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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.instFiniteDimensionalCarrierPiObjModuleCatOfFinite {k : Type v} [Field k] {J : Type} [Finite J] (Z : J → ModuleCat k) [∀ (j : J), FiniteDimensional k ↑(Z j)] :
FiniteDimensional k ↑(∏ᶜ fun (j : J) => Z j)
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  • ⋯ = ⋯
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instance FGModuleCat.instFiniteDimensionalCarrierLimitModuleCatCompForget₂ {J : Type} [CategoryTheory.SmallCategory J] [CategoryTheory.FinCategory J] {k : Type v} [Field k] (F : CategoryTheory.Functor J (FGModuleCat k)) :
FiniteDimensional k ↑(CategoryTheory.Limits.limit (F.comp (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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  • One or more equations did not get rendered due to their size.
Instances For
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    instance FGModuleCat.instCreatesLimitsOfShapeModuleCatForget₂ {J : Type} [CategoryTheory.SmallCategory J] [CategoryTheory.FinCategory J] {k : Type v} [Field k] :
    CategoryTheory.CreatesLimitsOfShape J (CategoryTheory.forget₂ (FGModuleCat k) (ModuleCat k))
    Equations
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    instance FGModuleCat.instHasLimitsOfShapeOfFinCategory {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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    • ⋯ = ⋯
    source
    instance FGModuleCat.instHasFiniteLimits {k : Type v} [Field k] :
    CategoryTheory.Limits.HasFiniteLimits (FGModuleCat k)
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    • ⋯ = ⋯
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    instance FGModuleCat.instPreservesFiniteLimitsModuleCatForget₂ {k : Type v} [Field k] :
    CategoryTheory.Limits.PreservesFiniteLimits (CategoryTheory.forget₂ (FGModuleCat k) (ModuleCat k))
    Equations
    • ⋯ = ⋯