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

Further, these limits are preserved by the forgetful functor --- that is, the underlying types are just the limits in the category of types.

source

instance MonCat.monoidObj {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J MonCat) (j : J) :

Monoid ((F.comp (CategoryTheory.forget MonCat)).obj j)

Equations

source

instance AddMonCat.addMonoidObj {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J AddMonCat) (j : J) :

AddMonoid ((F.comp (CategoryTheory.forget AddMonCat)).obj j)

Equations

source

def MonCat.sectionsSubmonoid {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J MonCat) :

Submonoid ((j : J) → ↑(F.obj j))

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

Equations

Instances For

source

def AddMonCat.sectionsAddSubmonoid {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J AddMonCat) :

AddSubmonoid ((j : J) → ↑(F.obj j))

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

Equations

Instances For

source

instance MonCat.sectionsMonoid {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J MonCat) :

Monoid ↑(F.comp (CategoryTheory.forget MonCat)).sections

Equations

source

instance AddMonCat.sectionsAddMonoid {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J AddMonCat) :

AddMonoid ↑(F.comp (CategoryTheory.forget AddMonCat)).sections

Equations

source

noncomputable instance MonCat.limitMonoid {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J MonCat) [Small.{u, max u v} ↑(F.comp (CategoryTheory.forget MonCat)).sections] :

Monoid (CategoryTheory.Limits.Types.Small.limitCone (F.comp (CategoryTheory.forget MonCat))).pt

Equations

source

noncomputable instance AddMonCat.limitAddMonoid {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J AddMonCat) [Small.{u, max u v} ↑(F.comp (CategoryTheory.forget AddMonCat)).sections] :

AddMonoid (CategoryTheory.Limits.Types.Small.limitCone (F.comp (CategoryTheory.forget AddMonCat))).pt

Equations

source

noncomputable def MonCat.limitπMonoidHom {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J MonCat) [Small.{u, max u v} ↑(F.comp (CategoryTheory.forget MonCat)).sections] (j : J) :

(CategoryTheory.Limits.Types.Small.limitCone (F.comp (CategoryTheory.forget MonCat))).pt →* (F.comp (CategoryTheory.forget MonCat)).obj j

limit.π (F ⋙ forget MonCat) j as a MonoidHom.

Equations

Instances For

source

noncomputable def AddMonCat.limitπAddMonoidHom {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J AddMonCat) [Small.{u, max u v} ↑(F.comp (CategoryTheory.forget AddMonCat)).sections] (j : J) :

(CategoryTheory.Limits.Types.Small.limitCone (F.comp (CategoryTheory.forget AddMonCat))).pt →+ (F.comp (CategoryTheory.forget AddMonCat)).obj j

limit.π (F ⋙ forget AddMonCat) j as an AddMonoidHom.

Equations

Instances For

source

noncomputable def MonCat.HasLimits.limitCone {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J MonCat) [Small.{u, max u v} ↑(F.comp (CategoryTheory.forget MonCat)).sections] :

CategoryTheory.Limits.Cone F

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

Equations

Instances For

source

noncomputable def AddMonCat.HasLimits.limitCone {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J AddMonCat) [Small.{u, max u v} ↑(F.comp (CategoryTheory.forget AddMonCat)).sections] :

CategoryTheory.Limits.Cone F

(Internal use only; use the limits API.)

Equations

Instances For

source

noncomputable def MonCat.HasLimits.limitConeIsLimit {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J MonCat) [Small.{u, max u v} ↑(F.comp (CategoryTheory.forget MonCat)).sections] :

CategoryTheory.Limits.IsLimit (limitCone F)

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

Equations

Instances For

source

noncomputable def AddMonCat.HasLimits.limitConeIsLimit {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J AddMonCat) [Small.{u, max u v} ↑(F.comp (CategoryTheory.forget AddMonCat)).sections] :

CategoryTheory.Limits.IsLimit (limitCone F)

(Internal use only; use the limits API.)

Equations

Instances For

source

instance MonCat.HasLimits.hasLimit {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J MonCat) [Small.{u, max u v} ↑(F.comp (CategoryTheory.forget MonCat)).sections] :

CategoryTheory.Limits.HasLimit F

If (F ⋙ forget MonCat).sections is u-small, F has a limit.

source

instance AddMonCat.HasLimits.hasLimit {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J AddMonCat) [Small.{u, max u v} ↑(F.comp (CategoryTheory.forget AddMonCat)).sections] :

CategoryTheory.Limits.HasLimit F

If (F ⋙ forget AddMonCat).sections is u-small, F has a limit.

source

instance MonCat.HasLimits.hasLimitsOfShape {J : Type v} [CategoryTheory.Category.{w, v} J] [Small.{u, v} J] :

