory.functor_to_types.map_comp_apply | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id_apply (a : F.obj X) : (F.map (𝟙 X)) a = a | by simp [types_id] | lemma | category_theory.functor_to_types.map_id_apply | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
naturality (f : X ⟶ Y) (x : F.obj X) : σ.app Y ((F.map f) x) = (G.map f) (σ.app X x) | congr_fun (σ.naturality f) x | lemma | category_theory.functor_to_types.naturality | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (x : F.obj X) : (σ ≫ τ).app X x = τ.app X (σ.app X x) | rfl | lemma | category_theory.functor_to_types.comp | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hcomp (x : (I ⋙ F).obj W) :
(ρ ◫ σ).app W x = (G.map (ρ.app W)) (σ.app (I.obj W) x) | rfl | lemma | category_theory.functor_to_types.hcomp | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_inv_map_hom_apply (f : X ≅ Y) (x : F.obj X) : F.map f.inv (F.map f.hom x) = x | congr_fun (F.map_iso f).hom_inv_id x | lemma | category_theory.functor_to_types.map_inv_map_hom_apply | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_hom_map_inv_apply (f : X ≅ Y) (y : F.obj Y) : F.map f.hom (F.map f.inv y) = y | congr_fun (F.map_iso f).inv_hom_id y | lemma | category_theory.functor_to_types.map_hom_map_inv_apply | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_inv_id_app_apply (α : F ≅ G) (X) (x) : α.inv.app X (α.hom.app X x) = x | congr_fun (α.hom_inv_id_app X) x | lemma | category_theory.functor_to_types.hom_inv_id_app_apply | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_hom_id_app_apply (α : F ≅ G) (X) (x) : α.hom.app X (α.inv.app X x) = x | congr_fun (α.inv_hom_id_app X) x | lemma | category_theory.functor_to_types.inv_hom_id_app_apply | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ulift_trivial (V : Type u) : ulift.{u} V ≅ V | by tidy | def | category_theory.ulift_trivial | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | The isomorphism between a `Type` which has been `ulift`ed to the same universe,
and the original type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ulift_functor : Type u ⥤ Type (max u v) | { obj := λ X, ulift.{v} X,
map := λ X Y f, λ x : ulift.{v} X, ulift.up (f x.down) } | def | category_theory.ulift_functor | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | The functor embedding `Type u` into `Type (max u v)`.
Write this as `ulift_functor.{5 2}` to get `Type 2 ⥤ Type 5`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ulift_functor_map {X Y : Type u} (f : X ⟶ Y) (x : ulift.{v} X) :
ulift_functor.map f x = ulift.up (f x.down) | rfl | lemma | category_theory.ulift_functor_map | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ulift_functor_full : full.{u} ulift_functor | { preimage := λ X Y f x, (f (ulift.up x)).down } | instance | category_theory.ulift_functor_full | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ulift_functor_faithful : faithful ulift_functor | { map_injective' := λ X Y f g p, funext $ λ x,
congr_arg ulift.down ((congr_fun p (ulift.up x)) : ((ulift.up (f x)) = (ulift.up (g x)))) } | instance | category_theory.ulift_functor_faithful | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ulift_functor_trivial : ulift_functor.{u u} ≅ 𝟭 _ | nat_iso.of_components ulift_trivial (by tidy) | def | category_theory.ulift_functor_trivial | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | The functor embedding `Type u` into `Type u` via `ulift` is isomorphic to the identity functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_of_element {X : Type u} (x : X) : punit ⟶ X | λ _, x | def | category_theory.hom_of_element | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_of_element_eq_iff {X : Type u} (x y : X) :
hom_of_element x = hom_of_element y ↔ x = y | ⟨λ H, congr_fun H punit.star, by cc⟩ | lemma | category_theory.hom_of_element_eq_iff | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_iff_injective {X Y : Type u} (f : X ⟶ Y) : mono f ↔ function.injective f | begin
split,
{ intros H x x' h,
resetI,
rw ←hom_of_element_eq_iff at ⊢ h,
exact (cancel_mono f).mp h },
{ exact λ H, ⟨λ Z, H.comp_left⟩ }
end | lemma | category_theory.mono_iff_injective | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | A morphism in `Type` is a monomorphism if and only if it is injective.
