nLab essential image

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 $F: A\to B$ between categories or $n$-categories is the smallest replete subcategory of the target $n$-category $B$ containing the image of $F$. (The image is, in turn, the smallest subcategory which contains all the $n$-cells which are strictly the images of $n$-cells in $A$.)

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

Considerations of the principle of equivalence

Note that the property of “belonging to the image” (said of an object or morphism) breaks the principle of equivalence of category theory; 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” respects the principle of equivalence.

Of course, the property of “being equal to the essential image” (said of a subcategory) violates the principle of equivalence, as is the property of “being replete”. But $D$ is a replete subcategory of $C$ if and only if the property of belonging to $D$ (said of an object or morphism) does not violate the principle of equivalence.

Remarks

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