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.

@[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) :

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

@[simp]
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) :

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

The category of types has punit as a terminal object.

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The category of types has pempty as an initial object.

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

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

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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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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 u) :
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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@[simp]
theorem category_theory.limits.types.product_iso_hom_comp_eval {J : Type u} (F : J → Type u) (j : J) :
@[simp]
theorem category_theory.limits.types.product_iso_inv_comp_π {J : Type u} (F : J → Type u) (j : J) :
@[simp]
theorem category_theory.limits.types.product_iso_inv_comp_π_apply {J : Type u} (F : J → Type u) (j : J) (x : (Π (j : J), F j)) :

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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@[simp]
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)) :

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.

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)

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) :

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

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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.

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' '' 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.

@[nolint]
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.

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