See <https://stacks.math.columbia.edu/tag/003C>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
injective_of_mono {X Y : Type u} (f : X ⟶ Y) [hf : mono f] : function.injective f | (mono_iff_injective f).1 hf | lemma | category_theory.injective_of_mono | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_iff_surjective {X Y : Type u} (f : X ⟶ Y) : epi f ↔ function.surjective f | begin
split,
{ rintros ⟨H⟩,
refine function.surjective_of_right_cancellable_Prop (λ g₁ g₂ hg, _),
rw [← equiv.ulift.symm.injective.comp_left.eq_iff],
apply H,
change ulift.up ∘ (g₁ ∘ f) = ulift.up ∘ (g₂ ∘ f),
rw hg },
{ exact λ H, ⟨λ Z, H.injective_comp_right⟩ }
end | lemma | category_theory.epi_iff_surjective | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [
"function.surjective_of_right_cancellable_Prop"
] | A morphism in `Type` is an epimorphism if and only if it is surjective.
See <https://stacks.math.columbia.edu/tag/003C>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
surjective_of_epi {X Y : Type u} (f : X ⟶ Y) [hf : epi f] : function.surjective f | (epi_iff_surjective f).1 hf | lemma | category_theory.surjective_of_epi | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_type_functor (m : Type u → Type v) [_root_.functor m] [is_lawful_functor m] :
Type u ⥤ Type v | { obj := m,
map := λα β, _root_.functor.map,
map_id' := assume α, _root_.functor.map_id,
map_comp' := assume α β γ f g, funext $ assume a, is_lawful_functor.comp_map f g _ } | def | category_theory.of_type_functor | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | `of_type_functor m` converts from Lean's `Type`-based `category` to `category_theory`. This
allows us to use these functors in category theory. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_type_functor_obj : (of_type_functor m).obj = m | rfl | lemma | category_theory.of_type_functor_obj | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_type_functor_map {α β} (f : α → β) :
(of_type_functor m).map f = (_root_.functor.map f : m α → m β) | rfl | lemma | category_theory.of_type_functor_map | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_iso (e : X ≃ Y) : X ≅ Y | { hom := e.to_fun,
inv := e.inv_fun,
hom_inv_id' := funext e.left_inv,
inv_hom_id' := funext e.right_inv } | def | equiv.to_iso | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | Any equivalence between types in the same universe gives
a categorical isomorphism between those types. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_iso_hom {e : X ≃ Y} : e.to_iso.hom = e | rfl | lemma | equiv.to_iso_hom | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_iso_inv {e : X ≃ Y} : e.to_iso.inv = e.symm | rfl | lemma | equiv.to_iso_inv | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_equiv (i : X ≅ Y) : X ≃ Y | { to_fun := i.hom,
inv_fun := i.inv,
left_inv := λ x, congr_fun i.hom_inv_id x,
right_inv := λ y, congr_fun i.inv_hom_id y } | def | category_theory.iso.to_equiv | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [
"inv_fun"
] | Any isomorphism between types gives an equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_equiv_fun (i : X ≅ Y) : (i.to_equiv : X → Y) = i.hom | rfl | lemma | category_theory.iso.to_equiv_fun | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_equiv_symm_fun (i : X ≅ Y) : (i.to_equiv.symm : Y → X) = i.inv | rfl | lemma | category_theory.iso.to_equiv_symm_fun | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_equiv_id (X : Type u) : (iso.refl X).to_equiv = equiv.refl X | rfl | lemma | category_theory.iso.to_equiv_id | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [
"equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_equiv_comp {X Y Z : Type u} (f : X ≅ Y) (g : Y ≅ Z) :
