Documentation

Mathlib.CategoryTheory.Limits.Shapes.Equalizers

Equalizers and coequalizers #

This file defines (co)equalizers as special cases of (co)limits.

An equalizer is the categorical generalization of the subobject {a ∈ A | f(a) = g(a)} known from abelian groups or modules. It is a limit cone over the diagram formed by f and g.

A coequalizer is the dual concept.

Main definitions #

Each of these has a dual.

Main statements #

Implementation notes #

As with the other special shapes in the limits library, all the definitions here are given as abbreviations of the general statements for limits, so all the simp lemmas and theorems about general limits can be used.

References #

The type of objects for the diagram indexing a (co)equalizer.

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    • CategoryTheory.Limits.instDecidableEqWalkingParallelPairHom = CategoryTheory.Limits.decEqWalkingParallelPairHom✝

    The functor WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ sending left to left and right to right.

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      The equivalence WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ sending left to left and right to right.

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        parallelPair f g is the diagram in C consisting of the two morphisms f and g with common domain and codomain.

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          Construct a morphism between parallel pairs.

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            def CategoryTheory.Limits.parallelPair.eqOfHomEq {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {f : X Y} {g : X Y} {f' : X Y} {g' : X Y} (hf : f = f') (hg : g = g') :

            Construct a natural isomorphism between parallelPair f g and parallelPair f' g' given equalities f = f' and g = g'.

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              @[reducible, inline]
              abbrev CategoryTheory.Limits.Fork {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} (f : X Y) (g : X Y) :
              Type (max (max 0 u) v)

              A fork on f and g is just a Cone (parallelPair f g).

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                @[reducible, inline]
                abbrev CategoryTheory.Limits.Cofork {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} (f : X Y) (g : X Y) :
                Type (max (max 0 u) v)

                A cofork on f and g is just a Cocone (parallelPair f g).

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                  A fork t on the parallel pair f g : X ⟶ Y consists of two morphisms t.π.app zero : t.pt ⟶ X and t.π.app one : t.pt ⟶ Y. Of these, only the first one is interesting, and we give it the shorter name Fork.ι t.

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                    A cofork t on the parallelPair f g : X ⟶ Y consists of two morphisms t.ι.app zero : X ⟶ t.pt and t.ι.app one : Y ⟶ t.pt. Of these, only the second one is interesting, and we give it the shorter name Cofork.π t.

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                      A fork on f g : X ⟶ Y is determined by the morphism ι : P ⟶ X satisfying ι ≫ f = ι ≫ g.

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                        A cofork on f g : X ⟶ Y is determined by the morphism π : Y ⟶ P satisfying f ≫ π = g ≫ π.

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                          @[simp]

                          To check whether two maps are equalized by both maps of a fork, it suffices to check it for the first map

                          To check whether two maps are coequalized by both maps of a cofork, it suffices to check it for the second map

                          theorem CategoryTheory.Limits.Fork.IsLimit.hom_ext {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {f : X Y} {g : X Y} {s : CategoryTheory.Limits.Fork f g} (hs : CategoryTheory.Limits.IsLimit s) {W : C} {k : W s.pt} {l : W s.pt} (h : CategoryTheory.CategoryStruct.comp k s = CategoryTheory.CategoryStruct.comp l s) :
                          k = l

                          If s is a limit fork over f and g, then a morphism k : W ⟶ X satisfying k ≫ f = k ≫ g induces a morphism l : W ⟶ s.pt such that l ≫ fork.ι s = k.

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                            If s is a limit fork over f and g, then a morphism k : W ⟶ X satisfying k ≫ f = k ≫ g induces a morphism l : W ⟶ s.pt such that l ≫ fork.ι s = k.

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                              If s is a colimit cofork over f and g, then a morphism k : Y ⟶ W satisfying f ≫ k = g ≫ k induces a morphism l : s.pt ⟶ W such that cofork.π s ≫ l = k.

