The category of (commutative) rings 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.

instance SemiRingCat.semiringObj {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J SemiRingCatMax) (j : J) : Semiring ((CategoryTheory.Functor.comp F (CategoryTheory.forget SemiRingCat)).obj j)

def SemiRingCat.sectionsSubsemiring {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J SemiRingCatMax) : Subsemiring ((j : J) → ↑(F.obj j))

The flat sections of a functor into SemiRingCat form a subsemiring of all sections.

instance SemiRingCat.limitSemiring {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J SemiRingCatMax) : Semiring (CategoryTheory.Limits.Types.limitCone (CategoryTheory.Functor.comp F (CategoryTheory.forget SemiRingCat))).pt

def SemiRingCat.limitπRingHom {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J SemiRingCatMax) (j : J) : (CategoryTheory.Limits.Types.limitCone (CategoryTheory.Functor.comp F (CategoryTheory.forget SemiRingCat))).pt →+* (CategoryTheory.Functor.comp F (CategoryTheory.forget SemiRingCat)).obj j

limit.π (F ⋙ forget SemiRingCat) j as a RingHom.

def SemiRingCat.HasLimits.limitCone {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J SemiRingCatMax) : CategoryTheory.Limits.Cone F

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

def SemiRingCat.HasLimits.limitConeIsLimit {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J SemiRingCatMax) : CategoryTheory.Limits.IsLimit (SemiRingCat.HasLimits.limitCone F)

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

instance SemiRingCat.hasLimitsOfSize : CategoryTheory.Limits.HasLimitsOfSize.{v, v, max u v, max (u + 1) (v + 1)} SemiRingCatMax

The category of rings has all limits.

instance SemiRingCat.hasLimits : CategoryTheory.Limits.HasLimits SemiRingCat

def SemiRingCat.forget₂AddCommMonPreservesLimitsAux {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J SemiRingCatMax) : CategoryTheory.Limits.IsLimit ((CategoryTheory.forget₂ SemiRingCat AddCommMonCat).mapCone (SemiRingCat.HasLimits.limitCone F))

Auxiliary lemma to prove the cone induced by limitCone is a limit cone.

instance SemiRingCat.forget₂AddCommMonPreservesLimitsOfSize : CategoryTheory.Limits.PreservesLimitsOfSize.{v, v, max u v, max u v, max (u + 1) (v + 1), max (u + 1) (v + 1)} (CategoryTheory.forget₂ SemiRingCat AddCommMonCat)

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

instance SemiRingCat.forget₂AddCommMonPreservesLimits : CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget₂ SemiRingCat AddCommMonCat)

def SemiRingCat.forget₂MonPreservesLimitsAux {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J SemiRingCatMax) : CategoryTheory.Limits.IsLimit ((CategoryTheory.forget₂ SemiRingCat MonCat).mapCone (SemiRingCat.HasLimits.limitCone F))

An auxiliary declaration to speed up typechecking.

instance SemiRingCat.forget₂MonPreservesLimitsOfSize : CategoryTheory.Limits.PreservesLimitsOfSize.{v, v, max u v, max u v, max (u + 1) (v + 1), max (u + 1) (v + 1)} (CategoryTheory.forget₂ SemiRingCat MonCat)

The forgetful functor from semirings to monoids preserves all limits.

instance SemiRingCat.forget₂MonPreservesLimits : CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget₂ SemiRingCat MonCat)

instance SemiRingCat.forgetPreservesLimitsOfSize : CategoryTheory.Limits.PreservesLimitsOfSize.{v, v, max u v, max u v, max (u + 1) (v + 1), max (u + 1) (v + 1)} (CategoryTheory.forget SemiRingCat)

The forgetful functor from semirings to types preserves all limits.

instance SemiRingCat.forgetPreservesLimits : CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget SemiRingCat)

@[inline, reducible]

abbrev CommSemiRingCatMax : Type ((max u1 u2) + 1)

An alias for CommSemiring.{max u v}, to deal with unification issues.