(f ≪≫ g).to_equiv = f.to_equiv.trans (g.to_equiv) | rfl | lemma | category_theory.iso.to_equiv_comp | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_iff_bijective {X Y : Type u} (f : X ⟶ Y) : is_iso f ↔ function.bijective f | iff.intro
(λ i, (by exactI as_iso f : X ≅ Y).to_equiv.bijective)
(λ b, is_iso.of_iso (equiv.of_bijective f b).to_iso) | lemma | category_theory.is_iso_iff_bijective | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [
"equiv.of_bijective"
] | A morphism in `Type u` is an isomorphism if and only if it is bijective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_iso_iso {X Y : Type u} : (X ≃ Y) ≅ (X ≅ Y) | { hom := λ e, e.to_iso,
inv := λ i, i.to_equiv, } | def | equiv_iso_iso | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | Equivalences (between types in the same universe) are the same as (isomorphic to) isomorphisms
of types. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_equiv_iso {X Y : Type u} : (X ≃ Y) ≃ (X ≅ Y) | (equiv_iso_iso).to_equiv | def | equiv_equiv_iso | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [
"equiv_iso_iso"
] | Equivalences (between types in the same universe) are the same as (equivalent to) isomorphisms
of types. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_equiv_iso_hom {X Y : Type u} (e : X ≃ Y) :
equiv_equiv_iso e = e.to_iso | rfl | lemma | equiv_equiv_iso_hom | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [
"equiv_equiv_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_equiv_iso_inv {X Y : Type u} (e : X ≅ Y) :
equiv_equiv_iso.symm e = e.to_equiv | rfl | lemma | equiv_equiv_iso_inv | category_theory | src/category_theory/types.lean | [
"category_theory.epi_mono",
"category_theory.functor.fully_faithful",
"logic.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_left (F : C ⥤ D) {G H : D ⥤ E} (α : G ⟶ H) : (F ⋙ G) ⟶ (F ⋙ H) | { app := λ X, α.app (F.obj X),
naturality' := λ X Y f, by rw [functor.comp_map, functor.comp_map, α.naturality] } | def | category_theory.whisker_left | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | If `α : G ⟶ H` then
`whisker_left F α : (F ⋙ G) ⟶ (F ⋙ H)` has components `α.app (F.obj X)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whisker_right {G H : C ⥤ D} (α : G ⟶ H) (F : D ⥤ E) : (G ⋙ F) ⟶ (H ⋙ F) | { app := λ X, F.map (α.app X),
naturality' := λ X Y f,
by rw [functor.comp_map, functor.comp_map, ←F.map_comp, ←F.map_comp, α.naturality] } | def | category_theory.whisker_right | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | If `α : G ⟶ H` then
`whisker_right α F : (G ⋙ F) ⟶ (G ⋙ F)` has components `F.map (α.app X)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whiskering_left : (C ⥤ D) ⥤ ((D ⥤ E) ⥤ (C ⥤ E)) | { obj := λ F,
{ obj := λ G, F ⋙ G,
map := λ G H α, whisker_left F α },
map := λ F G τ,
{ app := λ H,
{ app := λ c, H.map (τ.app c),
naturality' := λ X Y f, begin dsimp, rw [←H.map_comp, ←H.map_comp, ←τ.naturality] end },
naturality' := λ X Y f, begin ext, dsimp, rw [f.naturality] end } } | def | category_theory.whiskering_left | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | Left-composition gives a functor `(C ⥤ D) ⥤ ((D ⥤ E) ⥤ (C ⥤ E))`.
`(whiskering_left.obj F).obj G` is `F ⋙ G`, and
`(whiskering_left.obj F).map α` is `whisker_left F α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whiskering_right : (D ⥤ E) ⥤ ((C ⥤ D) ⥤ (C ⥤ E)) | { obj := λ H,
{ obj := λ F, F ⋙ H,
map := λ _ _ α, whisker_right α H },
map := λ G H τ,
{ app := λ F,
{ app := λ c, τ.app (F.obj c),
naturality' := λ X Y f, begin dsimp, rw [τ.naturality] end },
naturality' := λ X Y f, begin ext, dsimp, rw [←nat_trans.naturality] end } } | def | category_theory.whiskering_right | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | Right-composition gives a functor `(D ⥤ E) ⥤ ((C ⥤ D) ⥤ (C ⥤ E))`.
