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Mathlib.Algebra.Homology.Square

Relation between pullback/pushout squares and kernel/cokernel sequences #

This file is the bundled counterpart of Algebra.Homology.CommSq. The same results are obtained here for squares sq : Square C where C is an additive category.

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@[reducible, inline]
noncomputable abbrev CategoryTheory.Square.cokernelCofork {C : Type u_1} [Category.{u_2, u_1} C] [Preadditive C] (sq : Square C) [Limits.HasBinaryBiproduct sq.X₂ sq.X₃] :
Limits.CokernelCofork (Limits.biprod.lift sq.f₁₂ (-sq.f₁₃))

The cokernel cofork attached to a commutative square in a preadditive category.

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      noncomputable def CategoryTheory.Square.isPushoutEquivIsColimitCokernelCofork {C : Type u_1} [Category.{u_2, u_1} C] [Preadditive C] (sq : Square C) [Limits.HasBinaryBiproduct sq.X₂ sq.X₃] :
      sq.IsPushout ≃ Limits.IsColimit sq.cokernelCofork

      A commutative square in a preadditive category is a pushout square iff the corresponding diagram X₁ ⟶ X₂ ⊞ X₃ ⟶ X₄ ⟶ 0 makes X₄ a cokernel.

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          noncomputable def CategoryTheory.Square.IsPushout.isColimitCokernelCofork {C : Type u_1} [Category.{u_2, u_1} C] [Preadditive C] {sq : Square C} [Limits.HasBinaryBiproduct sq.X₂ sq.X₃] (h : sq.IsPushout) :
          Limits.IsColimit sq.cokernelCofork

          The colimit cokernel cofork attached to a pushout square.

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              @[reducible, inline]
              noncomputable abbrev CategoryTheory.Square.kernelFork {C : Type u_1} [Category.{u_2, u_1} C] [Preadditive C] (sq : Square C) [Limits.HasBinaryBiproduct sq.X₂ sq.X₃] :
              Limits.KernelFork (Limits.biprod.desc sq.f₂₄ (-sq.f₃₄))

              The kernel fork attached to a commutative square in a preadditive category.

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                  noncomputable def CategoryTheory.Square.isPullbackEquivIsLimitKernelFork {C : Type u_1} [Category.{u_2, u_1} C] [Preadditive C] (sq : Square C) [Limits.HasBinaryBiproduct sq.X₂ sq.X₃] :
                  sq.IsPullback ≃ Limits.IsLimit sq.kernelFork

                  A commutative square in a preadditive category is a pullback square iff the corresponding diagram 0 ⟶ X₁ ⟶ X₂ ⊞ X₃ ⟶ X₄ ⟶ 0 makes X₁ a kernel.

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                      noncomputable def CategoryTheory.Square.IsPullback.isLimitKernelFork {C : Type u_1} [Category.{u_2, u_1} C] [Preadditive C] {sq : Square C} [Limits.HasBinaryBiproduct sq.X₂ sq.X₃] (h : sq.IsPullback) :
                      Limits.IsLimit sq.kernelFork

                      The limit kernel fork attached to a pullback square.

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