# mathlib3documentation

algebra.category.Group.limits

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

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

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def Group.group_obj {J : Type v} (F : J Group) (j : J) :
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def Group.sections_subgroup {J : Type v} (F : J Group) :
subgroup (Π (j : J), (F.obj j))

The flat sections of a functor into Group form a subgroup of all sections.

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add_subgroup (Π (j : J), (F.obj j))

The flat sections of a functor into AddGroup form an additive subgroup of all sections.

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def Group.limit_group {J : Type v} (F : J Group) :
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noncomputable def AddGroup.forget₂.creates_limit {J : Type v} (F : J AddGroup) :

We show that the forgetful functor AddGroup ⥤ AddMon creates limits.

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

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noncomputable def Group.forget₂.creates_limit {J : Type v} (F : J Group) :

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

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

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noncomputable def AddGroup.limit_cone {J : Type v} (F : J AddGroup) :

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

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noncomputable def Group.limit_cone {J : Type v} (F : J Group) :

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

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noncomputable def Group.limit_cone_is_limit {J : Type v} (F : J Group) :

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

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noncomputable def AddGroup.limit_cone_is_limit {J : Type v} (F : J AddGroup) :

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

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The category of additive groups has all limits.

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The category of groups has all limits.

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The forgetful functor from groups to monoids preserves all limits.

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

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

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

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

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

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The forgetful functor from groups to types preserves all limits.

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

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def CommGroup.comm_group_obj {J : Type v} (F : J CommGroup) (j : J) :
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noncomputable def CommGroup.forget₂.creates_limit {J : Type v} (F : J CommGroup) :

We show that the forgetful functor CommGroup ⥤ Group creates limits.

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

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noncomputable def AddCommGroup.forget₂.creates_limit {J : Type v} (F : J AddCommGroup) :

We show that the forgetful functor AddCommGroup ⥤ AddGroup creates limits.

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

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noncomputable def CommGroup.limit_cone {J : Type v} (F : J CommGroup) :

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

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noncomputable def AddCommGroup.limit_cone {J : Type v} (F : J AddCommGroup) :

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

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noncomputable def AddCommGroup.limit_cone_is_limit {J : Type v} (F : J AddCommGroup) :

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

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noncomputable def CommGroup.limit_cone_is_limit {J : Type v} (F : J CommGroup) :

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

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The category of additive commutative groups has all limits.

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The category of commutative groups has all limits.

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

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

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An auxiliary declaration to speed up typechecking.

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An auxiliary declaration to speed up typechecking.

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

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

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

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

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noncomputable def AddCommGroup.kernel_iso_ker {G H : AddCommGroup} (f : G H) :

The categorical kernel of a morphism in AddCommGroup agrees with the usual group-theoretical kernel.

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noncomputable def AddCommGroup.kernel_iso_ker_over {G H : AddCommGroup} (f : G H) :

The categorical kernel inclusion for f : G ⟶ H, as an object over G, agrees with the subtype map.

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