`(whiskering_right.obj H).obj F` is `F ⋙ H`, and
`(whiskering_right.obj H).map α` is `whisker_right α H`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
faithful_whiskering_right_obj {F : D ⥤ E} [faithful F] :
faithful ((whiskering_right C D E).obj F) | { map_injective' := λ G H α β hαβ, nat_trans.ext _ _ $ funext $ λ X,
functor.map_injective _ $ congr_fun (congr_arg nat_trans.app hαβ) X } | instance | category_theory.faithful_whiskering_right_obj | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_left_id (F : C ⥤ D) {G : D ⥤ E} :
whisker_left F (nat_trans.id G) = nat_trans.id (F.comp G) | rfl | lemma | category_theory.whisker_left_id | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_left_id' (F : C ⥤ D) {G : D ⥤ E} :
whisker_left F (𝟙 G) = 𝟙 (F.comp G) | rfl | lemma | category_theory.whisker_left_id' | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_right_id {G : C ⥤ D} (F : D ⥤ E) :
whisker_right (nat_trans.id G) F = nat_trans.id (G.comp F) | ((whiskering_right C D E).obj F).map_id _ | lemma | category_theory.whisker_right_id | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_right_id' {G : C ⥤ D} (F : D ⥤ E) :
whisker_right (𝟙 G) F = 𝟙 (G.comp F) | ((whiskering_right C D E).obj F).map_id _ | lemma | category_theory.whisker_right_id' | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_left_comp (F : C ⥤ D) {G H K : D ⥤ E} (α : G ⟶ H) (β : H ⟶ K) :
whisker_left F (α ≫ β) = (whisker_left F α) ≫ (whisker_left F β) | rfl | lemma | category_theory.whisker_left_comp | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_right_comp {G H K : C ⥤ D} (α : G ⟶ H) (β : H ⟶ K) (F : D ⥤ E) :
whisker_right (α ≫ β) F = (whisker_right α F) ≫ (whisker_right β F) | ((whiskering_right C D E).obj F).map_comp α β | lemma | category_theory.whisker_right_comp | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [
"map_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_whisker_left (F : C ⥤ D) {G H : D ⥤ E} (α : G ≅ H) : (F ⋙ G) ≅ (F ⋙ H) | ((whiskering_left C D E).obj F).map_iso α | def | category_theory.iso_whisker_left | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | If `α : G ≅ H` is a natural isomorphism then
`iso_whisker_left F α : (F ⋙ G) ≅ (F ⋙ H)` has components `α.app (F.obj X)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_whisker_left_hom (F : C ⥤ D) {G H : D ⥤ E} (α : G ≅ H) :
(iso_whisker_left F α).hom = whisker_left F α.hom | rfl | lemma | category_theory.iso_whisker_left_hom | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_whisker_left_inv (F : C ⥤ D) {G H : D ⥤ E} (α : G ≅ H) :
(iso_whisker_left F α).inv = whisker_left F α.inv | rfl | lemma | category_theory.iso_whisker_left_inv | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_whisker_right {G H : C ⥤ D} (α : G ≅ H) (F : D ⥤ E) : (G ⋙ F) ≅ (H ⋙ F) | ((whiskering_right C D E).obj F).map_iso α | def | category_theory.iso_whisker_right | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | If `α : G ≅ H` then
`iso_whisker_right α F : (G ⋙ F) ≅ (H ⋙ F)` has components `F.map_iso (α.app X)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_whisker_right_hom {G H : C ⥤ D} (α : G ≅ H) (F : D ⥤ E) :
(iso_whisker_right α F).hom = whisker_right α.hom F | rfl | lemma | category_theory.iso_whisker_right_hom | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_whisker_right_inv {G H : C ⥤ D} (α : G ≅ H) (F : D ⥤ E) :
(iso_whisker_right α F).inv = whisker_right α.inv F | rfl | lemma | category_theory.iso_whisker_right_inv | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_whisker_left (F : C ⥤ D) {G H : D ⥤ E} (α : G ⟶ H) [is_iso α] :
is_iso (whisker_left F α) | is_iso.of_iso (iso_whisker_left F (as_iso α)) | instance | category_theory.is_iso_whisker_left | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_whisker_right {G H : C ⥤ D} (α : G ⟶ H) (F : D ⥤ E) [is_iso α] :
is_iso (whisker_right α F) | is_iso.of_iso (iso_whisker_right (as_iso α) F) | instance | category_theory.is_iso_whisker_right | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_left_twice (F : B ⥤ C) (G : C ⥤ D) {H K : D ⥤ E} (α : H ⟶ K) :
whisker_left F (whisker_left G α) = whisker_left (F ⋙ G) α | rfl | lemma | category_theory.whisker_left_twice | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_right_twice {H K : B ⥤ C} (F : C ⥤ D) (G : D ⥤ E) (α : H ⟶ K) :
