mathlib documentation

category_theory.limits.shapes.types

Special shapes for limits in Type. #

The general shape (co)limits defined in category_theory.limits.types are intended for use through the limits API, and the actual implementation should mostly be considered "sealed".

In this file, we provide definitions of the "standard" special shapes of limits in Type, giving the expected definitional implementation:

We first construct terms of is_limit and limit_cone, and then provide isomorphisms with the types generated by the has_limit API.

As an example, when setting up the monoidal category structure on Type we use the types_has_terminal and types_has_binary_products instances.

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@[simp]
theorem category_theory.limits.types.pi_lift_π_apply {β : Type u} (f : β → Type u) {P : Type u} (s : Π (b : β), P ⟶ f b) (b : β) (x : P) :
category_theory.limits.pi.π f b (category_theory.limits.pi.lift s x) = s b x

A restatement of types.lift_π_apply that uses pi.π and pi.lift.

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theorem category_theory.limits.types.pi_map_π_apply {β : Type u} {f g : β → Type u} (α : Π (j : β), f j ⟶ g j) (b : β) (x : ∏ λ (j : β), f j) :
category_theory.limits.pi.π g b (category_theory.limits.pi.map α x) = α b (category_theory.limits.pi.π f b x)

A restatement of types.map_π_apply that uses pi.π and pi.map.

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def category_theory.limits.types.terminal_limit_cone  :
category_theory.limits.limit_cone (category_theory.functor.empty (Type u))

The category of types has punit as a terminal object.

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noncomputable def category_theory.limits.types.terminal_iso  :
⊤_ Type u ≅ punit

The terminal object in Type u is punit.

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noncomputable def category_theory.limits.types.is_terminal_punit  :
category_theory.limits.is_terminal punit

The terminal object in Type u is punit.

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def category_theory.limits.types.initial_colimit_cocone  :
category_theory.limits.colimit_cocone (category_theory.functor.empty (Type u))

The category of types has pempty as an initial object.

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noncomputable def category_theory.limits.types.initial_iso  :
⊥_ Type u ≅ pempty

The initial object in Type u is pempty.

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noncomputable def category_theory.limits.types.is_initial_punit  :
category_theory.limits.is_initial pempty

The initial object in Type u is pempty.

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def category_theory.limits.types.binary_product_cone (X Y : Type u) :
category_theory.limits.binary_fan X Y

The product type X × Y forms a cone for the binary product of X and Y.

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theorem category_theory.limits.types.binary_product_cone_X (X Y : Type u) :
(category_theory.limits.types.binary_product_cone X Y).X = (X × Y)
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theorem category_theory.limits.types.binary_product_cone_fst (X Y : Type u) :
(category_theory.limits.types.binary_product_cone X Y).fst = prod.fst
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theorem category_theory.limits.types.binary_product_cone_snd (X Y : Type u) :
(category_theory.limits.types.binary_product_cone X Y).snd = prod.snd
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theorem category_theory.limits.types.binary_product_limit_lift (X Y : Type u) (s : category_theory.limits.binary_fan X Y) (x : s.X) :
(category_theory.limits.types.binary_product_limit X Y).lift s x = (s.fst x, s.snd x)
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def category_theory.limits.types.binary_product_limit (X Y : Type u) :
category_theory.limits.is_limit (category_theory.limits.types.binary_product_cone X Y)

The product type X × Y is a binary product for X and Y.

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def category_theory.limits.types.binary_product_limit_cone (X Y : Type u) :
category_theory.limits.limit_cone (category_theory.limits.pair X Y)

The category of types has X × Y, the usual cartesian product, as the binary product of X and Y.

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theorem category_theory.limits.types.binary_product_limit_cone_cone (X Y : Type u) :
(category_theory.limits.types.binary_product_limit_cone X Y).cone = category_theory.limits.types.binary_product_cone X Y
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theorem category_theory.limits.types.binary_product_limit_cone_is_limit (X Y : Type u) :
(category_theory.limits.types.binary_product_limit_cone X Y).is_limit = category_theory.limits.types.binary_product_limit X Y
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noncomputable def category_theory.limits.types.binary_product_iso (X Y : Type u) :
X ⨯ Y ≅ X × Y

The categorical binary product in Type u is cartesian product.