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                                If s is a colimit cofork over f and g, then a morphism k : Y ⟶ W satisfying f ≫ k = g ≫ k induces a morphism l : s.pt ⟶ W such that cofork.π s ≫ l = k.

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                                  @[simp]
                                  theorem CategoryTheory.Limits.Fork.IsLimit.mk_lift {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {f : X Y} {g : X Y} (t : CategoryTheory.Limits.Fork f g) (lift : (s : CategoryTheory.Limits.Fork f g) → s.pt t.pt) (fac : ∀ (s : CategoryTheory.Limits.Fork f g), CategoryTheory.CategoryStruct.comp (lift s) t = s) (uniq : ∀ (s : CategoryTheory.Limits.Fork f g) (m : s.pt t.pt), CategoryTheory.CategoryStruct.comp m t = sm = lift s) (s : CategoryTheory.Limits.Fork f g) :
                                  (CategoryTheory.Limits.Fork.IsLimit.mk t lift fac uniq).lift s = lift s
                                  def CategoryTheory.Limits.Fork.IsLimit.mk {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {f : X Y} {g : X Y} (t : CategoryTheory.Limits.Fork f g) (lift : (s : CategoryTheory.Limits.Fork f g) → s.pt t.pt) (fac : ∀ (s : CategoryTheory.Limits.Fork f g), CategoryTheory.CategoryStruct.comp (lift s) t = s) (uniq : ∀ (s : CategoryTheory.Limits.Fork f g) (m : s.pt t.pt), CategoryTheory.CategoryStruct.comp m t = sm = lift s) :

                                  This is a slightly more convenient method to verify that a fork is a limit cone. It only asks for a proof of facts that carry any mathematical content

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                                    def CategoryTheory.Limits.Cofork.IsColimit.mk {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {f : X Y} {g : X Y} (t : CategoryTheory.Limits.Cofork f g) (desc : (s : CategoryTheory.Limits.Cofork f g) → t.pt s.pt) (fac : ∀ (s : CategoryTheory.Limits.Cofork f g), CategoryTheory.CategoryStruct.comp t (desc s) = s) (uniq : ∀ (s : CategoryTheory.Limits.Cofork f g) (m : t.pt s.pt), CategoryTheory.CategoryStruct.comp t m = sm = desc s) :

                                    This is a slightly more convenient method to verify that a cofork is a colimit cocone. It only asks for a proof of facts that carry any mathematical content

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                                      This is another convenient method to verify that a fork is a limit cone. It only asks for a proof of facts that carry any mathematical content, and allows access to the same s for all parts.

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                                        noncomputable def CategoryTheory.Limits.Fork.IsLimit.ofExistsUnique {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {f : X Y} {g : X Y} {t : CategoryTheory.Limits.Fork f g} (hs : ∀ (s : CategoryTheory.Limits.Fork f g), ∃! l : s.pt t.pt, CategoryTheory.CategoryStruct.comp l t = s) :

                                        Noncomputably make a limit cone from the existence of unique factorizations.

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                                          Noncomputably make a colimit cocone from the existence of unique factorizations.

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                                            Given a limit cone for the pair f g : X ⟶ Y, for any Z, morphisms from Z to its point are in bijection with morphisms h : Z ⟶ X such that h ≫ f = h ≫ g. Further, this bijection is natural in Z: see Fork.IsLimit.homIso_natural. This is a special case of IsLimit.homIso', often useful to construct adjunctions.

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                                              Given a colimit cocone for the pair f g : X ⟶ Y, for any Z, morphisms from the cocone point to Z are in bijection with morphisms h : Y ⟶ Z such that f ≫ h = g ≫ h. Further, this bijection is natural in Z: see Cofork.IsColimit.homIso_natural. This is a special case of IsColimit.homIso', often useful to construct adjunctions.

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                                                This is a helper construction that can be useful when verifying that a category has all equalizers. Given F : WalkingParallelPair ⥤ C, which is really the same as parallelPair (F.map left) (F.map right), and a fork on F.map left and F.map right, we get a cone on F.

