File: Mathlib/Algebra/Category/AlgebraCat/Limits.lean

full_name: AlgebraCat.semiringObj
code: instance semiringObj (j) : Semiring ((F ⋙ forget (AlgebraCat R)).obj j) 
goal: Semiring ((F ⋙ CategoryTheory.forget (AlgebraCat R)).obj j)

full_name: AlgebraCat.algebraObj
code: instance algebraObj (j) :
    Algebra R ((F ⋙ forget (AlgebraCat R)).obj j) 
goal: Algebra R ((F ⋙ CategoryTheory.forget (AlgebraCat R)).obj j)

full_name: AlgebraCat.sectionsSubalgebra
code: /-- The flat sections of a functor into `AlgebraCat R` form a submodule of all sections.
-/
def sectionsSubalgebra : Subalgebra R (∀ j, F.obj j) 
goal: Subalgebra R ((j : J) → ↑(F.obj j))

full_name: AlgebraCat.instRingElemForallObjCompForgetSections
code: instance (F : J ⥤ AlgebraCat.{w} R) : Ring (F ⋙ forget _).sections 
goal: Ring ↑(F ⋙ CategoryTheory.forget (AlgebraCat R)).sections

full_name: AlgebraCat.instAlgebraElemForallObjCompForgetSections
code: instance (F : J ⥤ AlgebraCat.{w} R) : Algebra R (F ⋙ forget _).sections 
goal: Algebra R ↑(F ⋙ CategoryTheory.forget (AlgebraCat R)).sections

full_name: AlgebraCat.instSmallSubtypeForallCarrierObjMemSubalgebraSectionsSubalgebra
code: instance : Small.{w} (sectionsSubalgebra F) 
goal: Small.{w, max v w} ↥(AlgebraCat.sectionsSubalgebra F)

full_name: AlgebraCat.limitSemiring
code: instance limitSemiring :
    Ring.{w} (Types.Small.limitCone.{v, w} (F ⋙ forget (AlgebraCat.{w} R))).pt 
goal: Ring (CategoryTheory.Limits.Types.Small.limitCone (F ⋙ CategoryTheory.forget (AlgebraCat R))).pt

full_name: AlgebraCat.limitAlgebra
code: instance limitAlgebra :
    Algebra R (Types.Small.limitCone (F ⋙ forget (AlgebraCat.{w} R))).pt 
goal: Algebra R (CategoryTheory.Limits.Types.Small.limitCone (F ⋙ CategoryTheory.forget (AlgebraCat R))).pt

full_name: AlgebraCat.limitπAlgHom
code: /-- `limit.π (F ⋙ forget (AlgebraCat R)) j` as a `AlgHom`. -/
def limitπAlgHom (j) :
    (Types.Small.limitCone (F ⋙ forget (AlgebraCat R))).pt →ₐ[R]
      (F ⋙ forget (AlgebraCat.{w} R)).obj j 
goal: (CategoryTheory.Limits.Types.Small.limitCone (F ⋙ CategoryTheory.forget (AlgebraCat R))).pt →ₐ[R]
  (F ⋙ CategoryTheory.forget (AlgebraCat R)).obj j

full_name: AlgebraCat.HasLimits.limitCone
code: /-- Construction of a limit cone in `AlgebraCat R`.
(Internal use only; use the limits API.)
-/
def limitCone : Cone F 
goal: CategoryTheory.Limits.Cone F

full_name: AlgebraCat.HasLimits.limitConeIsLimit
code: /-- Witness that the limit cone in `AlgebraCat R` is a limit cone.
(Internal use only; use the limits API.)
-/
def limitConeIsLimit : IsLimit (limitCone.{v, w} F) 
goal: CategoryTheory.Limits.IsLimit (AlgebraCat.HasLimits.limitCone F)

full_name: AlgebraCat.hasLimitsOfSize
code: /-- The category of R-algebras has all limits. -/
lemma hasLimitsOfSize [UnivLE.{v, w}] : HasLimitsOfSize.{t, v} (AlgebraCat.{w} R) 
goal: CategoryTheory.Limits.HasLimitsOfSize.{t, v, w, max (w + 1) u} (AlgebraCat R)

full_name: AlgebraCat.hasLimits
code: instance hasLimits : HasLimits (AlgebraCat.{w} R) 
goal: CategoryTheory.Limits.HasLimits (AlgebraCat R)

full_name: AlgebraCat.forget₂RingPreservesLimitsOfSize
code: /-- The forgetful functor from R-algebras to rings preserves all limits.
-/
instance forget₂RingPreservesLimitsOfSize [UnivLE.{v, w}] :
    PreservesLimitsOfSize.{t, v} (forget₂ (AlgebraCat.{w} R) RingCat.{w}) 
goal: CategoryTheory.Limits.PreservesLimitsOfSize.{t, v, w, w, max u (w + 1), w + 1}
  (CategoryTheory.forget₂ (AlgebraCat R) RingCat)

full_name: AlgebraCat.forget₂RingPreservesLimits
code: instance forget₂RingPreservesLimits : PreservesLimits (forget₂ (AlgebraCat R) RingCat.{w}) 
goal: CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget₂ (AlgebraCat R) RingCat)

full_name: AlgebraCat.forget₂ModulePreservesLimitsOfSize
code: /-- The forgetful functor from R-algebras to R-modules preserves all limits.
-/
instance forget₂ModulePreservesLimitsOfSize [UnivLE.{v, w}] : PreservesLimitsOfSize.{t, v}
    (forget₂ (AlgebraCat.{w} R) (ModuleCat.{w} R)) 
goal: CategoryTheory.Limits.PreservesLimitsOfSize.{t, v, w, w, max u (w + 1), max u (w + 1)}
  (CategoryTheory.forget₂ (AlgebraCat R) (ModuleCat R))

full_name: AlgebraCat.forget₂ModulePreservesLimits
code: instance forget₂ModulePreservesLimits :
    PreservesLimits (forget₂ (AlgebraCat R) (ModuleCat.{w} R)) 
goal: CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget₂ (AlgebraCat R) (ModuleCat R))

full_name: AlgebraCat.forgetPreservesLimitsOfSize
code: /-- The forgetful functor from R-algebras to types preserves all limits.
-/
instance forgetPreservesLimitsOfSize [UnivLE.{v, w}] :
    PreservesLimitsOfSize.{t, v} (forget (AlgebraCat.{w} R)) 
goal: CategoryTheory.Limits.PreservesLimitsOfSize.{t, v, w, w, max u (w + 1), w + 1} (CategoryTheory.forget (AlgebraCat R))

full_name: AlgebraCat.forgetPreservesLimits
code: instance forgetPreservesLimits : PreservesLimits (forget (AlgebraCat.{w} R)) 
goal: CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget (AlgebraCat R))