instance CommSemiRingCat.commSemiringObj {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommSemiRingCatMax) (j : J) : CommSemiring ((CategoryTheory.Functor.comp F (CategoryTheory.forget CommSemiRingCat)).obj j)

instance CommSemiRingCat.limitCommSemiring {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommSemiRingCatMax) : CommSemiring (CategoryTheory.Limits.Types.limitCone (CategoryTheory.Functor.comp F (CategoryTheory.forget CommSemiRingCat))).pt

instance CommSemiRingCat.instCreatesLimitCommSemiRingCatMaxInstCommSemiRingCatLargeCategorySemiRingCatMaxInstSemiRingCatLargeCategoryForget₂InstConcreteCategoryCommSemiRingCatInstCommSemiRingCatLargeCategoryInstConcreteCategorySemiRingCatInstSemiRingCatLargeCategoryHasForgetToSemiRingCat {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommSemiRingCatMax) : CategoryTheory.CreatesLimit F (CategoryTheory.forget₂ CommSemiRingCatMax SemiRingCatMax)

We show that the forgetful functor CommSemiRingCat ⥤ SemiRingCat creates limits.

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

def CommSemiRingCat.limitCone {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommSemiRingCatMax) : CategoryTheory.Limits.Cone F

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

def CommSemiRingCat.limitConeIsLimit {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommSemiRingCatMax) : CategoryTheory.Limits.IsLimit (CommSemiRingCat.limitCone F)

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

instance CommSemiRingCat.hasLimitsOfSize : CategoryTheory.Limits.HasLimitsOfSize.{v, v, max u v, max (u + 1) (v + 1)} CommSemiRingCatMax

The category of rings has all limits.

instance CommSemiRingCat.hasLimits : CategoryTheory.Limits.HasLimits CommSemiRingCat

instance CommSemiRingCat.forget₂SemiRingPreservesLimitsOfSize : CategoryTheory.Limits.PreservesLimitsOfSize.{v, v, max u v, max u v, max (u + 1) (v + 1), max (u + 1) (v + 1)} (CategoryTheory.forget₂ CommSemiRingCat SemiRingCat)

The forgetful functor from rings to semirings preserves all limits.

instance CommSemiRingCat.forget₂SemiRingPreservesLimits : CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget₂ CommSemiRingCat SemiRingCat)

instance CommSemiRingCat.forgetPreservesLimitsOfSize : CategoryTheory.Limits.PreservesLimitsOfSize.{v, v, max u v, max u v, max (u + 1) (v + 1), max (u + 1) (v + 1)} (CategoryTheory.forget CommSemiRingCatMax)

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

instance CommSemiRingCat.forgetPreservesLimits : CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget CommSemiRingCat)

@[inline, reducible]

abbrev RingCatMax : Type ((max u1 u2) + 1)

An alias for RingCat.{max u v}, to deal around unification issues.

instance RingCat.ringObj {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J RingCatMax) (j : J) : Ring ((CategoryTheory.Functor.comp F (CategoryTheory.forget RingCat)).obj j)

def RingCat.sectionsSubring {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J RingCatMax) : Subring ((j : J) → ↑(F.obj j))

The flat sections of a functor into RingCat form a subring of all sections.

instance RingCat.limitRing {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J RingCatMax) : Ring (CategoryTheory.Limits.Types.limitCone (CategoryTheory.Functor.comp F (CategoryTheory.forget RingCatMax))).pt

instance RingCat.instCreatesLimitRingCatMaxInstRingCatLargeCategorySemiRingCatMaxInstSemiRingCatLargeCategoryForget₂InstConcreteCategoryRingCatInstRingCatLargeCategoryInstConcreteCategorySemiRingCatInstSemiRingCatLargeCategoryHasForgetToSemiRingCat {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J RingCatMax) : CategoryTheory.CreatesLimit F (CategoryTheory.forget₂ RingCatMax SemiRingCatMax)

We show that the forgetful functor CommRingCat ⥤ RingCat creates limits.

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

def RingCat.limitCone {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J RingCatMax) : CategoryTheory.Limits.Cone F

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

def RingCat.limitConeIsLimit {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J RingCatMax) : CategoryTheory.Limits.IsLimit (RingCat.limitCone F)

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

instance RingCat.hasLimitsOfSize : CategoryTheory.Limits.HasLimitsOfSize.{v, v, max u v, max (u + 1) (v + 1)} RingCat

The category of rings has all limits.

instance RingCat.hasLimits : CategoryTheory.Limits.HasLimits RingCat

instance RingCat.forget₂SemiRingPreservesLimitsOfSize : CategoryTheory.Limits.PreservesLimitsOfSize.{v, v, max u v, max u v, max (u + 1) (v + 1), max (u + 1) (v + 1)} (CategoryTheory.forget₂ RingCat SemiRingCat)

The forgetful functor from rings to semirings preserves all limits.

