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Mathlib.Algebra.Category.FGModuleCat.Limits

forget₂ (FGModuleCat K) (ModuleCat K) creates all finite limits. #

And hence FGModuleCat K has all finite limits.

Future work #

After generalising FGModuleCat to allow the ring and the module to live in different universes, generalize this construction so we can take limits over smaller diagrams, as is done for the other algebraic categories.

Analogous constructions for Noetherian modules.

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instance FGModuleCat.instFiniteDimensionalCarrierToRingToDivisionRingPiObjModuleCatModuleCategoryHasLimitDiscreteDiscreteCategoryFunctorSmall_subtypeForAllObjToQuiverToCategoryStructTypeToQuiverToCategoryStructTypesToPrefunctorCompForgetModuleConcreteCategorySmall_PiInstSmallDiscreteUnivLE_of_maxSmall_selfUnivLE_of_maxMemSetInstMembershipSetSectionsIsAddCommGroupIsModule {k : Type v} [Field k] {J : Type} [Finite J] (Z : J → ModuleCat k) [∀ (j : J), FiniteDimensional k ↑(Z j)] :
FiniteDimensional k ↑(∏ fun (j : J) => Z j)
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  • ⋯ = ⋯
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instance FGModuleCat.instFiniteDimensionalCarrierToRingToDivisionRingLimitModuleCatModuleCategoryCompFGModuleCatInstLargeCategoryFGModuleCatForget₂InstConcreteCategoryFGModuleCatInstLargeCategoryFGModuleCatModuleConcreteCategoryInstHasForget₂FGModuleCatModuleCatInstLargeCategoryFGModuleCatInstConcreteCategoryFGModuleCatInstLargeCategoryFGModuleCatModuleCategoryModuleConcreteCategoryHasLimitSmall_subtypeForAllObjToQuiverToCategoryStructTypeToQuiverToCategoryStructTypesToPrefunctorForgetSmall_PiUnivLE_of_maxSmall_selfUnivLE_of_maxMemSetInstMembershipSetSectionsIsAddCommGroupIsModule {J : Type} [CategoryTheory.SmallCategory J] [CategoryTheory.FinCategory J] {k : Type v} [Field k] (F : CategoryTheory.Functor J (FGModuleCat k)) :
FiniteDimensional k ↑(CategoryTheory.Limits.limit (CategoryTheory.Functor.comp F (CategoryTheory.forget₂ (FGModuleCat k) (ModuleCat k))))

Finite limits of finite dimensional vectors spaces are finite dimensional, because we can realise them as subobjects of a finite product.

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  • ⋯ = ⋯
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def FGModuleCat.forget₂CreatesLimit {J : Type} [CategoryTheory.SmallCategory J] [CategoryTheory.FinCategory J] {k : Type v} [Field k] (F : CategoryTheory.Functor J (FGModuleCat k)) :
CategoryTheory.CreatesLimit F (CategoryTheory.forget₂ (FGModuleCat k) (ModuleCat k))

The forgetful functor from FGModuleCat k to ModuleCat k creates all finite limits.

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Instances For
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    instance FGModuleCat.instCreatesLimitsOfShapeFGModuleCatToRingToDivisionRingInstLargeCategoryFGModuleCatModuleCatModuleCategoryForget₂InstConcreteCategoryFGModuleCatInstLargeCategoryFGModuleCatModuleConcreteCategoryInstHasForget₂FGModuleCatModuleCatInstLargeCategoryFGModuleCatInstConcreteCategoryFGModuleCatInstLargeCategoryFGModuleCatModuleCategoryModuleConcreteCategory {J : Type} [CategoryTheory.SmallCategory J] [CategoryTheory.FinCategory J] {k : Type v} [Field k] :
    CategoryTheory.CreatesLimitsOfShape J (CategoryTheory.forget₂ (FGModuleCat k) (ModuleCat k))
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    instance FGModuleCat.instHasLimitsOfShapeFGModuleCatToRingToDivisionRingInstLargeCategoryFGModuleCat {k : Type v} [Field k] (J : Type) [CategoryTheory.Category.{0, 0} J] [CategoryTheory.FinCategory J] :
    CategoryTheory.Limits.HasLimitsOfShape J (FGModuleCat k)
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    • ⋯ = ⋯
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    instance FGModuleCat.instHasFiniteLimitsFGModuleCatToRingToDivisionRingInstLargeCategoryFGModuleCat {k : Type v} [Field k] :
    CategoryTheory.Limits.HasFiniteLimits (FGModuleCat k)
    Equations
    • ⋯ = ⋯
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    instance FGModuleCat.instPreservesFiniteLimitsFGModuleCatToRingToDivisionRingInstLargeCategoryFGModuleCatModuleCatModuleCategoryForget₂InstConcreteCategoryFGModuleCatInstLargeCategoryFGModuleCatModuleConcreteCategoryInstHasForget₂FGModuleCatModuleCatInstLargeCategoryFGModuleCatInstConcreteCategoryFGModuleCatInstLargeCategoryFGModuleCatModuleCategoryModuleConcreteCategory {k : Type v} [Field k] :
    CategoryTheory.Limits.PreservesFiniteLimits (CategoryTheory.forget₂ (FGModuleCat k) (ModuleCat k))
    Equations
    • One or more equations did not get rendered due to their size.