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theorem category_theory.limits.types.binary_product_iso_hom_comp_fst_apply (X Y : Type u) (x : ↥(X ⨯ Y)) :
((category_theory.limits.types.binary_product_iso X Y).hom x).fst = category_theory.limits.prod.fst x
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theorem category_theory.limits.types.binary_product_iso_hom_comp_fst (X Y : Type u) :
(category_theory.limits.types.binary_product_iso X Y).hom ≫ prod.fst = category_theory.limits.prod.fst
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theorem category_theory.limits.types.binary_product_iso_hom_comp_snd (X Y : Type u) :
(category_theory.limits.types.binary_product_iso X Y).hom ≫ prod.snd = category_theory.limits.prod.snd
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theorem category_theory.limits.types.binary_product_iso_hom_comp_snd_apply (X Y : Type u) (x : ↥(X ⨯ Y)) :
((category_theory.limits.types.binary_product_iso X Y).hom x).snd = category_theory.limits.prod.snd x
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theorem category_theory.limits.types.binary_product_iso_inv_comp_fst_apply (X Y : Type u) (x : ↥(X × Y)) :
category_theory.limits.prod.fst ((category_theory.limits.types.binary_product_iso X Y).inv x) = x.fst
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theorem category_theory.limits.types.binary_product_iso_inv_comp_fst (X Y : Type u) :
(category_theory.limits.types.binary_product_iso X Y).inv ≫ category_theory.limits.prod.fst = prod.fst
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theorem category_theory.limits.types.binary_product_iso_inv_comp_snd (X Y : Type u) :
(category_theory.limits.types.binary_product_iso X Y).inv ≫ category_theory.limits.prod.snd = prod.snd
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theorem category_theory.limits.types.binary_product_iso_inv_comp_snd_apply (X Y : Type u) (x : ↥(X × Y)) :
category_theory.limits.prod.snd ((category_theory.limits.types.binary_product_iso X Y).inv x) = x.snd
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def category_theory.limits.types.binary_product_functor  :
Type u ⥤ Type u ⥤ Type u

The functor which sends X, Y to the product type X × Y.

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theorem category_theory.limits.types.binary_product_functor_obj_obj (X Y : Type u) :
(category_theory.limits.types.binary_product_functor.obj X).obj Y = (X × Y)
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theorem category_theory.limits.types.binary_product_functor_map_app (X₁ X₂ : Type u) (f : X₁ ⟶ X₂) (Y : Type u) :
(category_theory.limits.types.binary_product_functor.map f).app Y = (category_theory.limits.types.binary_product_limit X₂ Y).lift (category_theory.limits.binary_fan.mk (prod.fst ≫ f) prod.snd)
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theorem category_theory.limits.types.binary_product_functor_obj_map (X Y₁ Y₂ : Type u) (f : Y₁ ⟶ Y₂) :
(category_theory.limits.types.binary_product_functor.obj X).map f = (category_theory.limits.types.binary_product_limit X Y₂).lift (category_theory.limits.binary_fan.mk prod.fst (prod.snd ≫ f))
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noncomputable def category_theory.limits.types.binary_product_iso_prod  :
category_theory.limits.types.binary_product_functor ≅ category_theory.limits.prod.functor

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.

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theorem category_theory.limits.types.binary_coproduct_cocone_X (X Y : Type u) :
(category_theory.limits.types.binary_coproduct_cocone X Y).X = (X ⊕ Y)
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def category_theory.limits.types.binary_coproduct_cocone (X Y : Type u) :
category_theory.limits.cocone (category_theory.limits.pair X Y)

The sum type X ⊕ Y forms a cocone for the binary coproduct of X and Y.

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theorem category_theory.limits.types.binary_coproduct_cocone_ι_app (X Y : Type u) (j : category_theory.discrete category_theory.limits.walking_pair) (ᾰ : (category_theory.limits.pair X Y).obj j) :
(category_theory.limits.types.binary_coproduct_cocone X Y).ι.app j ᾰ = category_theory.discrete.rec (category_theory.limits.walking_pair.rec sum.inl sum.inr) j ᾰ
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def category_theory.limits.types.binary_coproduct_colimit (X Y : Type u) :
category_theory.limits.is_colimit (category_theory.limits.types.binary_coproduct_cocone X Y)

The sum type X ⊕ Y is a binary coproduct for X and Y.

