nLab
essential image

Idea

The notion of essential image is an adaptation of the notion of image from a 1-categorical to the 2-categorical context Cat, i.e. to the image of functors.

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 $F$, which is in turn the smallest subcategory which contains all the $n$-cells which are strictly the images of $n$-cells in $A$.

Mike Shulman: I don’t think this is well-defined. If the inclusion of the image is not pseudomonic, then it doesn’t necessarily have a repletion. I don’t remember how I’ve heard “essential image” used in the past, but I think at least sometimes it means the essential full image, i.e. the repletion of the full image. If it doesn’t mean that, then I don’t know what it can mean for functors whose image is not pseudomonic.

Toby: I know the ‘weak coimage?’ and the ‘full image’ as described in the last section of this work for week 1 of John’s 2004 spring seminar.

Mike Shulman: Yes, but I don’t think either of those is helpful. The full image is, well, full, so its repletion would be the essential full image. And the weak coimage need not be pseudomonic in general.

Note that the property of belonging to the image is evil; of two equivalent objects, one may belong while the other does not. Passing to the essential image precisely removes this evil.

Zoran: I disagree that it removes. Usual image in Set splits maps into epi onto image and mono into target. If in Cat one splits the functor into functor into image then the rest can be split into equivalence from image to essential image and a functor form essential image into final category which is fully faithful and replete. Repletness is a part which is still “evil”.

Toby: It is the property of belonging to the image (said of an object or morphism) which is evil, while the property of belonging to the essential image (said of an object or morphism) is not evil. Of course the property of being equal to the essential image (said of a subcategory) is evil; it must be, as it refers to equality of something other than top-level morphisms. Even the property of being replete (again, said of a subcategory) is evil … but $D$ is a replete subcategory of $C$ if and only if the property of belonging to $D$ (said of an object or morphism) is not evil.

Note that:

• the image 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 $0\le k\le n$.