whisker_right (whisker_right α F) G = whisker_right α (F ⋙ G) | rfl | lemma | category_theory.whisker_right_twice | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_right_left (F : B ⥤ C) {G H : C ⥤ D} (α : G ⟶ H) (K : D ⥤ E) :
whisker_right (whisker_left F α) K = whisker_left F (whisker_right α K) | rfl | lemma | category_theory.whisker_right_left | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_unitor (F : A ⥤ B) : ((𝟭 A) ⋙ F) ≅ F | { hom := { app := λ X, 𝟙 (F.obj X) },
inv := { app := λ X, 𝟙 (F.obj X) } } | def | category_theory.functor.left_unitor | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | The left unitor, a natural isomorphism `((𝟭 _) ⋙ F) ≅ F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_unitor (F : A ⥤ B) : (F ⋙ (𝟭 B)) ≅ F | { hom := { app := λ X, 𝟙 (F.obj X) },
inv := { app := λ X, 𝟙 (F.obj X) } } | def | category_theory.functor.right_unitor | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | The right unitor, a natural isomorphism `(F ⋙ (𝟭 B)) ≅ F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associator (F : A ⥤ B) (G : B ⥤ C) (H : C ⥤ D) : ((F ⋙ G) ⋙ H) ≅ (F ⋙ (G ⋙ H)) | { hom := { app := λ _, 𝟙 _ },
inv := { app := λ _, 𝟙 _ } } | def | category_theory.functor.associator | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | The associator for functors, a natural isomorphism `((F ⋙ G) ⋙ H) ≅ (F ⋙ (G ⋙ H))`.
(In fact, `iso.refl _` will work here, but it tends to make Lean slow later,
and it's usually best to insert explicit associators.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
assoc (F : A ⥤ B) (G : B ⥤ C) (H : C ⥤ D) : ((F ⋙ G) ⋙ H) = (F ⋙ (G ⋙ H)) | rfl | lemma | category_theory.functor.assoc | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
triangle (F : A ⥤ B) (G : B ⥤ C) :
(associator F (𝟭 B) G).hom ≫ (whisker_left F (left_unitor G).hom) =
(whisker_right (right_unitor F).hom G) | by { ext, dsimp, simp } | lemma | category_theory.functor.triangle | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pentagon :
(whisker_right (associator F G H).hom K) ≫
(associator F (G ⋙ H) K).hom ≫
(whisker_left F (associator G H K).hom) =
((associator (F ⋙ G) H K).hom ≫ (associator F G (H ⋙ K)).hom) | by { ext, dsimp, simp } | lemma | category_theory.functor.pentagon | category_theory | src/category_theory/whiskering.lean | [
"category_theory.isomorphism",
"category_theory.functor.category",
"category_theory.functor.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_terminal : Type u
| of : C → with_terminal
| star : with_terminal | inductive | category_theory.with_terminal | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [] | Formally adjoin a terminal object to a category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_initial : Type u
| of : C → with_initial
| star : with_initial | inductive | category_theory.with_initial | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [] | Formally adjoin an initial object to a category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom : with_terminal C → with_terminal C → Type v | | (of X) (of Y) := X ⟶ Y
| star (of X) := pempty
| _ star := punit | def | category_theory.with_terminal.hom | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [
"pempty"
] | Morphisms for `with_terminal C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id : Π (X : with_terminal C), hom X X | | (of X) := 𝟙 _
| star := punit.star | def | category_theory.with_terminal.id | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [] | Identity morphisms for `with_terminal C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp : Π {X Y Z : with_terminal C}, hom X Y → hom Y Z → hom X Z | | (of X) (of Y) (of Z) := λ f g, f ≫ g
| (of X) _ star := λ f g, punit.star
| star (of X) _ := λ f g, pempty.elim f
| _ star (of Y) := λ f g, pempty.elim g
| star star star := λ _ _, punit.star | def | category_theory.with_terminal.comp | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [
"pempty.elim"
] | Composition of morphisms for `with_terminal C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
incl : C ⥤ (with_terminal C) | { obj := of,
