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Mathlib.LinearAlgebra.QuadraticForm.QuadraticModuleCat

The category of quadratic modules #

structure QuadraticModuleCat (R : Type u) [CommRing R] extends ModuleCat R :
Type (max u (v + 1))

The category of quadratic modules; modules with an associated quadratic form

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    The object in the category of quadratic R-modules associated to a quadratic R-module.

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      @[simp]
      theorem QuadraticModuleCat.of_form {R : Type u} [CommRing R] {X : Type v} [AddCommGroup X] [Module R X] (Q : QuadraticForm R X) :
      (of Q).form = Q
      structure QuadraticModuleCat.Hom {R : Type u} [CommRing R] (V W : QuadraticModuleCat R) :

      A type alias for QuadraticForm.LinearIsometry to avoid confusion between the categorical and algebraic spellings of composition.

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        theorem QuadraticModuleCat.Hom.ext {R : Type u} {inst✝ : CommRing R} {V W : QuadraticModuleCat R} {x y : V.Hom W} (toIsometry : x.toIsometry = y.toIsometry) :
        x = y
        @[reducible, inline]
        abbrev QuadraticModuleCat.ofHom {R : Type u} [CommRing R] {X : Type v} [AddCommGroup X] [Module R X] {Q₁ Q₂ : QuadraticForm R X} (f : Q₁ →qᵢ Q₂) :
        of Q₁ of Q₂

        Typecheck a QuadraticForm.Isometry as a morphism in Module R.

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          def QuadraticModuleCat.ofIso {R : Type u} [CommRing R] {X Y : Type v} [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] {Q₁ : QuadraticForm R X} {Q₂ : QuadraticForm R Y} (e : QuadraticMap.IsometryEquiv Q₁ Q₂) :
          of Q₁ of Q₂

          Build an isomorphism in the category QuadraticModuleCat R from a QuadraticForm.IsometryEquiv.

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            theorem QuadraticModuleCat.ofIso_symm {R : Type u} [CommRing R] {X Y : Type v} [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] {Q₁ : QuadraticForm R X} {Q₂ : QuadraticForm R Y} (e : QuadraticMap.IsometryEquiv Q₁ Q₂) :
            @[simp]
            theorem QuadraticModuleCat.ofIso_trans {R : Type u} [CommRing R] {X Y Z : Type v} [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] [AddCommGroup Z] [Module R Z] {Q₁ : QuadraticForm R X} {Q₂ : QuadraticForm R Y} {Q₃ : QuadraticForm R Z} (e : QuadraticMap.IsometryEquiv Q₁ Q₂) (f : QuadraticMap.IsometryEquiv Q₂ Q₃) :

            Build a QuadraticForm.IsometryEquiv from an isomorphism in the category QuadraticModuleCat R.

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