The image of a function $f\colon A\to B$ between sets is the subset of $B$ consisting of all those elements $b\in B$ that are of the form $f(a)$ for some $a\in A$. This notion can be generalized from Set to other categories, as follows.
To discuss images in a category $C$, we must first fix a notion of subobject or embedding in $C$. (Sometimes we want these to be the monomorphisms, but sometimes we want the regular monomorphisms instead.) Then the image of a morphism $f\colon A\to B$ in $C$ is a universal factorization of $f$ into a composite $A \to im(f) \to B$ such that $im(f)\to B$ is a subobject, of the specified sort.
Note that in this generality, a given morphism may or may not have an image, although if it does, it is unique up to isomorphism by universality. In many cases, images can be constructed out of limits and colimits in the ambient category. In particular, in a regular category, images (relative to all monomorphisms) can be constructed as the quotient object of a kernel pair.
Let $C$ be a category, let $M\subset Mono(C)$ be a subclass of the monomorphisms in $C$, and let $f: c \to d$ be a morphism in $C$. The image of $f$ is the smallest $M$-subobject $im(f) \hookrightarrow d$ through which $f$ factors (if it exists). The factorizing morphism $c \to im(f)$ is sometimes called the corestriction of $f$ (or coastriction, see mathoverflow):
In other words, it is a factorization $c \overset{e}{\to} im(f) \overset{m}{\to} d$ of $f$ (i.e. $f = m e$) such that $m\in M$, and given any other factorization $f = m' e'$ with $m'\in M$, we have $m \subseteq m'$ as subobjects of $C$ (i.e. $m$ factors through $m'$, $m = m' k$ for some $k$). Such a factorization is unique up to unique isomorphism, if it exists.
(For instance if $M$ is the class of regular monomorphisms, then the $M$-image is the regular image. See below for more.)
This can be phrased equivalently as follows. Let $C/d$ be the slice category of $C$ over $d$, and let $M/d$ be its full subcategory whose objects are $M$-morphisms into $d$. If all images exist in $C$, then taking the image of a map $f: c \to d$ provides a left adjoint
to the inclusion $M/d \hookrightarrow C/d$. More generally, an image of a single morphism $f\colon c\to d$ is a universal arrow from $f$ to this inclusion.
In Set, for $M=$ monomorphisms = injections, this reproduces the ordinary notion of image.
More generally in any topos the (epi,mono) factorization system gives the factorization through the image $f \colon A \overset{epi}{\to} im(f) \overset{mono}{\to} B$.
Similarly in an abelian category (by definition) (epi,mono) factorization gives factorization through the image.
In algebraic categories such as Grp, for $M=$ monomorphisms, this also reproduces the ordinary notions of image.
In Top, for $M=$ subspace inclusions, the $M$-image is the set-theoretic image topologized as a subspace of the codomain. On the other hand, for $M=$ injective continuous maps, the $M$-image is the set-theoretic image topologized as a quotient space of the domain. For more basic details see at Introduction to Topology -- 1 here.
A regular category can be defined as a finitely complete category in which all images exist for $M=$ monomorphisms, and such images are moreover stable under pullback. In particular, this includes any topos.
In Cat (considered as a 1-category), the image of a functor $F\colon A\to B$ is the smallest subcategory of $B$ which contains images through $F$ of all morphisms in $A$. Note that some of the morphisms in the image may not be images of any morphism in $A$; all morphisms in the image of $F$ are compositions in $B$ of $B$-composable sequences of images of morphisms in $A$, but these themselves do not necessarily form $A$-composable sequences of morphisms in $A$.
Usually it is better to treat $Cat$ as a 2-category, in which case one can use a more 2-categorical notion of image. See, for instance, full image, essential image, and replete image.
(images preserve unions but not in general intersections)
Let $f \colon X \longrightarrow Y$ be a function between sets. Let $\{S_i \subset X\}_{i \in I}$ be a set of subsets of $X$. Then.
$im_f\left( \underset{i \in I}{\cup} S_i\right) = \left(\underset{i \in I}{\cup} im_f(S_i)\right)$
$im_f\left( \underset{i \in I}{\cap} S_i\right) \subset \left(\underset{i \in I}{\cap} im_f(S_i)\right)$
The inclusion in the second item is in general proper. If $f$ is an injective function and $I$ is inhabited then this is a bijection:
For details see at interactions of images and pre-images with unions and intersections.
Suppose that $M$ is closed under composition, and that $f = m e$ is an image factorization relative to $M$. Then $e$ has the property that if $e = n g$ for some $n\in M$, then $n$ is an isomorphism — for then we would have $f = (m n) g$ and so by universality of images, $m$ would factor through $m n$. In particular, if $M$ is the class of all monomorphisms and $C$ has equalizers, then $e$ is an extremal epimorphism.