                                                If you're thinking about using this, have a look at hasEqualizers_of_hasLimit_parallelPair, which you may find to be an easier way of achieving your goal.

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                                                  This is a helper construction that can be useful when verifying that a category has all coequalizers. Given F : WalkingParallelPair ⥤ C, which is really the same as parallelPair (F.map left) (F.map right), and a cofork on F.map left and F.map right, we get a cocone on F.

                                                  If you're thinking about using this, have a look at hasCoequalizers_of_hasColimit_parallelPair, which you may find to be an easier way of achieving your goal.

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                                                    Given F : WalkingParallelPair ⥤ C, which is really the same as parallelPair (F.map left) (F.map right) and a cone on F, we get a fork on F.map left and F.map right.

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                                                      Given F : WalkingParallelPair ⥤ C, which is really the same as parallelPair (F.map left) (F.map right) and a cocone on F, we get a cofork on F.map left and F.map right.

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                                                        @[simp]
                                                        theorem CategoryTheory.Limits.Fork.mkHom_hom {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {f : X Y} {g : X Y} {s : CategoryTheory.Limits.Fork f g} {t : CategoryTheory.Limits.Fork f g} (k : s.pt t.pt) (w : CategoryTheory.CategoryStruct.comp k t = s) :
                                                        def CategoryTheory.Limits.Fork.mkHom {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {f : X Y} {g : X Y} {s : CategoryTheory.Limits.Fork f g} {t : CategoryTheory.Limits.Fork f g} (k : s.pt t.pt) (w : CategoryTheory.CategoryStruct.comp k t = s) :
                                                        s t

                                                        Helper function for constructing morphisms between equalizer forks.

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                                                          def CategoryTheory.Limits.Fork.ext {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {f : X Y} {g : X Y} {s : CategoryTheory.Limits.Fork f g} {t : CategoryTheory.Limits.Fork f g} (i : s.pt t.pt) (w : autoParam (CategoryTheory.CategoryStruct.comp i.hom t = s) _auto✝) :
                                                          s t

                                                          To construct an isomorphism between forks, it suffices to give an isomorphism between the cone points and check that it commutes with the ι morphisms.

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                                                            Every fork is isomorphic to one of the form Fork.of_ι _ _.

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                                                              def CategoryTheory.Limits.Fork.isLimitOfIsos {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {f : X Y} {g : X Y} {X' : C} {Y' : C} (c : CategoryTheory.Limits.Fork f g) (hc : CategoryTheory.Limits.IsLimit c) {f' : X' Y'} {g' : X' Y'} (c' : CategoryTheory.Limits.Fork f' g') (e₀ : X X') (e₁ : Y Y') (e : c.pt c'.pt) (comm₁ : autoParam (CategoryTheory.CategoryStruct.comp e₀.hom f' = CategoryTheory.CategoryStruct.comp f e₁.hom) _auto✝) (comm₂ : autoParam (CategoryTheory.CategoryStruct.comp e₀.hom g' = CategoryTheory.CategoryStruct.comp g e₁.hom) _auto✝) (comm₃ : autoParam (CategoryTheory.CategoryStruct.comp e.hom c' = CategoryTheory.CategoryStruct.comp c e₀.hom) _auto✝) :

                                                              Given two forks with isomorphic components in such a way that the natural diagrams commute, then if one is a limit, then the other one is as well.

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                                                                @[simp]
                                                                def CategoryTheory.Limits.Cofork.mkHom {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {f : X Y} {g : X Y} {s : CategoryTheory.Limits.Cofork f g} {t : CategoryTheory.Limits.Cofork f g} (k : s.pt t.pt) (w : CategoryTheory.CategoryStruct.comp s k = t) :
                                                                s t

                                                                Helper function for constructing morphisms between coequalizer coforks.