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theorem category_theory.limits.types.binary_coproduct_colimit_desc (X Y : Type u) (s : category_theory.limits.binary_cofan X Y) (ᾰ : X ⊕ Y) :
(category_theory.limits.types.binary_coproduct_colimit X Y).desc s ᾰ = sum.elim s.inl s.inr ᾰ
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def category_theory.limits.types.binary_coproduct_colimit_cocone (X Y : Type u) :
category_theory.limits.colimit_cocone (category_theory.limits.pair X Y)

The category of types has X ⊕ Y, as the binary coproduct of X and Y.

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noncomputable def category_theory.limits.types.binary_coproduct_iso (X Y : Type u) :
X ⨿ Y ≅ X ⊕ Y

The categorical binary coproduct in Type u is the sum X ⊕ Y.

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theorem category_theory.limits.types.binary_coproduct_iso_inl_comp_hom_apply (X Y : Type u) (x : ↥X) :
(category_theory.limits.types.binary_coproduct_iso X Y).hom (category_theory.limits.coprod.inl x) = sum.inl x
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theorem category_theory.limits.types.binary_coproduct_iso_inl_comp_hom (X Y : Type u) :
category_theory.limits.coprod.inl ≫ (category_theory.limits.types.binary_coproduct_iso X Y).hom = sum.inl
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theorem category_theory.limits.types.binary_coproduct_iso_inr_comp_hom_apply (X Y : Type u) (x : ↥Y) :
(category_theory.limits.types.binary_coproduct_iso X Y).hom (category_theory.limits.coprod.inr x) = sum.inr x
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theorem category_theory.limits.types.binary_coproduct_iso_inr_comp_hom (X Y : Type u) :
category_theory.limits.coprod.inr ≫ (category_theory.limits.types.binary_coproduct_iso X Y).hom = sum.inr
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theorem category_theory.limits.types.binary_coproduct_iso_inl_comp_inv (X Y : Type u) :
category_theory.as_hom sum.inl ≫ (category_theory.limits.types.binary_coproduct_iso X Y).inv = category_theory.limits.coprod.inl
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theorem category_theory.limits.types.binary_coproduct_iso_inl_comp_inv_apply (X Y : Type u) (x : ↥X) :
(category_theory.limits.types.binary_coproduct_iso X Y).inv (category_theory.as_hom sum.inl x) = category_theory.limits.coprod.inl x
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theorem category_theory.limits.types.binary_coproduct_iso_inr_comp_inv_apply (X Y : Type u) (x : ↥Y) :
(category_theory.limits.types.binary_coproduct_iso X Y).inv (category_theory.as_hom sum.inr x) = category_theory.limits.coprod.inr x
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theorem category_theory.limits.types.binary_coproduct_iso_inr_comp_inv (X Y : Type u) :
category_theory.as_hom sum.inr ≫ (category_theory.limits.types.binary_coproduct_iso X Y).inv = category_theory.limits.coprod.inr
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theorem category_theory.limits.types.binary_cofan_is_colimit_iff {X Y : Type u} (c : category_theory.limits.binary_cofan X Y) :
nonempty (category_theory.limits.is_colimit c) ↔ function.injective c.inl ∧ function.injective c.inr ∧ is_compl (set.range c.inl) (set.range c.inr)
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noncomputable def category_theory.limits.types.is_coprod_of_mono {X Y : Type u} (f : X ⟶ Y) [category_theory.mono f] :
category_theory.limits.is_colimit (category_theory.limits.binary_cofan.mk f subtype.val)

Any monomorphism in Type is an coproduct injection.

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def category_theory.limits.types.product_limit_cone {J : Type u} (F : J → Type (max u v)) :
category_theory.limits.limit_cone (category_theory.discrete.functor F)

The category of types has Π j, f j as the product of a type family f : J → Type.

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noncomputable def category_theory.limits.types.product_iso {J : Type u} (F : J → Type (max u v)) :
∏ F ≅ Π (j : J), F j

The categorical product in Type u is the type theoretic product Π j, F j.