instance RingCat.forget₂SemiRingPreservesLimits : CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget₂ RingCat SemiRingCat)

def RingCat.forget₂AddCommGroupPreservesLimitsAux {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J RingCatMax) : CategoryTheory.Limits.IsLimit ((CategoryTheory.forget₂ RingCatMax AddCommGroupCat).mapCone (RingCat.limitCone F))

An auxiliary declaration to speed up typechecking.

instance RingCat.forget₂AddCommGroupPreservesLimitsOfSize : CategoryTheory.Limits.PreservesLimitsOfSize.{v, v, max u v, max u v, max (u + 1) (v + 1), max (u + 1) (v + 1)} (CategoryTheory.forget₂ RingCatMax AddCommGroupCat)

The forgetful functor from rings to additive commutative groups preserves all limits.

instance RingCat.forget₂AddCommGroupPreservesLimits : CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget₂ RingCat AddCommGroupCat)

instance RingCat.forgetPreservesLimitsOfSize : CategoryTheory.Limits.PreservesLimitsOfSize.{v, v, max u v, max u v, max (u + 1) (v + 1), max (u + 1) (v + 1)} (CategoryTheory.forget RingCatMax)

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

instance RingCat.forgetPreservesLimits : CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget RingCat)

@[inline, reducible]

abbrev CommRingCatMax : Type ((max u1 u2) + 1)

An alias for CommRingCat.{max u v}, to deal around unification issues.

instance CommRingCat.commRingObj {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCatMax) (j : J) : CommRing ((CategoryTheory.Functor.comp F (CategoryTheory.forget CommRingCat)).obj j)

instance CommRingCat.limitCommRing {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCatMax) : CommRing (CategoryTheory.Limits.Types.limitCone (CategoryTheory.Functor.comp F (CategoryTheory.forget CommRingCatMax))).pt

instance CommRingCat.instCreatesLimitCommRingCatMaxInstCommRingCatLargeCategoryRingCatMaxInstRingCatLargeCategoryForget₂InstConcreteCategoryCommRingCatInstCommRingCatLargeCategoryInstConcreteCategoryRingCatInstRingCatLargeCategoryHasForgetToRingCat {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCatMax) : CategoryTheory.CreatesLimit F (CategoryTheory.forget₂ CommRingCatMax RingCatMax)

We show that the forgetful functor CommRingCat ⥤ RingCat creates limits.

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

def CommRingCat.limitCone {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCatMax) : CategoryTheory.Limits.Cone F

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

def CommRingCat.limitConeIsLimit {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCatMax) : CategoryTheory.Limits.IsLimit (CommRingCat.limitCone F)

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

instance CommRingCat.hasLimitsOfSize : CategoryTheory.Limits.HasLimitsOfSize.{v, v, max u v, max (u + 1) (v + 1)} CommRingCatMax

The category of commutative rings has all limits.

instance CommRingCat.hasLimits : CategoryTheory.Limits.HasLimits CommRingCat

instance CommRingCat.forget₂RingPreservesLimitsOfSize : CategoryTheory.Limits.PreservesLimitsOfSize.{v, v, max u v, max u v, max (u + 1) (v + 1), max (u + 1) (v + 1)} (CategoryTheory.forget₂ CommRingCat RingCat)

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

instance CommRingCat.forget₂RingPreservesLimits : CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget₂ CommRingCat RingCat)

def CommRingCat.forget₂CommSemiRingPreservesLimitsAux {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCatMax) : CategoryTheory.Limits.IsLimit ((CategoryTheory.forget₂ CommRingCat CommSemiRingCat).mapCone (CommRingCat.limitCone F))

An auxiliary declaration to speed up typechecking.

instance CommRingCat.forget₂CommSemiRingPreservesLimitsOfSize : CategoryTheory.Limits.PreservesLimitsOfSize.{v, v, max u v, max u v, max (u + 1) (v + 1), max (u + 1) (v + 1)} (CategoryTheory.forget₂ CommRingCat CommSemiRingCat)

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

instance CommRingCat.forget₂CommSemiRingPreservesLimits : CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget₂ CommRingCat CommSemiRingCat)

instance CommRingCat.forgetPreservesLimitsOfSize : CategoryTheory.Limits.PreservesLimitsOfSize.{v, v, max u v, max u v, max (u + 1) (v + 1), max (u + 1) (v + 1)} (CategoryTheory.forget CommRingCat)

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

instance CommRingCat.forgetPreservesLimits : CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget CommRingCat)