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                                                                  theorem CategoryTheory.Limits.Fork.hom_comp_ι {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {f : X Y} {g : X Y} {s : CategoryTheory.Limits.Fork f✝ g} {t : CategoryTheory.Limits.Fork f✝ g} (f : s t) :
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                                                                  theorem CategoryTheory.Limits.Fork.π_comp_hom {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {f : X Y} {g : X Y} {s : CategoryTheory.Limits.Cofork f✝ g} {t : CategoryTheory.Limits.Cofork f✝ g} (f : s t) :
                                                                  def CategoryTheory.Limits.Cofork.ext {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {f : X Y} {g : X Y} {s : CategoryTheory.Limits.Cofork f g} {t : CategoryTheory.Limits.Cofork f g} (i : s.pt t.pt) (w : autoParam (CategoryTheory.CategoryStruct.comp s i.hom = t) _auto✝) :
                                                                  s t

                                                                  To construct an isomorphism between coforks, it suffices to give an isomorphism between the cocone points and check that it commutes with the π morphisms.

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                                                                    Every cofork is isomorphic to one of the form Cofork.ofπ _ _.

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                                                                      @[reducible, inline]
                                                                      abbrev CategoryTheory.Limits.HasEqualizer {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} (f : X Y) (g : X Y) :

                                                                      HasEqualizer f g represents a particular choice of limiting cone for the parallel pair of morphisms f and g.

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                                                                        @[reducible, inline]
                                                                        noncomputable abbrev CategoryTheory.Limits.equalizer {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} (f : X Y) (g : X Y) [CategoryTheory.Limits.HasEqualizer f g] :
                                                                        C

                                                                        If an equalizer of f and g exists, we can access an arbitrary choice of such by saying equalizer f g.

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                                                                          @[reducible, inline]

                                                                          If an equalizer of f and g exists, we can access the inclusion equalizer f g ⟶ X by saying equalizer.ι f g.

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                                                                            @[reducible, inline]

                                                                            An equalizer cone for a parallel pair f and g

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                                                                              The equalizer built from equalizer.ι f g is limiting.

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                                                                                @[reducible, inline]

                                                                                A morphism k : W ⟶ X satisfying k ≫ f = k ≫ g factors through the equalizer of f and g via equalizer.lift : W ⟶ equalizer f g.

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                                                                                  A morphism k : W ⟶ X satisfying k ≫ f = k ≫ g induces a morphism l : W ⟶ equalizer f g satisfying l ≫ equalizer.ι f g = k.

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                                                                                    Two maps into an equalizer are equal if they are equal when composed with the equalizer map.

                                                                                    An equalizer morphism is a monomorphism

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                                                                                    The equalizer morphism in any limit cone is a monomorphism.

                                                                                    def CategoryTheory.Limits.idFork {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {f : X Y} {g : X Y} (h : f = g) :

                                                                                    The identity determines a cone on the equalizer diagram of f and g if f = g.

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                                                                                      The identity on X is an equalizer of (f, g), if f = g.

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                                                                                        Every equalizer of (f, g), where f = g, is an isomorphism.

                                                                                        The equalizer of (f, g), where f = g, is an isomorphism.

                                                                                        Every equalizer of (f, f) is an isomorphism.

                                                                                        An equalizer that is an epimorphism is an isomorphism.

                                                                                        theorem CategoryTheory.Limits.eq_of_epi_fork_ι {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {f : X Y} {g : X Y} (t : CategoryTheory.Limits.Fork f g) [CategoryTheory.Epi t] :
                                                                                        f = g

                                                                                        Two morphisms are equal if there is a fork whose inclusion is epi.

                                                                                        If the equalizer of two morphisms is an epimorphism, then the two morphisms are equal.

                                                                                        The equalizer inclusion for (f, f) is an isomorphism.

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                                                                                        The equalizer of a morphism with itself is isomorphic to the source.

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                                                                                          @[reducible, inline]
                                                                                          abbrev CategoryTheory.Limits.HasCoequalizer {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} (f : X Y) (g : X Y) :

                                                                                          HasCoequalizer f g represents a particular choice of colimiting cocone for the parallel pair of morphisms f and g.