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theorem category_theory.limits.types.product_iso_hom_comp_eval {J : Type u} (F : J → Type (max u v)) (j : J) :
((category_theory.limits.types.product_iso F).hom ≫ λ (f : Π (j : J), F j), f j) = category_theory.limits.pi.π F j
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theorem category_theory.limits.types.product_iso_hom_comp_eval_apply {J : Type u} (F : J → Type (max u v)) (j : J) (x : ↥∏ F) :
(category_theory.limits.types.product_iso F).hom x j = category_theory.limits.pi.π F j x
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theorem category_theory.limits.types.product_iso_inv_comp_π {J : Type u} (F : J → Type (max u v)) (j : J) :
(category_theory.limits.types.product_iso F).inv ≫ category_theory.limits.pi.π F j = λ (f : Π (j : J), F j), f j
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theorem category_theory.limits.types.product_iso_inv_comp_π_apply {J : Type u} (F : J → Type (max u v)) (j : J) (x : ↥(Π (j : J), F j)) :
category_theory.limits.pi.π F j ((category_theory.limits.types.product_iso F).inv x) = x j
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def category_theory.limits.types.coproduct_colimit_cocone {J : Type u} (F : J → Type u) :
category_theory.limits.colimit_cocone (category_theory.discrete.functor F)

The category of types has Σ j, f j as the coproduct of a type family f : J → Type.

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noncomputable def category_theory.limits.types.coproduct_iso {J : Type u} (F : J → Type u) :
∐ F ≅ Σ (j : J), F j

The categorical coproduct in Type u is the type theoretic coproduct Σ j, F j.

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theorem category_theory.limits.types.coproduct_iso_ι_comp_hom_apply {J : Type u} (F : J → Type u) (j : J) (x : ↥(F j)) :
(category_theory.limits.types.coproduct_iso F).hom (category_theory.limits.sigma.ι F j x) = ⟨j, x⟩
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theorem category_theory.limits.types.coproduct_iso_ι_comp_hom {J : Type u} (F : J → Type u) (j : J) :
category_theory.limits.sigma.ι F j ≫ (category_theory.limits.types.coproduct_iso F).hom = λ (x : F j), ⟨j, x⟩
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theorem category_theory.limits.types.coproduct_iso_mk_comp_inv_apply {J : Type u} (F : J → Type u) (j : J) (x : ↥(F j)) :
(category_theory.limits.types.coproduct_iso F).inv (category_theory.as_hom (sigma.mk j) x) = category_theory.limits.sigma.ι F j x
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theorem category_theory.limits.types.coproduct_iso_mk_comp_inv {J : Type u} (F : J → Type u) (j : J) :
category_theory.as_hom (λ (x : F j), ⟨j, x⟩) ≫ (category_theory.limits.types.coproduct_iso F).inv = category_theory.limits.sigma.ι F j
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noncomputable def category_theory.limits.types.type_equalizer_of_unique {X Y Z : Type u} (f : X ⟶ Y) {g h : Y ⟶ Z} (w : f ≫ g = f ≫ h) (t : ∀ (y : Y), g y = h y → (∃! (x : X), f x = y)) :
category_theory.limits.is_limit (category_theory.limits.fork.of_ι f w)

Show the given fork in Type u is an equalizer given that any element in the "difference kernel" comes from X. The converse of unique_of_type_equalizer.

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theorem category_theory.limits.types.unique_of_type_equalizer {X Y Z : Type u} (f : X ⟶ Y) {g h : Y ⟶ Z} (w : f ≫ g = f ≫ h) (t : category_theory.limits.is_limit (category_theory.limits.fork.of_ι f w)) (y : Y) (hy : g y = h y) :
∃! (x : X), f x = y

The converse of type_equalizer_of_unique.

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theorem category_theory.limits.types.type_equalizer_iff_unique {X Y Z : Type u} (f : X ⟶ Y) {g h : Y ⟶ Z} (w : f ≫ g = f ≫ h) :
nonempty (category_theory.limits.is_limit (category_theory.limits.fork.of_ι f w)) ↔ ∀ (y : Y), g y = h y → (∃! (x : X), f x = y)
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def category_theory.limits.types.equalizer_limit {Y Z : Type u} {g h : Y ⟶ Z} :
category_theory.limits.limit_cone (category_theory.limits.parallel_pair g h)

Show that the subtype {x : Y // g x = h x} is an equalizer for the pair (g,h).