map := λ X Y f, f } | def | category_theory.with_terminal.incl | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [] | The inclusion from `C` into `with_terminal C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map {D : Type*} [category D] (F : C ⥤ D) : with_terminal C ⥤ with_terminal D | { obj := λ X,
match X with
| of x := of $ F.obj x
| star := star
end,
map := λ X Y f,
match X, Y, f with
| of x, of y, f := F.map f
| of x, star, punit.star := punit.star
| star, star, punit.star := punit.star
end } | def | category_theory.with_terminal.map | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [] | Map `with_terminal` with respect to a functor `F : C ⥤ D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_terminal : limits.is_terminal (star : with_terminal C) | limits.is_terminal.of_unique _ | def | category_theory.with_terminal.star_terminal | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [] | `with_terminal.star` is terminal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift {D : Type*} [category D] {Z : D} (F : C ⥤ D) (M : Π (x : C), F.obj x ⟶ Z)
(hM : ∀ (x y : C) (f : x ⟶ y), F.map f ≫ M y = M x) :
(with_terminal C) ⥤ D | { obj := λ X,
match X with
| of x := F.obj x
| star := Z
end,
map := λ X Y f,
match X, Y, f with
| of x, of y, f := F.map f
| of x, star, punit.star := M x
| star, star, punit.star := 𝟙 Z
end } | def | category_theory.with_terminal.lift | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [
"lift"
] | Lift a functor `F : C ⥤ D` to `with_term C ⥤ D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
incl_lift {D : Type*} [category D] {Z : D} (F : C ⥤ D) (M : Π (x : C), F.obj x ⟶ Z)
(hM : ∀ (x y : C) (f : x ⟶ y), F.map f ≫ M y = M x) :
incl ⋙ lift F M hM ≅ F | { hom := { app := λ X, 𝟙 _ },
inv := { app := λ X, 𝟙 _ } } | def | category_theory.with_terminal.incl_lift | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [
"lift"
] | The isomorphism between `incl ⋙ lift F _ _` with `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_star {D : Type*} [category D] {Z : D} (F : C ⥤ D) (M : Π (x : C), F.obj x ⟶ Z)
(hM : ∀ (x y : C) (f : x ⟶ y), F.map f ≫ M y = M x) :
(lift F M hM).obj star ≅ Z | eq_to_iso rfl | def | category_theory.with_terminal.lift_star | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [
"lift"
] | The isomorphism between `(lift F _ _).obj with_terminal.star` with `Z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_map_lift_star {D : Type*} [category D] {Z : D} (F : C ⥤ D) (M : Π (x : C), F.obj x ⟶ Z)
(hM : ∀ (x y : C) (f : x ⟶ y), F.map f ≫ M y = M x) (x : C) :
(lift F M hM).map (star_terminal.from (incl.obj x)) ≫ (lift_star F M hM).hom =
(incl_lift F M hM).hom.app x ≫ M x | begin
erw [category.id_comp, category.comp_id],
refl,
end | lemma | category_theory.with_terminal.lift_map_lift_star | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_unique {D : Type*} [category D] {Z : D} (F : C ⥤ D)
(M : Π (x : C), F.obj x ⟶ Z) (hM : ∀ (x y : C) (f : x ⟶ y), F.map f ≫ M y = M x)
(G : with_terminal C ⥤ D) (h : incl ⋙ G ≅ F) (hG : G.obj star ≅ Z)
(hh : ∀ x : C, G.map (star_terminal.from (incl.obj x)) ≫ hG.hom = h.hom.app x ≫ M x) :
G ≅ lift F M hM | nat_iso.of_components (λ X,
match X with
| of x := h.app x
| star := hG
end)
begin
rintro (X|X) (Y|Y) f,
{ apply h.hom.naturality },
{ cases f, exact hh _ },
{ cases f, },
{ cases f,
change G.map (𝟙 _) ≫ hG.hom = hG.hom ≫ 𝟙 _,
simp }
end | def | category_theory.with_terminal.lift_unique | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [
"lift",
"lift_unique"
] | The uniqueness of `lift`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_to_terminal {D : Type*} [category D] {Z : D} (F : C ⥤ D) (hZ : limits.is_terminal Z) :
with_terminal C ⥤ D | lift F (λ x, hZ.from _) (λ x y f, hZ.hom_ext _ _) | def | category_theory.with_terminal.lift_to_terminal | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [
"lift"
] | A variant of `lift` with `Z` a terminal object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
incl_lift_to_terminal {D : Type*} [category D] {Z : D} (F : C ⥤ D) (hZ : limits.is_terminal Z) :
