Documentation

Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects

Zero objects #

A category "has a zero object" if it has an object which is both initial and terminal. Having a zero object provides zero morphisms, as the unique morphisms factoring through the zero object; see CategoryTheory.Limits.Shapes.ZeroMorphisms.

References #

An object X in a category is a zero object if for every object Y there is a unique morphism to : X → Y and a unique morphism from : Y → X.

This is a characteristic predicate for has_zero_object.

  • unique_to : ∀ (Y : C), Nonempty (Unique (X Y))

    there are unique morphisms to the object

  • unique_from : ∀ (Y : C), Nonempty (Unique (Y X))

    there are unique morphisms from the object

Instances For

    there are unique morphisms to the object

    there are unique morphisms from the object

    If h : IsZero X, then h.to_ Y is a choice of unique morphism X → Y.

    Equations
    • h.to_ Y = default
    Instances For

      If h : is_zero X, then h.from_ Y is a choice of unique morphism Y → X.

      Equations
      • h.from_ Y = default
      Instances For
        theorem CategoryTheory.Limits.IsZero.eq_from {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} (h : CategoryTheory.Limits.IsZero X) (f : Y X) :
        f = h.from_ Y
        theorem CategoryTheory.Limits.IsZero.from_eq {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} (h : CategoryTheory.Limits.IsZero X) (f : Y X) :
        h.from_ Y = f
        theorem CategoryTheory.Limits.IsZero.eq_of_src {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} (hX : CategoryTheory.Limits.IsZero X) (f : X Y) (g : X Y) :
        f = g
        theorem CategoryTheory.Limits.IsZero.eq_of_tgt {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} (hX : CategoryTheory.Limits.IsZero X) (f : Y X) (g : Y X) :
        f = g

        Any two zero objects are isomorphic.

        Equations
        • hX.iso hY = { hom := hX.to_ Y, inv := hX.from_ Y, hom_inv_id := , inv_hom_id := }
        Instances For

          The (unique) isomorphism between any initial object and the zero object.

          Equations
          • hX.isoIsInitial hY = hX.isInitial.uniqueUpToIso hY
          Instances For

            The (unique) isomorphism between any terminal object and the zero object.

            Equations
            • hX.isoIsTerminal hY = hX.isTerminal.uniqueUpToIso hY
            Instances For

              A category "has a zero object" if it has an object which is both initial and terminal.

              Instances

                Construct a Zero C for a category with a zero object. This can not be a global instance as it will trigger for every Zero C typeclass search.

                Equations
                Instances For

                  Every zero object is isomorphic to the zero object.

                  Equations
                  • hX.isoZero = hX.iso
                  Instances For

                    There is a unique morphism from the zero object to any object X.

                    Equations
                    Instances For

                      There is a unique morphism from any object X to the zero object.

                      Equations
                      Instances For

                        A zero object is in particular initial.

                        Equations
                        • CategoryTheory.Limits.HasZeroObject.zeroIsInitial = .isInitial
                        Instances For

                          A zero object is in particular terminal.

                          Equations
                          • CategoryTheory.Limits.HasZeroObject.zeroIsTerminal = .isTerminal
                          Instances For
                            @[instance 10]

                            A zero object is in particular initial.

                            Equations
                            • =
                            @[instance 10]

                            A zero object is in particular terminal.

                            Equations
                            • =

                            The (unique) isomorphism between any initial object and the zero object.

                            Equations
                            Instances For

                              The (unique) isomorphism between any terminal object and the zero object.

                              Equations
                              Instances For

                                The (unique) isomorphism between the chosen initial object and the chosen zero object.

                                Equations
                                • CategoryTheory.Limits.HasZeroObject.zeroIsoInitial = CategoryTheory.Limits.HasZeroObject.zeroIsInitial.uniqueUpToIso CategoryTheory.Limits.initialIsInitial
                                Instances For

                                  The (unique) isomorphism between the chosen terminal object and the chosen zero object.

                                  Equations
                                  • CategoryTheory.Limits.HasZeroObject.zeroIsoTerminal = CategoryTheory.Limits.HasZeroObject.zeroIsTerminal.uniqueUpToIso CategoryTheory.Limits.terminalIsTerminal
                                  Instances For