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noncomputable def category_theory.limits.types.equalizer_iso {Y Z : Type u} (g h : Y ⟶ Z) :
category_theory.limits.equalizer g h ≅ {x // g x = h x}

The categorical equalizer in Type u is {x : Y // g x = h x}.

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theorem category_theory.limits.types.equalizer_iso_hom_comp_subtype {Y Z : Type u} (g h : Y ⟶ Z) :
(category_theory.limits.types.equalizer_iso g h).hom ≫ subtype.val = category_theory.limits.equalizer.ι g h
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theorem category_theory.limits.types.equalizer_iso_hom_comp_subtype_apply {Y Z : Type u} (g h : Y ⟶ Z) (x : ↥(category_theory.limits.equalizer g h)) :
((category_theory.limits.types.equalizer_iso g h).hom x).val = category_theory.limits.equalizer.ι g h x
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theorem category_theory.limits.types.equalizer_iso_inv_comp_ι {Y Z : Type u} (g h : Y ⟶ Z) :
(category_theory.limits.types.equalizer_iso g h).inv ≫ category_theory.limits.equalizer.ι g h = subtype.val
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theorem category_theory.limits.types.equalizer_iso_inv_comp_ι_apply {Y Z : Type u} (g h : Y ⟶ Z) (x : ↥{x // g x = h x}) :
category_theory.limits.equalizer.ι g h ((category_theory.limits.types.equalizer_iso g h).inv x) = x.val
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inductive category_theory.limits.types.coequalizer_rel {X Y : Type u} (f g : X ⟶ Y) :
Y → Y → Prop

(Implementation) The relation to be quotiented to obtain the coequalizer.

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def category_theory.limits.types.coequalizer_colimit {X Y : Type u} (f g : X ⟶ Y) :
category_theory.limits.colimit_cocone (category_theory.limits.parallel_pair f g)

Show that the quotient by the relation generated by f(x) ~ g(x) is a coequalizer for the pair (f, g).

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theorem category_theory.limits.types.coequalizer_preimage_image_eq_of_preimage_eq {X Y Z : Type u} (f g : X ⟶ Y) (π : Y ⟶ Z) (e : f ≫ π = g ≫ π) (h : category_theory.limits.is_colimit (category_theory.limits.cofork.of_π π e)) (U : set Y) (H : f ⁻¹' U = g ⁻¹' U) :
π ⁻¹' (π '' U) = U

If π : Y ⟶ Z is an equalizer for (f, g), and U ⊆ Y such that f ⁻¹' U = g ⁻¹' U, then π ⁻¹' (π '' U) = U.

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noncomputable def category_theory.limits.types.coequalizer_iso {X Y : Type u} (f g : X ⟶ Y) :
category_theory.limits.coequalizer f g ≅ quot (category_theory.limits.types.coequalizer_rel f g)

The categorical coequalizer in Type u is the quotient by f g ~ g x.

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theorem category_theory.limits.types.coequalizer_iso_π_comp_hom_apply {X Y : Type u} (f g : X ⟶ Y) (x : ↥Y) :
(category_theory.limits.types.coequalizer_iso f g).hom (category_theory.limits.coequalizer.π f g x) = quot.mk (category_theory.limits.types.coequalizer_rel f g) x
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@[simp]
theorem category_theory.limits.types.coequalizer_iso_π_comp_hom {X Y : Type u} (f g : X ⟶ Y) :
category_theory.limits.coequalizer.π f g ≫ (category_theory.limits.types.coequalizer_iso f g).hom = quot.mk (category_theory.limits.types.coequalizer_rel f g)
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@[simp]
theorem category_theory.limits.types.coequalizer_iso_quot_comp_inv_apply {X Y : Type u} (f g : X ⟶ Y) (x : ↥Y) :
(category_theory.limits.types.coequalizer_iso f g).inv (category_theory.as_hom (quot.mk (category_theory.limits.types.coequalizer_rel f g)) x) = category_theory.limits.coequalizer.π f g x
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@[simp]
theorem category_theory.limits.types.coequalizer_iso_quot_comp_inv {X Y : Type u} (f g : X ⟶ Y) :
category_theory.as_hom (quot.mk (category_theory.limits.types.coequalizer_rel f g)) ≫ (category_theory.limits.types.coequalizer_iso f g).inv = category_theory.limits.coequalizer.π f g
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@[nolint, reducible]
def category_theory.limits.types.pullback_obj {X Y Z : Type u} (f : X ⟶ Z) (g : Y ⟶ Z) :
Type u

The usual explicit pullback in the category of types, as a subtype of the product. The full limit_cone data is bundled as pullback_limit_cone f g.

