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Mathlib.CategoryTheory.ObjectProperty.ColimitsOfShape

Objects that are colimits of objects satisfying a certain property #

Given a property of objects P : ObjectProperty C and a category J, we introduce two properties of objects P.strictColimitsOfShape J and P.colimitsOfShape J. The former contains exactly the objects of the form colimit F for any functor F : J ⥤ C that has a colimit and such that F.obj j satisfies P for any j, while the latter contains all the objects that are isomorphic to these "chosen" objects colimit F.

Under certain circumstances, the type of objects satisfying P.strictColimitsOfShape J is small: the main reason this variant is introduced is to deduce that the full subcategory of P.colimitsOfShape J is essentially small.

TODO #

The property of objects that are equal to colimit F for some functor F : J ⥤ C where all F.obj j satisfy P.

Instances For
    structure CategoryTheory.ObjectProperty.ColimitOfShape {C : Type u_1} [Category.{u_2, u_1} C] (P : ObjectProperty C) (J : Type u') [Category.{v', u'} J] (X : C) extends CategoryTheory.Limits.ColimitPresentation J X :
    Type (max (max (max u' u_1) u_2) v')

    A structure expressing that X : C is the colimit of a functor diag : J ⥤ C such that P (diag.obj j) holds for all j.

    Instances For
      noncomputable def CategoryTheory.ObjectProperty.ColimitOfShape.colimit {C : Type u_1} [Category.{u_2, u_1} C] {P : ObjectProperty C} {J : Type u'} [Category.{v', u'} J] (F : Functor J C) [Limits.HasColimit F] (hF : ∀ (j : J), P (F.obj j)) :

      If F : J ⥤ C is a functor that has a colimit and is such that for all j, F.obj j satisfies a property P, then this structure expresses that colimit F is indeed a colimit of objects satisfying P.

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        If X is a colimit indexed by J of objects satisfying a property P, then any object that is isomorphic to X also is.

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        • h.ofIso e = { toColimitPresentation := h.ofIso e, prop_diag_obj := }
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          If X is a colimit indexed by J of objects satisfying a property P, it is also a colimit indexed by J of objects satisfying Q if P ≤ Q.

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            The property of objects that are the point of a colimit cocone for a functor F : J ⥤ C where all objects F.obj j satisfy P.

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