The category of R-algebras 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.

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instance AlgebraCat.semiringObj {R : Type u} [CommRing R] {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J (AlgebraCatMax R)) (j : J) :

Semiring ((CategoryTheory.Functor.comp F (CategoryTheory.forget (AlgebraCat R))).obj j)

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instance AlgebraCat.algebraObj {R : Type u} [CommRing R] {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J (AlgebraCatMax R)) (j : J) :

Algebra R ((CategoryTheory.Functor.comp F (CategoryTheory.forget (AlgebraCat R))).obj j)

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def AlgebraCat.sectionsSubalgebra {R : Type u} [CommRing R] {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J (AlgebraCatMax R)) :

Subalgebra R ((j : J) → ↑(F.obj j))

The flat sections of a functor into AlgebraCat R form a submodule of all sections.

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instance AlgebraCat.limitSemiring {R : Type u} [CommRing R] {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J (AlgebraCatMax R)) :

Ring (CategoryTheory.Limits.Types.limitCone (CategoryTheory.Functor.comp F (CategoryTheory.forget (AlgebraCatMax R)))).pt

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instance AlgebraCat.limitAlgebra {R : Type u} [CommRing R] {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J (AlgebraCatMax R)) :

Algebra R (CategoryTheory.Limits.Types.limitCone (CategoryTheory.Functor.comp F (CategoryTheory.forget (AlgebraCatMax R)))).pt

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def AlgebraCat.limitπAlgHom {R : Type u} [CommRing R] {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J (AlgebraCatMax R)) (j : J) :

(CategoryTheory.Limits.Types.limitCone (CategoryTheory.Functor.comp F (CategoryTheory.forget (AlgebraCat R)))).pt →ₐ[R] (CategoryTheory.Functor.comp F (CategoryTheory.forget (AlgebraCatMax R))).obj j

limit.π (F ⋙ forget (AlgebraCat R)) j as a AlgHom.

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def AlgebraCat.HasLimits.limitCone {R : Type u} [CommRing R] {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J (AlgebraCatMax R)) :

CategoryTheory.Limits.Cone F

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

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def AlgebraCat.HasLimits.limitConeIsLimit {R : Type u} [CommRing R] {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J (AlgebraCatMax R)) :

CategoryTheory.Limits.IsLimit (AlgebraCat.HasLimits.limitCone F)

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

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theorem AlgebraCat.hasLimitsOfSize {R : Type u} [CommRing R] :

CategoryTheory.Limits.HasLimitsOfSize.{v, v, max v w, max (max (v + 1) (w + 1)) u} (AlgebraCatMax R)

The category of R-algebras has all limits.

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instance AlgebraCat.hasLimits {R : Type u} [CommRing R] :

CategoryTheory.Limits.HasLimits (AlgebraCat R)

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instance AlgebraCat.forget₂RingPreservesLimitsOfSize {R : Type u} [CommRing R] :

CategoryTheory.Limits.PreservesLimitsOfSize.{v, v, max v w, max v w, max (max u (v + 1)) (w + 1), max (v + 1) (w + 1)} (CategoryTheory.forget₂ (AlgebraCatMax R) RingCatMax)

The forgetful functor from R-algebras to rings preserves all limits.

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instance AlgebraCat.forget₂RingPreservesLimits {R : Type u} [CommRing R] :

CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget₂ (AlgebraCat R) RingCat)

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instance AlgebraCat.forget₂ModulePreservesLimitsOfSize {R : Type u} [CommRing R] :

CategoryTheory.Limits.PreservesLimitsOfSize.{v, v, max v w, max v w, max (max u (v + 1)) (w + 1), max (max u (v + 1)) (w + 1)} (CategoryTheory.forget₂ (AlgebraCatMax R) (ModuleCatMax R))

The forgetful functor from R-algebras to R-modules preserves all limits.

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instance AlgebraCat.forget₂ModulePreservesLimits {R : Type u} [CommRing R] :

CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget₂ (AlgebraCat R) (ModuleCat R))

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instance AlgebraCat.forgetPreservesLimitsOfSize {R : Type u} [CommRing R] :

CategoryTheory.Limits.PreservesLimitsOfSize.{v, v, max v w, max v w, max (max u (v + 1)) (w + 1), max (v + 1) (w + 1)} (CategoryTheory.forget (AlgebraCatMax R))

The forgetful functor from R-algebras to types preserves all limits.

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instance AlgebraCat.forgetPreservesLimits {R : Type u} [CommRing R] :

CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget (AlgebraCat R))