incl ⋙ lift_to_terminal F hZ ≅ F | incl_lift _ _ _ | def | category_theory.with_terminal.incl_lift_to_terminal | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [] | A variant of `incl_lift` with `Z` a terminal object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_to_terminal_unique {D : Type*} [category D] {Z : D} (F : C ⥤ D) (hZ : limits.is_terminal Z)
(G : with_terminal C ⥤ D) (h : incl ⋙ G ≅ F) (hG : G.obj star ≅ Z) :
G ≅ lift_to_terminal F hZ | lift_unique F (λ z, hZ.from _) (λ x y f, hZ.hom_ext _ _) G h hG (λ x, hZ.hom_ext _ _) | def | category_theory.with_terminal.lift_to_terminal_unique | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [
"lift_unique"
] | A variant of `lift_unique` with `Z` a terminal object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_from (X : C) : incl.obj X ⟶ star | star_terminal.from _ | def | category_theory.with_terminal.hom_from | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [] | Constructs a morphism to `star` from `of X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_of_from_star {X : with_terminal C} (f : star ⟶ X) : is_iso f | by tidy | instance | category_theory.with_terminal.is_iso_of_from_star | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom : with_initial C → with_initial C → Type v | | (of X) (of Y) := X ⟶ Y
| (of X) _ := pempty
| star _ := punit | def | category_theory.with_initial.hom | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [
"pempty"
] | Morphisms for `with_initial C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id : Π (X : with_initial C), hom X X | | (of X) := 𝟙 _
| star := punit.star | def | category_theory.with_initial.id | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [] | Identity morphisms for `with_initial C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp : Π {X Y Z : with_initial C}, hom X Y → hom Y Z → hom X Z | | (of X) (of Y) (of Z) := λ f g, f ≫ g
| star _ (of X) := λ f g, punit.star
| _ (of X) star := λ f g, pempty.elim g
| (of Y) star _ := λ f g, pempty.elim f
| star star star := λ _ _, punit.star | def | category_theory.with_initial.comp | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [
"pempty.elim"
] | Composition of morphisms for `with_initial C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
incl : C ⥤ (with_initial C) | { obj := of,
map := λ X Y f, f } | def | category_theory.with_initial.incl | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [] | The inclusion of `C` into `with_initial C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map {D : Type*} [category D] (F : C ⥤ D) : with_initial C ⥤ with_initial D | { obj := λ X,
match X with
| of x := of $ F.obj x
| star := star
end,
map := λ X Y f,
match X, Y, f with
| of x, of y, f := F.map f
| star, of x, punit.star := punit.star
| star, star, punit.star := punit.star
end } | def | category_theory.with_initial.map | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [] | Map `with_initial` with respect to a functor `F : C ⥤ D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_initial : limits.is_initial (star : with_initial C) | limits.is_initial.of_unique _ | def | category_theory.with_initial.star_initial | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [] | `with_initial.star` is initial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift {D : Type*} [category D] {Z : D} (F : C ⥤ D) (M : Π (x : C), Z ⟶ F.obj x)
(hM : ∀ (x y : C) (f : x ⟶ y), M x ≫ F.map f = M y) :
(with_initial C) ⥤ D | { obj := λ X,
match X with
| of x := F.obj x
| star := Z
end,
map := λ X Y f,
match X, Y, f with
| of x, of y, f := F.map f
| star, of x, punit.star := M _
| star, star, punit.star := 𝟙 _
end } | def | category_theory.with_initial.lift | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [
"lift"
] | Lift a functor `F : C ⥤ D` to `with_initial C ⥤ D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
incl_lift {D : Type*} [category D] {Z : D} (F : C ⥤ D)
(M : Π (x : C), Z ⟶ F.obj x) (hM : ∀ (x y : C) (f : x ⟶ y), M x ≫ F.map f = M y) :
incl ⋙ lift F M hM ≅ F | { hom := { app := λ X, 𝟙 _ },
inv := { app := λ X, 𝟙 _ } } | def | category_theory.with_initial.incl_lift | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [
"lift"
] | The isomorphism between `incl ⋙ lift F _ _` with `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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