mathlib docs: algebra.category.Mon.limits

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@[instance]

def AddMon.add_monoid_obj {J : Type u} [category_theory.small_category J] (F : J ⥤ AddMon) (j : J) :

add_monoid ((F ⋙ category_theory.forget AddMon).obj j)

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@[instance]

def Mon.monoid_obj {J : Type u} [category_theory.small_category J] (F : J ⥤ Mon) (j : J) :

monoid ((F ⋙ category_theory.forget Mon).obj j)

Equations

Mon.monoid_obj F j = id (F.obj j).monoid

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def AddMon.sections_add_submonoid {J : Type u} [category_theory.small_category J] (F : J ⥤ AddMon) :

add_submonoid (Π (j : J), ↥(F.obj j))

The flat sections of a functor into AddMon form an additive submonoid of all sections.

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def Mon.sections_submonoid {J : Type u} [category_theory.small_category J] (F : J ⥤ Mon) :

submonoid (Π (j : J), ↥(F.obj j))

The flat sections of a functor into Mon form a submonoid of all sections.

Equations

Mon.sections_submonoid F = {carrier := (F ⋙ category_theory.forget Mon).sections, one_mem' := _, mul_mem' := _}

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@[instance]

def AddMon.limit_add_monoid {J : Type u} [category_theory.small_category J] (F : J ⥤ AddMon) :

add_monoid (category_theory.limits.types.limit_cone (F ⋙ category_theory.forget AddMon)).X

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@[instance]

def Mon.limit_monoid {J : Type u} [category_theory.small_category J] (F : J ⥤ Mon) :

monoid (category_theory.limits.types.limit_cone (F ⋙ category_theory.forget Mon)).X

Equations

Mon.limit_monoid F = (Mon.sections_submonoid F).to_monoid

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def AddMon.limit_π_add_monoid_hom {J : Type u} [category_theory.small_category J] (F : J ⥤ AddMon) (j : J) :

(category_theory.limits.types.limit_cone (F ⋙ category_theory.forget AddMon)).X →+ (F ⋙ category_theory.forget AddMon).obj j

limit.π (F ⋙ forget AddMon) j as an add_monoid_hom.

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def Mon.limit_π_monoid_hom {J : Type u} [category_theory.small_category J] (F : J ⥤ Mon) (j : J) :

(category_theory.limits.types.limit_cone (F ⋙ category_theory.forget Mon)).X →* (F ⋙ category_theory.forget Mon).obj j

limit.π (F ⋙ forget Mon) j as a monoid_hom.

Equations

Mon.limit_π_monoid_hom F j = {to_fun := (category_theory.limits.types.limit_cone (F ⋙ category_theory.forget Mon)).π.app j, map_one' := _, map_mul' := _}

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def Mon.has_limits.limit_cone {J : Type u} [category_theory.small_category J] (F : J ⥤ Mon) :

category_theory.limits.cone F

Construction of a limit cone in Mon. (Internal use only; use the limits API.)

Equations

Mon.has_limits.limit_cone F = {X := Mon.of (category_theory.limits.types.limit_cone (F ⋙ category_theory.forget Mon)).X (Mon.limit_monoid F), π := {app := Mon.limit_π_monoid_hom F, naturality' := _}}

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def AddMon.has_limits.limit_cone {J : Type u} [category_theory.small_category J] (F : J ⥤ AddMon) :

category_theory.limits.cone F

(Internal use only; use the limits API.)

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def Mon.has_limits.limit_cone_is_limit {J : Type u} [category_theory.small_category J] (F : J ⥤ Mon) :

category_theory.limits.is_limit (Mon.has_limits.limit_cone F)

Witness that the limit cone in Mon is a limit cone. (Internal use only; use the limits API.)

