Documentation

Mathlib.CategoryTheory.Preadditive.Basic

Preadditive categories #

A preadditive category is a category in which X ⟶ Y is an abelian group in such a way that composition of morphisms is linear in both variables.

This file contains a definition of preadditive category that directly encodes the definition given above. The definition could also be phrased as follows: A preadditive category is a category enriched over the category of Abelian groups. Once the general framework to state this in Lean is available, the contents of this file should become obsolete.

Main results #

Implementation notes #

The simp normal form for negation and composition is to push negations as far as possible to the outside. For example, f ≫ (-g) and (-f) ≫ g both become -(f ≫ g), and (-f) ≫ (-g) is simplified to f ≫ g.

References #

Tags #

additive, preadditive, Hom group, Ab-category, Ab-enriched

A category is called preadditive if P ⟶ Q is an abelian group such that composition is linear in both variables.

Instances
    @[simp]

    A category is called preadditive if P ⟶ Q is an abelian group such that composition is linear in both variables.

    @[simp]

    A category is called preadditive if P ⟶ Q is an abelian group such that composition is linear in both variables.

    def CategoryTheory.Preadditive.leftComp {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {P : C} {Q : C} (R : C) (f : P Q) :
    (Q R) →+ (P R)

    Composition by a fixed left argument as a group homomorphism

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      def CategoryTheory.Preadditive.rightComp {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (P : C) {Q : C} {R : C} (g : Q R) :
      (P Q) →+ (P R)

      Composition by a fixed right argument as a group homomorphism

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        Composition as a bilinear group homomorphism

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          theorem CategoryTheory.Preadditive.comp_sum_assoc {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {P : C} {Q : C} {R : C} {J : Type u_1} (s : Finset J) (f : P Q) (g : J(Q R)) {Z : C} (h : R Z) :
          theorem CategoryTheory.Preadditive.comp_sum {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {P : C} {Q : C} {R : C} {J : Type u_1} (s : Finset J) (f : P Q) (g : J(Q R)) :
          theorem CategoryTheory.Preadditive.sum_comp_assoc {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {P : C} {Q : C} {R : C} {J : Type u_1} (s : Finset J) (f : J(P Q)) (g : Q R) {Z : C} (h : R Z) :
          theorem CategoryTheory.Preadditive.sum_comp {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {P : C} {Q : C} {R : C} {J : Type u_1} (s : Finset J) (f : J(P Q)) (g : Q R) :

          Porting note: adding this before the ring instance allowed moduleEndRight to find the correct Monoid structure on End. Moved both down after preadditiveHasZeroMorphisms to make use of them

          Equations
          • CategoryTheory.Preadditive.instSemiringEnd = Semiring.mk Monoid.npow

          Porting note: It looks like Ring's parent classes changed in Lean 4 so the previous instance needed modification. Was following my nose here.

          Equations
          • CategoryTheory.Preadditive.instRingEnd = Ring.mk SubNegMonoid.zsmul
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          • CategoryTheory.Preadditive.moduleEndRight = Module.mk

          Map a kernel cone on the difference of two morphisms to the equalizer fork.

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            Map any equalizer fork to a cone on the difference of the two morphisms.

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              A kernel of f - g is an equalizer of f and g.

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              • One or more equations did not get rendered due to their size.
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                An equalizer of f and g is a kernel of f - g.

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                • One or more equations did not get rendered due to their size.
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                  A preadditive category has an equalizer for f and g if it has a kernel for f - g.

                  A preadditive category has a kernel for f - g if it has an equalizer for f and g.

                  Map any coequalizer cofork to a cocone on the difference of the two morphisms.

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                    A cokernel of f - g is a coequalizer of f and g.

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                    • One or more equations did not get rendered due to their size.
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                      A coequalizer of f and g is a cokernel of f - g.

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                      • One or more equations did not get rendered due to their size.
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                        A preadditive category has a coequalizer for f and g if it has a cokernel for f - g.

                        A preadditive category has a cokernel for f - g if it has a coequalizer for f and g.

                        Equations
                        • CategoryTheory.Preadditive.instSMulUnitsIntIso = { smul := fun (a : ˣ) (e : X Y) => { hom := a e.hom, inv := a⁻¹ e.inv, hom_inv_id := , inv_hom_id := } }
                        @[simp]
                        theorem CategoryTheory.Preadditive.smul_iso_hom {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : C} {Y : C} (a : ˣ) (e : X Y) :
                        (a e).hom = a e.hom
                        @[simp]
                        theorem CategoryTheory.Preadditive.smul_iso_inv {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : C} {Y : C} (a : ˣ) (e : X Y) :
                        (a e).inv = a⁻¹ e.inv
                        Equations
                        • CategoryTheory.Preadditive.instNegIso = { neg := fun (e : X Y) => { hom := -e.hom, inv := -e.inv, hom_inv_id := , inv_hom_id := } }
                        @[simp]
                        theorem CategoryTheory.Preadditive.neg_iso_hom {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : C} {Y : C} (e : X Y) :
                        (-e).hom = -e.hom
                        @[simp]
                        theorem CategoryTheory.Preadditive.neg_iso_inv {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {X : C} {Y : C} (e : X Y) :
                        (-e).inv = -e.inv