If $C$ has pullbacks and $M$ is closed under pullbacks, then we can say more: $e$ is orthogonal to $M$. For if
is a commutative square with $n\in M$, then the pullback $k^*n$ is an $M$-morphism through which $e$ factors. Hence $k^*n$ must be an isomorphism, and so the square admits a diagonal filler, which is unique since $n\in M$ is monic. It follows that if all $M$-images exist in $C$, then $M$ is the right class of an orthogonal factorization system, and $M$-images are precisely the factorizations in this OFS.
Conversely, it is easy to see that if $(E,M)$ is an OFS on a category $C$, then all $M$-images exist and are given by the factorizations of the OFS. Therefore, to give a notion of image is more or less equivalent to giving an orthogonal factorization system.
Duality
Note that the notion of factorization system is self-dual. Therefore, if $(E,M)$ is a factorization system and $c \overset{e}{\to} a \overset{m}{\to} d$ is an $(E,M)$-factorization of $f\colon c\to d$, then not only is $m$ the $M$-image of $f$ (the least $M$-subobject through which $f$ factors), but dually $e$ is also the $E$-coimage of $f$, i.e. the greatest $E$-quotient through which $f$ factors.
However, see below for additional remarks on the usage of the terms “image” and “coimage.”
Suppose that the category $C$ admits finite limits and colimits, and that $M=RegMono$ consists of the regular monomorphisms. Then the $M$-image of a morphism $f : c \to d$ may be constructed as
where $d \sqcup_c d$ denotes the pushout
In other words, the regular image is the equalizer of the cokernel pair. To see that this is in fact the $RegMono$-image, we first note that it is of course a regular monomorphism by definition, and then invoke the fact that in a category with finite limits and colimits, a monomorphism is regular if and only if it is the equalizer of its cokernel pair.
Dually, the regular coimage of a morphism is the coequalizer of its kernel pair. In Set (and more generally in any topos) these two constructions coincide, but in general they are distinct. For example, in Top the regular image is the set-theoretic image topologized as a subspace of the codomain, while the regular coimage is the set-theoretic image topologized as a quotient space of the domain.
Note that some authors drop the “regular” and simply call these constructions the image and coimage respectively. This can be confusing, however, since in many cases (such as in any regular category) the regular coimage coincides with the $M$-image for $M=Mono$ the class of all monomorphisms, which it is also natural to simply call the image.
Suppose that $M_1$ and $M_2$ are two classes with $M_1\subseteq M_2$. If $f$ has both an $M_1$-image $im_1(f)$ and an $M_2$-image $im_2(f)$, then by universality, the latter must factor through the former. The correspondence between images and factorization systems also extends to pairs; see ternary factorization system.
As a special case of this, we have:
If $C$ has finite limits and colimits, then there is a unique map
from its regular coimage to its image such that
Because $f$ coequalizes $c \times_d c \rightrightarrows c$, a morphism $h$ in
exists uniquely.
Because $c \to coim f$ is epi it follows that $h$ equalizes $d \rightrightarrows d \sqcup_c d$ and hence $u$ in the diagram
exists uniquely.
If this map $u$ is an isomorphism, then $f$ is sometimes called a strict morphism. In particular, if $C$ has finite limits and colimits and every morphism is a strict morphism, then the regular image and regular coimage factorizations coincide and give an epi-mono factorization system.
In higher category theory there are generalizations of the notion of image, such as these:
However, it is not clear that either serves as the proper categorification of the notion described above.
There are several properties we might want a ‘higher image’ to have. For example, in an $2$-category, we might want isomorphic 1-cells to have equivalent images. In Cat, we might want the image of a functor between discrete categories to be its image as a function. One fruitful direction is to study a factorization system in a 2-category.
A (regular) $(\infty,1)$-image of a morphism $f : c \to d$ in an (∞,1)-category with (∞,1)-limits and -colimits should be defined to be the (∞,1)-limit over the Cech co-nerve of $f$:
Notice that
this reduces to the above equalizer definition in the case that the ambient $(\infty,1)$-category is just an ordinary category;
this implies that the inclusion $im f \to d$ is a regular monomorphism in the $(\infty,1)$-category sense (described here).
For more see n-image.
Applied to the $(\infty,1)$-category ∞Grpd this gives a notion of image of (∞,1)-functors between ∞-groupoids and hence a notion of image of functors between groupoids, 2-functors between 2-groupoids, etc.