Equations

Mon.has_limits.limit_cone_is_limit F = category_theory.limits.is_limit.of_faithful (category_theory.forget Mon) (category_theory.limits.types.limit_cone_is_limit (F ⋙ category_theory.forget Mon)) (λ (s : category_theory.limits.cone F), {to_fun := λ (v : ((category_theory.forget Mon).map_cone s).X), ⟨λ (j : J), ((category_theory.forget Mon).map_cone s).π.app j v, _⟩, map_one' := _, map_mul' := _}) _

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def AddMon.has_limits.limit_cone_is_limit {J : Type u} [category_theory.small_category J] (F : J ⥤ AddMon) :

category_theory.limits.is_limit (AddMon.has_limits.limit_cone F)

(Internal use only; use the limits API.)

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@[instance]

def Mon.has_limits :

category_theory.limits.has_limits Mon

The category of monoids has all limits.

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@[instance]

def AddMon.has_limits :

category_theory.limits.has_limits AddMon

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@[instance]

def AddMon.forget_preserves_limits :

category_theory.limits.preserves_limits (category_theory.forget AddMon)

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@[instance]

def Mon.forget_preserves_limits :

category_theory.limits.preserves_limits (category_theory.forget Mon)

The forgetful functor from monoids to types preserves all limits. (That is, the underlying types could have been computed instead as limits in the category of types.)

Equations

Mon.forget_preserves_limits = {preserves_limits_of_shape := λ (J : Type u_1) (𝒥 : category_theory.small_category J), {preserves_limit := λ (F : J ⥤ Mon), category_theory.limits.preserves_limit_of_preserves_limit_cone (Mon.has_limits.limit_cone_is_limit F) (category_theory.limits.types.limit_cone_is_limit (F ⋙ category_theory.forget Mon))}}

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@[instance]

def CommMon.comm_monoid_obj {J : Type u} [category_theory.small_category J] (F : J ⥤ CommMon) (j : J) :

comm_monoid ((F ⋙ category_theory.forget CommMon).obj j)

Equations

CommMon.comm_monoid_obj F j = id (F.obj j).comm_monoid

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@[instance]

def AddCommMon.add_comm_monoid_obj {J : Type u} [category_theory.small_category J] (F : J ⥤ AddCommMon) (j : J) :

add_comm_monoid ((F ⋙ category_theory.forget AddCommMon).obj j)

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@[instance]

def AddCommMon.limit_add_comm_monoid {J : Type u} [category_theory.small_category J] (F : J ⥤ AddCommMon) :

add_comm_monoid (category_theory.limits.types.limit_cone (F ⋙ category_theory.forget AddCommMon)).X

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@[instance]

def CommMon.limit_comm_monoid {J : Type u} [category_theory.small_category J] (F : J ⥤ CommMon) :

comm_monoid (category_theory.limits.types.limit_cone (F ⋙ category_theory.forget CommMon)).X

Equations

CommMon.limit_comm_monoid F = (Mon.sections_submonoid (F ⋙ category_theory.forget₂ CommMon Mon)).to_comm_monoid

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@[instance]

def AddCommMon.category_theory.creates_limit {J : Type u} [category_theory.small_category J] (F : J ⥤ AddCommMon) :

category_theory.creates_limit F (category_theory.forget₂ AddCommMon AddMon)

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@[instance]

def CommMon.category_theory.creates_limit {J : Type u} [category_theory.small_category J] (F : J ⥤ CommMon) :

category_theory.creates_limit F (category_theory.forget₂ CommMon Mon)

We show that the forgetful functor CommMon ⥤ Mon creates limits.

All we need to do is notice that the limit point has a comm_monoid instance available, and then reuse the existing limit.

