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Mathlib.Algebra.Category.Ring.Constructions

Constructions of (co)limits in CommRingCat #

In this file we provide the explicit (co)cones for various (co)limits in CommRingCat, including

The explicit cocone with tensor products as the fibered product in CommRingCat.

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@[simp]
theorem CommRingCat.pushoutCocone_pt (R A B : Type u) [CommRing R] [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] :

Verify that the pushout_cocone is indeed the colimit.

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The tensor product A ⊗[ℤ] B is a coproduct for A and B.

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The limit cone of the tensor product A ⊗[ℤ] B in CommRingCat.

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The product in CommRingCat is the cartesian product. This is the binary fan.

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@[simp]
theorem CommRingCat.prodFan_pt (A B : CommRingCat) :
(A.prodFan B).pt = of (A × B)

The product in CommRingCat is the cartesian product.

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noncomputable def CommRingCat.piFan {ι : Type u} (R : ιCommRingCat) :

The categorical product of rings is the cartesian product of rings. This is its Fan.

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@[simp]
theorem CommRingCat.piFan_pt {ι : Type u} (R : ιCommRingCat) :
(piFan R).pt = of ((i : ι) → (R i))
noncomputable def CommRingCat.piFanIsLimit {ι : Type u} (R : ιCommRingCat) :

The categorical product of rings is the cartesian product of rings.

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noncomputable def CommRingCat.piIsoPi {ι : Type u} (R : ιCommRingCat) :
∏ᶜ R of ((i : ι) → (R i))

The categorical product and the usual product agrees

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noncomputable def RingEquiv.piEquivPi {ι : Type u} (R : ιType u) [(i : ι) → CommRing (R i)] :
↑(∏ᶜ fun (i : ι) => CommRingCat.of (R i)) ≃+* ((i : ι) → R i)

The categorical product and the usual product agrees

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noncomputable def CommRingCat.equalizerFork {A B : CommRingCat} (f g : A B) :

The equalizer in CommRingCat is the equalizer as sets. This is the equalizer fork.

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The equalizer in CommRingCat is the equalizer as sets.

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noncomputable def CommRingCat.pullbackCone {A B C : CommRingCat} (f : A C) (g : B C) :

In the category of CommRingCat, the pullback of f : A ⟶ C and g : B ⟶ C is the eqLocus of the two maps A × B ⟶ C. This is the constructed pullback cone.

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noncomputable def CommRingCat.pullbackConeIsLimit {A B C : CommRingCat} (f : A C) (g : B C) :

The constructed pullback cone is indeed the limit.

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