CategoryTheory.Limits.HasLimitsOfShape J MonCat

If J is u-small, MonCat.{u} has limits of shape J.

source

instance AddMonCat.HasLimits.hasLimitsOfShape {J : Type v} [CategoryTheory.Category.{w, v} J] [Small.{u, v} J] :

CategoryTheory.Limits.HasLimitsOfShape J AddMonCat

If J is u-small, AddMonCat.{u} has limits of shape J.

source

instance MonCat.hasLimitsOfSize [UnivLE.{v, u}] :

CategoryTheory.Limits.HasLimitsOfSize.{w, v, u, u + 1} MonCat

The category of monoids has all limits.

source

instance AddMonCat.hasLimitsOfSize [UnivLE.{v, u}] :

CategoryTheory.Limits.HasLimitsOfSize.{w, v, u, u + 1} AddMonCat

The category of additive monoids has all limits.

source

instance MonCat.hasLimits :

CategoryTheory.Limits.HasLimits MonCat

source

instance AddMonCat.hasLimits :

CategoryTheory.Limits.HasLimits AddMonCat

source

instance MonCat.forget_preservesLimitsOfShape {J : Type v} [CategoryTheory.Category.{w, v} J] [Small.{u, v} J] :

CategoryTheory.Limits.PreservesLimitsOfShape J (CategoryTheory.forget MonCat)

If J is u-small, the forgetful functor from MonCat.{u} preserves limits of shape J.

source

instance AddMonCat.forget_preservesLimitsOfShape {J : Type v} [CategoryTheory.Category.{w, v} J] [Small.{u, v} J] :

CategoryTheory.Limits.PreservesLimitsOfShape J (CategoryTheory.forget AddMonCat)

If J is u-small, the forgetful functor from AddMonCat.{u} preserves limits of shape J.

source

instance MonCat.forget_preservesLimitsOfSize [UnivLE.{v, u}] :

CategoryTheory.Limits.PreservesLimitsOfSize.{w, v, u, u, u + 1, u + 1} (CategoryTheory.forget MonCat)

The forgetful functor from monoids to types preserves all limits.

This means the underlying type of a limit can be computed as a limit in the category of types.

source

instance AddMonCat.forget_preservesLimitsOfSize [UnivLE.{v, u}] :

CategoryTheory.Limits.PreservesLimitsOfSize.{w, v, u, u, u + 1, u + 1} (CategoryTheory.forget AddMonCat)

The forgetful functor from additive monoids to types preserves all limits.

This means the underlying type of a limit can be computed as a limit in the category of types.

source

instance MonCat.forget_preservesLimits :

CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget MonCat)

source

instance AddMonCat.forget_preservesLimits :

CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget AddMonCat)

source

noncomputable instance MonCat.forget_createsLimit {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J MonCat) :

CategoryTheory.CreatesLimit F (CategoryTheory.forget MonCat)

Equations

source

noncomputable instance AddMonCat.forget_createsLimit {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J AddMonCat) :

CategoryTheory.CreatesLimit F (CategoryTheory.forget AddMonCat)

Equations

source

noncomputable instance MonCat.forget_createsLimitsOfShape {J : Type v} [CategoryTheory.Category.{w, v} J] :

CategoryTheory.CreatesLimitsOfShape J (CategoryTheory.forget MonCat)

Equations

source

noncomputable instance AddMonCat.forget_createsLimitsOfShape {J : Type v} [CategoryTheory.Category.{w, v} J] :

CategoryTheory.CreatesLimitsOfShape J (CategoryTheory.forget AddMonCat)

Equations

source

noncomputable instance MonCat.forget_createsLimitsOfSize :

CategoryTheory.CreatesLimitsOfSize.{w, v, u, u, u + 1, u + 1} (CategoryTheory.forget MonCat)

The forgetful functor from monoids to types preserves all limits.

Equations

source

noncomputable instance AddMonCat.forget_createsLimitsOfSize :

CategoryTheory.CreatesLimitsOfSize.{w, v, u, u, u + 1, u + 1} (CategoryTheory.forget AddMonCat)

The forgetful functor from additive monoids to types preserves all limits.