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@[reducible]
def category_theory.limits.types.pullback_cone {X Y Z : Type u} (f : X ⟶ Z) (g : Y ⟶ Z) :
category_theory.limits.pullback_cone f g

The explicit pullback cone on pullback_obj f g. This is bundled with the is_limit data as pullback_limit_cone f g.

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def category_theory.limits.types.pullback_limit_cone {X Y Z : Type u} (f : X ⟶ Z) (g : Y ⟶ Z) :
category_theory.limits.limit_cone (category_theory.limits.cospan f g)

The explicit pullback in the category of types, bundled up as a limit_cone for given f and g.

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@[simp]
theorem category_theory.limits.types.pullback_limit_cone_is_limit {X Y Z : Type u} (f : X ⟶ Z) (g : Y ⟶ Z) :
(category_theory.limits.types.pullback_limit_cone f g).is_limit = (category_theory.limits.types.pullback_cone f g).is_limit_aux (λ (s : category_theory.limits.pullback_cone f g) (x : s.X), ⟨(s.fst x, s.snd x), _⟩) _ _ _
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@[simp]
theorem category_theory.limits.types.pullback_limit_cone_cone {X Y Z : Type u} (f : X ⟶ Z) (g : Y ⟶ Z) :
(category_theory.limits.types.pullback_limit_cone f g).cone = category_theory.limits.types.pullback_cone f g
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noncomputable def category_theory.limits.types.pullback_cone_iso_pullback {X Y Z : Type u} (f : X ⟶ Z) (g : Y ⟶ Z) :
category_theory.limits.limit.cone (category_theory.limits.cospan f g) ≅ category_theory.limits.types.pullback_cone f g

The pullback cone given by the instance has_pullbacks (Type u) is isomorphic to the explicit pullback cone given by pullback_limit_cone.

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noncomputable def category_theory.limits.types.pullback_iso_pullback {X Y Z : Type u} (f : X ⟶ Z) (g : Y ⟶ Z) :
category_theory.limits.pullback f g ≅ category_theory.limits.types.pullback_obj f g

The pullback given by the instance has_pullbacks (Type u) is isomorphic to the explicit pullback object given by pullback_limit_obj.

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@[simp]
theorem category_theory.limits.types.pullback_iso_pullback_hom_fst {X Y Z : Type u} (f : X ⟶ Z) (g : Y ⟶ Z) (p : category_theory.limits.pullback f g) :
↑((category_theory.limits.types.pullback_iso_pullback f g).hom p).fst = category_theory.limits.pullback.fst p
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@[simp]
theorem category_theory.limits.types.pullback_iso_pullback_hom_snd {X Y Z : Type u} (f : X ⟶ Z) (g : Y ⟶ Z) (p : category_theory.limits.pullback f g) :
↑((category_theory.limits.types.pullback_iso_pullback f g).hom p).snd = category_theory.limits.pullback.snd p
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@[simp]
theorem category_theory.limits.types.pullback_iso_pullback_inv_fst {X Y Z : Type u} (f : X ⟶ Z) (g : Y ⟶ Z) :
(category_theory.limits.types.pullback_iso_pullback f g).inv ≫ category_theory.limits.pullback.fst = λ (p : category_theory.limits.types.pullback_obj f g), ↑p.fst
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@[simp]
theorem category_theory.limits.types.pullback_iso_pullback_inv_snd {X Y Z : Type u} (f : X ⟶ Z) (g : Y ⟶ Z) :
(category_theory.limits.types.pullback_iso_pullback f g).inv ≫ category_theory.limits.pullback.snd = λ (p : category_theory.limits.types.pullback_obj f g), ↑p.snd