Equations

CommMon.category_theory.creates_limit F = category_theory.creates_limit_of_reflects_iso (λ (c' : category_theory.limits.cone (F ⋙ category_theory.forget₂ CommMon Mon)) (t : category_theory.limits.is_limit c'), {to_liftable_cone := {lifted_cone := {X := CommMon.of (category_theory.limits.types.limit_cone (F ⋙ category_theory.forget CommMon)).X (CommMon.limit_comm_monoid F), π := {app := Mon.limit_π_monoid_hom (F ⋙ category_theory.forget₂ CommMon Mon), naturality' := _}}, valid_lift := (Mon.has_limits.limit_cone_is_limit (F ⋙ category_theory.forget₂ CommMon Mon)).unique_up_to_iso t}, makes_limit := category_theory.limits.is_limit.of_faithful (category_theory.forget₂ CommMon Mon) (Mon.has_limits.limit_cone_is_limit (F ⋙ category_theory.forget₂ CommMon Mon)) (λ (s : category_theory.limits.cone F), {to_fun := λ (v : ((category_theory.forget Mon).map_cone ((category_theory.forget₂ CommMon Mon).map_cone s)).X), ⟨λ (j : J), ((category_theory.forget Mon).map_cone ((category_theory.forget₂ CommMon Mon).map_cone s)).π.app j v, _⟩, map_one' := _, map_mul' := _}) _})

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def CommMon.limit_cone {J : Type u} [category_theory.small_category J] (F : J ⥤ CommMon) :

category_theory.limits.cone F

A choice of limit cone for a functor into CommMon. (Generally, you'll just want to use limit F.)

Equations

CommMon.limit_cone F = category_theory.lift_limit (category_theory.limits.limit.is_limit (F ⋙ category_theory.forget₂ CommMon Mon))

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def AddCommMon.limit_cone {J : Type u} [category_theory.small_category J] (F : J ⥤ AddCommMon) :

category_theory.limits.cone F

A choice of limit cone for a functor into CommMon. (Generally, you'll just want to use limit F.)

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def AddCommMon.limit_cone_is_limit {J : Type u} [category_theory.small_category J] (F : J ⥤ AddCommMon) :

category_theory.limits.is_limit (AddCommMon.limit_cone F)

The chosen cone is a limit cone. (Generally, you'll just want to use limit.cone F.)

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def CommMon.limit_cone_is_limit {J : Type u} [category_theory.small_category J] (F : J ⥤ CommMon) :

category_theory.limits.is_limit (CommMon.limit_cone F)

The chosen cone is a limit cone. (Generally, you'll just want to use limit.cone F.)

Equations

CommMon.limit_cone_is_limit F = category_theory.lifted_limit_is_limit (category_theory.limits.limit.is_limit (F ⋙ category_theory.forget₂ CommMon Mon))

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@[instance]

def CommMon.has_limits :

category_theory.limits.has_limits CommMon

The category of commutative monoids has all limits.

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@[instance]

def AddCommMon.has_limits :

category_theory.limits.has_limits AddCommMon

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@[instance]

def CommMon.forget₂_Mon_preserves_limits :

category_theory.limits.preserves_limits (category_theory.forget₂ CommMon Mon)

The forgetful functor from commutative monoids to monoids preserves all limits. (That is, the underlying monoid could have been computed instead as limits in the category of monoids.)

Equations

CommMon.forget₂_Mon_preserves_limits = {preserves_limits_of_shape := λ (J : Type u_1) (𝒥 : category_theory.small_category J), {preserves_limit := λ (F : J ⥤ CommMon), category_theory.preserves_limit_of_creates_limit_and_has_limit F (category_theory.forget₂ CommMon Mon)}}

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@[instance]

def AddCommMon.forget₂_AddMon_preserves_limits :

category_theory.limits.preserves_limits (category_theory.forget₂ AddCommMon AddMon)

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@[instance]

def AddCommMon.forget_preserves_limits :

category_theory.limits.preserves_limits (category_theory.forget AddCommMon)

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@[instance]

def CommMon.forget_preserves_limits :

category_theory.limits.preserves_limits (category_theory.forget CommMon)

The forgetful functor from commutative monoids to types preserves all limits. (That is, the underlying types could have been computed instead as limits in the category of types.)

Equations

CommMon.forget_preserves_limits = {preserves_limits_of_shape := λ (J : Type u_1) (𝒥 : category_theory.small_category J), {preserves_limit := λ (F : J ⥤ CommMon), category_theory.limits.comp_preserves_limit (category_theory.forget₂ CommMon Mon) (category_theory.forget Mon)}}

mathlib documentation

The category of (commutative) (additive) monoids has all limits