nLab
essential image

Context

Category theory

Higher category theory

higher category theory

Basic concepts

Basic theorems

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Morphisms

Functors

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Extra properties and structure

1-categorical presentations

The essential image of a functor

Idea

The notion of essential image is supposed to be an adaptation of the notion of image from a 1-categorical to the 2-categorical context Cat, i.e. to the image of functors. But some care has to be exercised.

For another definition of image of a functor, see (2,1)-image.

Definitions

(A concrete realization of) the essential image of a functor F:ABF: A\to B between categories or nn-categories is the smallest replete subcategory of the target nn-category BB containing the image of FF. (The image is, in turn, the smallest subcategory which contains all the nn-cells which are strictly the images of nn-cells in AA.)

Note that if FF is not pseudomonic, then its essential image, defined in this way, need not be equivalent to its ordinary image.

Considerations of evil

Note that the property of “belonging to the image” (said of an object or morphism) is evil; of two equivalent objects, one may belong while the other does not. Passing to the essential image removes this, so that the property of “belonging to the essential image” is no longer evil.

Of course, the property of “being equal to the essential image” (said of a subcategory) is evil, as is the property of “being replete”. But DD is a replete subcategory of CC if and only if the property of belonging to DD (said of an object or morphism) is not evil.

Remarks

  • the image of a functor F:ABF:A\to B may contain some morphisms or cells which are not the images of any cell in AA, namely the compositions of BB-composable chains of such image cells, whose preimage cells do not form any AA-composable chain;
  • the essential image in addition contains precisely all equivalent kk-cells to the kk-cells of the image for all 0kn0 \leq k \leq n.

Revised on February 4, 2011 17:24:14 by Zoran Škoda (161.53.130.104)