essential image

- homotopy hypothesis-theorem
- delooping hypothesis-theorem
- periodic table
- stabilization hypothesis-theorem
- exactness hypothesis
- holographic principle

- (n,r)-category
- Theta-space
- ∞-category/ω-category
- (∞,n)-category
- (∞,2)-category
- (∞,1)-category
- (∞,0)-category/∞-groupoid
- n-category = (n,n)-category
- n-poset = (n-1,n)-category
- n-groupoid = (n,0)-category

- categorification/decategorification
- geometric definition of higher category
- algebraic definition of higher category
- stable homotopy theory

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.

(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.

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.

- 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 $0 \leq k \leq n$.

Revised on September 25, 2017 09:21:16
by Matt Earnshaw
(144.82.8.59)