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Mathlib.CategoryTheory.EssentialImage

Essential image of a functor #

The essential image essImage of a functor consists of the objects in the target category which are isomorphic to an object in the image of the object function. This, for instance, allows us to talk about objects belonging to a subcategory expressed as a functor rather than a subtype, preserving the principle of equivalence. For example this lets us define exponential ideals.

The essential image can also be seen as a subcategory of the target category, and witnesses that a functor decomposes into an essentially surjective functor and a fully faithful functor. (TODO: show that this decomposition forms an orthogonal factorisation system).

The essential image of a functor F consists of those objects in the target category which are isomorphic to an object in the image of the function F.obj. In other words, this is the closure under isomorphism of the function F.obj. This is the "non-evil" way of describing the image of a functor.

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    Get the witnessing object that Y is in the subcategory given by F.

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      Being in the essential image is a "hygienic" property: it is preserved under isomorphism.

      If Y is in the essential image of F then it is in the essential image of F' as long as F ≅ F'.

      An object in the image is in the essential image.

      The essential image of a functor, interpreted as a full subcategory of the target category.

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        • CategoryTheory.Functor.instFullEssImageSubcategoryInstCategoryEssImageSubcategoryEssImageInclusion = inferInstance

        A functor F : C ⥤ D is essentially surjective if every object of D is in the essential image of F. In other words, for every Y : D, there is some X : C with F.obj X ≅ Y.

        See .

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          Given an essentially surjective functor, we can find a preimage for every object Y in the codomain. Applying the functor to this preimage will yield an object isomorphic to Y, see obj_obj_preimage_iso.

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            Applying an essentially surjective functor to a preimage of Y yields an object that is isomorphic to Y.

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