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                                                                                            @[reducible, inline]
                                                                                            noncomputable abbrev CategoryTheory.Limits.coequalizer {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} (f : X Y) (g : X Y) [CategoryTheory.Limits.HasCoequalizer f g] :
                                                                                            C

                                                                                            If a coequalizer of f and g exists, we can access an arbitrary choice of such by saying coequalizer f g.

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                                                                                              @[reducible, inline]

                                                                                              If a coequalizer of f and g exists, we can access the corresponding projection by saying coequalizer.π f g.

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                                                                                                @[reducible, inline]

                                                                                                An arbitrary choice of coequalizer cocone for a parallel pair f and g.

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                                                                                                  The cofork built from coequalizer.π f g is colimiting.

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                                                                                                    @[reducible, inline]

                                                                                                    Any morphism k : Y ⟶ W satisfying f ≫ k = g ≫ k factors through the coequalizer of f and g via coequalizer.desc : coequalizer f g ⟶ W.

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                                                                                                      Any morphism k : Y ⟶ W satisfying f ≫ k = g ≫ k induces a morphism l : coequalizer f g ⟶ W satisfying coequalizer.π ≫ g = l.

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                                                                                                        Two maps from a coequalizer are equal if they are equal when composed with the coequalizer map

                                                                                                        A coequalizer morphism is an epimorphism

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                                                                                                        The coequalizer morphism in any colimit cocone is an epimorphism.

                                                                                                        def CategoryTheory.Limits.idCofork {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} {f : X Y} {g : X Y} (h : f = g) :

                                                                                                        The identity determines a cocone on the coequalizer diagram of f and g, if f = g.

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                                                                                                          The identity on Y is a coequalizer of (f, g), where f = g.

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                                                                                                            Every coequalizer of (f, g), where f = g, is an isomorphism.

                                                                                                            The coequalizer of (f, g), where f = g, is an isomorphism.

                                                                                                            A coequalizer that is a monomorphism is an isomorphism.

                                                                                                            Two morphisms are equal if there is a cofork whose projection is mono.

                                                                                                            If the coequalizer of two morphisms is a monomorphism, then the two morphisms are equal.

                                                                                                            The coequalizer projection for (f, f) is an isomorphism.

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                                                                                                            The coequalizer of a morphism with itself is isomorphic to the target.

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                                                                                                              The comparison morphism for the equalizer of f,g. This is an isomorphism iff G preserves the equalizer of f,g; see CategoryTheory/Limits/Preserves/Shapes/Equalizers.lean

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                                                                                                                A split mono f equalizes (retraction f ≫ f) and (𝟙 Y). Here we build the cone, and show in isSplitMonoEqualizes that it is a limit cone.

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                                                                                                                  A split mono f equalizes (retraction f ≫ f) and (𝟙 Y).

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                                                                                                                    We show that the converse to isSplitMonoEqualizes is true: Whenever f equalizes (r ≫ f) and (𝟙 Y), then r is a retraction of f.

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                                                                                                                      The fork obtained by postcomposing an equalizer fork with a monomorphism is an equalizer.

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                                                                                                                        An equalizer of an idempotent morphism and the identity is split mono.

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                                                                                                                          The equalizer of an idempotent morphism and the identity is split mono.

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                                                                                                                            A split epi f coequalizes (f ≫ section_ f) and (𝟙 X). Here we build the cocone, and show in isSplitEpiCoequalizes that it is a colimit cocone.

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                                                                                                                              A split epi f coequalizes (f ≫ section_ f) and (𝟙 X).

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                                                                                                                                We show that the converse to isSplitEpiEqualizes is true: Whenever f coequalizes (f ≫ s) and (𝟙 X), then s is a section of f.

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                                                                                                                                  The cofork obtained by precomposing a coequalizer cofork with an epimorphism is a coequalizer.

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                                                                                                                                    The coequalizer of an idempotent morphism and the identity is split epi.

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