Equations

source

noncomputable instance MonCat.forget_createsLimits :

CategoryTheory.CreatesLimits (CategoryTheory.forget MonCat)

Equations

source

noncomputable instance AddMonCat.forget_createsLimits :

CategoryTheory.CreatesLimits (CategoryTheory.forget AddMonCat)

Equations

source

instance CommMonCat.commMonoidObj {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J CommMonCat) (j : J) :

CommMonoid ((F.comp (CategoryTheory.forget CommMonCat)).obj j)

Equations

source

instance AddCommMonCat.addCommMonoidObj {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J AddCommMonCat) (j : J) :

AddCommMonoid ((F.comp (CategoryTheory.forget AddCommMonCat)).obj j)

Equations

source

noncomputable instance CommMonCat.limitCommMonoid {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J CommMonCat) [Small.{u, max u v} ↑(F.comp (CategoryTheory.forget CommMonCat)).sections] :

CommMonoid (CategoryTheory.Limits.Types.Small.limitCone (F.comp (CategoryTheory.forget CommMonCat))).pt

Equations

source

noncomputable instance AddCommMonCat.limitAddCommMonoid {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J AddCommMonCat) [Small.{u, max u v} ↑(F.comp (CategoryTheory.forget AddCommMonCat)).sections] :

AddCommMonoid (CategoryTheory.Limits.Types.Small.limitCone (F.comp (CategoryTheory.forget AddCommMonCat))).pt

Equations

source

noncomputable instance CommMonCat.forget₂CreatesLimit {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J CommMonCat) [Small.{u, max u v} ↑(F.comp (CategoryTheory.forget CommMonCat)).sections] :

CategoryTheory.CreatesLimit F (CategoryTheory.forget₂ CommMonCat MonCat)

We show that the forgetful functor CommMonCat ⥤ MonCat creates limits.

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

Equations

source

noncomputable instance AddCommMonCat.forget₂CreatesLimit {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J AddCommMonCat) [Small.{u, max u v} ↑(F.comp (CategoryTheory.forget AddCommMonCat)).sections] :

CategoryTheory.CreatesLimit F (CategoryTheory.forget₂ AddCommMonCat AddMonCat)

We show that the forgetful functor AddCommMonCat ⥤ AddMonCat creates limits.

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

Equations

source

noncomputable def CommMonCat.limitCone {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J CommMonCat) [Small.{u, max u v} ↑(F.comp (CategoryTheory.forget CommMonCat)).sections] :

CategoryTheory.Limits.Cone F

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

Equations

Instances For

source

noncomputable def AddCommMonCat.limitCone {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J AddCommMonCat) [Small.{u, max u v} ↑(F.comp (CategoryTheory.forget AddCommMonCat)).sections] :

CategoryTheory.Limits.Cone F

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

Equations

Instances For

source

noncomputable def CommMonCat.limitConeIsLimit {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J CommMonCat) [Small.{u, max u v} ↑(F.comp (CategoryTheory.forget CommMonCat)).sections] :

CategoryTheory.Limits.IsLimit (limitCone F)

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

Equations

Instances For

source

noncomputable def AddCommMonCat.limitConeIsLimit {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J AddCommMonCat) [Small.{u, max u v} ↑(F.comp (CategoryTheory.forget AddCommMonCat)).sections] :

CategoryTheory.Limits.IsLimit (limitCone F)

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

Equations

Instances For

source

instance CommMonCat.hasLimit {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J CommMonCat) [Small.{u, max u v} ↑(F.comp (CategoryTheory.forget CommMonCat)).sections] :

CategoryTheory.Limits.HasLimit F

If (F ⋙ forget CommMonCat).sections is u-small, F has a limit.

source

instance AddCommMonCat.hasLimit {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J AddCommMonCat) [Small.{u, max u v} ↑(F.comp (CategoryTheory.forget AddCommMonCat)).sections] :

CategoryTheory.Limits.HasLimit F

If (F ⋙ forget AddCommMonCat).sections is u-small, F has a limit.

source

instance CommMonCat.hasLimitsOfShape {J : Type v} [CategoryTheory.Category.{w, v} J] [Small.{u, v} J] :

CategoryTheory.Limits.HasLimitsOfShape J CommMonCat

If J is u-small, CommMonCat.{u} has limits of shape J.

source

instance AddCommMonCat.hasLimitsOfShape {J : Type v} [CategoryTheory.Category.{w, v} J] [Small.{u, v} J] :

CategoryTheory.Limits.HasLimitsOfShape J AddCommMonCat

If J is u-small, AddCommMonCat.{u} has limits of shape J.

source

instance CommMonCat.hasLimitsOfSize [UnivLE.{v, u}] :

CategoryTheory.Limits.HasLimitsOfSize.{w, v, u, u + 1} CommMonCat

The category of commutative monoids has all limits.

source

instance AddCommMonCat.hasLimitsOfSize [UnivLE.{v, u}] :

CategoryTheory.Limits.HasLimitsOfSize.{w, v, u, u + 1} AddCommMonCat

The category of additive commutative monoids has all limits.

source

instance CommMonCat.hasLimits :

CategoryTheory.Limits.HasLimits CommMonCat

source

instance AddCommMonCat.hasLimits :

CategoryTheory.Limits.HasLimits AddCommMonCat

source

instance CommMonCat.forget₂Mon_preservesLimitsOfSize [UnivLE.{v, u}] :

CategoryTheory.Limits.PreservesLimitsOfSize.{w, v, u, u, u + 1, u + 1} (CategoryTheory.forget₂ CommMonCat MonCat)

The forgetful functor from commutative monoids to monoids preserves all limits.

