Documentation

Mathlib.CategoryTheory.Limits.IsLimit

Limits and colimits #

We set up the general theory of limits and colimits in a category. In this introduction we only describe the setup for limits; it is repeated, with slightly different names, for colimits.

The main structures defined in this file is

See also CategoryTheory.Limits.HasLimits which further builds:

Implementation #

At present we simply say everything twice, in order to handle both limits and colimits. It would be highly desirable to have some automation support, e.g. a @[dualize] attribute that behaves similarly to @[to_additive].

References #

A cone t on F is a limit cone if each cone on F admits a unique cone morphism to t.

See https://stacks.math.columbia.edu/tag/002E.

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    Given a natural transformation α : F ⟶ G, we give a morphism from the cone point of any cone over F to the cone point of a limit cone over G.

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      Restating the definition of a limit cone in terms of the ∃! operator.

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

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        Alternative constructor for isLimit, providing a morphism of cones rather than a morphism between the cone points and separately the factorisation condition.

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          Transport evidence that a cone is a limit cone across an isomorphism of cones.

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            If the canonical morphism from a cone point to a limiting cone point is an iso, then the first cone was limiting also.

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              Two morphisms into a limit are equal if their compositions with each cone morphism are equal.

              Given two functors which have equivalent categories of cones, we can transport a limiting cone across the equivalence.

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                A cone postcomposed with a natural isomorphism is a limit cone if and only if the original cone is.

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                  A cone postcomposed with the inverse of a natural isomorphism is a limit cone if and only if the original cone is.

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                    Constructing an equivalence IsLimit c ≃ IsLimit d from a natural isomorphism between the underlying functors, and then an isomorphism between c transported along this and d.

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                      The cone points of two limit cones for naturally isomorphic functors are themselves isomorphic.

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                        We can prove two cone points (s : Cone F).pt and (t : Cone G).pt are isomorphic if

                        • both cones are limit cones
                        • their indexing categories are equivalent via some e : J ≌ K,
                        • the triangle of functors commutes up to a natural isomorphism: e.functor ⋙ G ≅ F.

                        This is the most general form of uniqueness of cone points, allowing relabelling of both the indexing category (up to equivalence) and the functor (up to natural isomorphism).

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                          The universal property of a limit cone: a map W ⟶ X is the same as a cone on F with cone point W.

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                            The limit of F represents the functor taking W to the set of cones on F with cone point W.

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                              Another, more explicit, formulation of the universal property of a limit cone. See also homIso.

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                                If G : C → D is a faithful functor which sends t to a limit cone, then it suffices to check that the induced maps for the image of t can be lifted to maps of C.

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                                  If F and G are naturally isomorphic, then F.mapCone c being a limit implies G.mapCone c is also a limit.

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                                    A cone is a limit cone exactly if there is a unique cone morphism from any other cone.

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                                      If F.cones is represented by X, each morphism f : Y ⟶ X gives a cone with cone point Y.

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                                        If F.cones is represented by X, each cone s gives a morphism s.pt ⟶ X.

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                                          If F.cones is represented by X, the cone corresponding to the identity morphism on X will be a limit cone.

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                                            If F.cones is representable, then the cone corresponding to the identity morphism on the representing object is a limit cone.

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                                              A cocone t on F is a colimit cocone if each cocone on F admits a unique cocone morphism from t.

                                              See https://stacks.math.columbia.edu/tag/002F.

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                                                Given a natural transformation α : F ⟶ G, we give a morphism from the cocone point of a colimit cocone over F to the cocone point of any cocone over G.

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                                                  Restating the definition of a colimit cocone in terms of the ∃! operator.

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

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                                                    Alternative constructor for IsColimit, providing a morphism of cocones rather than a morphism between the cocone points and separately the factorisation condition.

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                                                      Transport evidence that a cocone is a colimit cocone across an isomorphism of cocones.

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                                                        If the canonical morphism to a cocone point from a colimiting cocone point is an iso, then the first cocone was colimiting also.

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                                                          Two morphisms out of a colimit are equal if their compositions with each cocone morphism are equal.

                                                          Given two functors which have equivalent categories of cocones, we can transport a colimiting cocone across the equivalence.

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                                                            A cocone precomposed with a natural isomorphism is a colimit cocone if and only if the original cocone is.

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                                                              A cocone precomposed with the inverse of a natural isomorphism is a colimit cocone if and only if the original cocone is.

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                                                                Constructing an equivalence is_colimit c ≃ is_colimit d from a natural isomorphism between the underlying functors, and then an isomorphism between c transported along this and d.

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                                                                  The cocone points of two colimit cocones for naturally isomorphic functors are themselves isomorphic.

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                                                                    We can prove two cocone points (s : Cocone F).pt and (t : Cocone G).pt are isomorphic if

                                                                    • both cocones are colimit cocones
                                                                    • their indexing categories are equivalent via some e : J ≌ K,
                                                                    • the triangle of functors commutes up to a natural isomorphism: e.functor ⋙ G ≅ F.

                                                                    This is the most general form of uniqueness of cocone points, allowing relabelling of both the indexing category (up to equivalence) and the functor (up to natural isomorphism).

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                                                                      The universal property of a colimit cocone: a map X ⟶ W is the same as a cocone on F with cone point W.

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                                                                        The colimit of F represents the functor taking W to the set of cocones on F with cone point W.

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                                                                          Another, more explicit, formulation of the universal property of a colimit cocone. See also homIso.

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                                                                            If G : C → D is a faithful functor which sends t to a colimit cocone, then it suffices to check that the induced maps for the image of t can be lifted to maps of C.

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                                                                              If F and G are naturally isomorphic, then F.mapCocone c being a colimit implies G.mapCocone c is also a colimit.

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                                                                                A cocone is a colimit cocone exactly if there is a unique cocone morphism from any other cocone.

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                                                                                  If F.cocones is corepresented by X, each morphism f : X ⟶ Y gives a cocone with cone point Y.

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                                                                                    If F.cocones is corepresented by X, each cocone s gives a morphism X ⟶ s.pt.

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                                                                                      If F.cocones is corepresented by X, the cocone corresponding to the identity morphism on X will be a colimit cocone.

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                                                                                        If F.cocones is corepresentable, then the cocone corresponding to the identity morphism on the representing object is a colimit cocone.

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