This means the underlying type of a limit can be computed as a limit in the category of monoids.

source

instance AddCommMonCat.forget₂AddMonPreservesLimitsOfSize [UnivLE.{v, u}] :

CategoryTheory.Limits.PreservesLimitsOfSize.{w, v, u, u, u + 1, u + 1} (CategoryTheory.forget₂ AddCommMonCat AddMonCat)

The forgetful functor from additive commutative monoids to additive monoids preserves all limits.

This means the underlying type of a limit can be computed as a limit in the category of additive monoids.

source

instance CommMonCat.forget₂Mon_preservesLimits :

CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget₂ CommMonCat MonCat)

source

instance AddCommMonCat.forget₂Mon_preservesLimits :

CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget₂ AddCommMonCat AddMonCat)

source

instance CommMonCat.forget_preservesLimitsOfShape {J : Type v} [CategoryTheory.Category.{w, v} J] [Small.{u, v} J] :

CategoryTheory.Limits.PreservesLimitsOfShape J (CategoryTheory.forget CommMonCat)

If J is u-small, the forgetful functor from CommMonCat.{u} preserves limits of shape J.

source

instance AddCommMonCat.forget_preservesLimitsOfShape {J : Type v} [CategoryTheory.Category.{w, v} J] [Small.{u, v} J] :

CategoryTheory.Limits.PreservesLimitsOfShape J (CategoryTheory.forget AddCommMonCat)

If J is u-small, the forgetful functor from AddCommMonCat.{u} preserves limits of shape J.

source

instance CommMonCat.forget_preservesLimitsOfSize [UnivLE.{v, u}] :

CategoryTheory.Limits.PreservesLimitsOfSize.{v, v, u, u, u + 1, u + 1} (CategoryTheory.forget CommMonCat)

The forgetful functor from commutative monoids to types preserves all limits.

This means the underlying type of a limit can be computed as a limit in the category of types.

source

instance AddCommMonCat.forget_preservesLimitsOfSize [UnivLE.{v, u}] :

CategoryTheory.Limits.PreservesLimitsOfSize.{v, v, u, u, u + 1, u + 1} (CategoryTheory.forget AddCommMonCat)

The forgetful functor from additive commutative monoids to types preserves all limits.

This means the underlying type of a limit can be computed as a limit in the category of types.

source

instance AddCommMonCat.forget_preservesLimits :

CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget AddCommMonCat)

source

instance CommMonCat.forget_preservesLimits :

CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget CommMonCat)

source

noncomputable instance CommMonCat.forget_createsLimit {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J CommMonCat) [Small.{u, max u v} ↑(F.comp (CategoryTheory.forget CommMonCat)).sections] :

CategoryTheory.CreatesLimit F (CategoryTheory.forget CommMonCat)

Equations

source

noncomputable instance AddCommMonCat.forget_createsLimit {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J AddCommMonCat) [Small.{u, max u v} ↑(F.comp (CategoryTheory.forget AddCommMonCat)).sections] :

CategoryTheory.CreatesLimit F (CategoryTheory.forget AddCommMonCat)

Equations

source

noncomputable instance CommMonCat.forget_createsLimitsOfShape {J : Type v} [CategoryTheory.Category.{w, v} J] :

CategoryTheory.CreatesLimitsOfShape J (CategoryTheory.forget MonCat)

Equations

source

noncomputable instance AddCommMonCat.forget_createsLimitsOfShape {J : Type v} [CategoryTheory.Category.{w, v} J] :

CategoryTheory.CreatesLimitsOfShape J (CategoryTheory.forget AddMonCat)

Equations

source

noncomputable instance CommMonCat.forget_createsLimitsOfSize :

CategoryTheory.CreatesLimitsOfSize.{w, v, u, u, u + 1, u + 1} (CategoryTheory.forget MonCat)

The forgetful functor from commutative monoids to types preserves all limits.

Equations

source

noncomputable instance AddCommMonCat.forget_createsLimitsOfSize :

CategoryTheory.CreatesLimitsOfSize.{w, v, u, u, u + 1, u + 1} (CategoryTheory.forget AddMonCat)

The forgetful functor from commutative additive monoids to types preserves all limits.

Equations

source

noncomputable instance CommMonCat.forget_createsLimits :

CategoryTheory.CreatesLimits (CategoryTheory.forget MonCat)

Equations

source

noncomputable instance AddCommMonCat.forget_createsLimits :

CategoryTheory.CreatesLimits (CategoryTheory.forget AddMonCat)

Equations

Documentation

Mathlib.Algebra.Category.MonCat.Limits

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