Image

Let $\mathcal{C}$ be a category and a $f:A\to B$ a morphism from objects $A$ to $B$ in $\mathcal{C}$ . An image of $f$ is a subobject $I$ of $B$ with a representing monomorphism $g:I\to B$ , such that

• $f$ factors through $g$ ; i.e. there is a morphism $h:A\to I$ such that $f=g\circ h$ : $$A \goto{f} B = A \goto{h} I \goto{g} B$$
• if $f$ factors through a monomorphism $s:J\to B$ : $$A \goto{f} B = A \goto{t} J \goto{s} B,$$ then there is a morphism $x:I\to J$ such that $$I \goto{g} B = I \goto{x} J \goto{s} B.$$

Example. In the category of sets, the image of a function $f:A\to B$ is just the image $f(A)$ with the canonical injection into $B$ .

In the literature, the first bullet is equivalent to saying that the subobject $g: C\to B$ allows $f$ . So the definition of the image of $f$ is the smallest subobject of $B$ that allows $f:A\to B$ .

Coimage

Dually, one defines the coimage of $f:A\to B$ as a quotient object $C$ of $A$ with a representing epimorphism $h: A\to C$ , such that

• $f$ factors through $h$ ; i.e. there is a morphism $g:C\to B$ such that $f=g\circ h$ : $$A \goto{f} B = A \goto{h} C \goto{g} B$$
• if $f$ factors through an epimorphism $t:A\to D$ : $$A \goto{f} B = A \goto{t} D \goto{s} B,$$ then there is a morphism $y:D\to C$ such that $$A \goto{h} C = A \goto{t} D \goto{y} C.$$

Remarks.

1. In the definition of image, since $g$ is a monomorphism, $x$ is a monomorphism. Furthermore, since $s$ is a monomorphism, $x$ is uniquely determined, and we have the following commutative diagram:
   
2. Dually, $y$ is a uniquely determined epimorphism satisfying the following commutative diagram:
   
3. If $f:A\to B$ has an image (dually, coimage), it is unique up to isomorphism. The image and coimage of $f$ are denoted by $\im(f)$ and $\coim(f)$ respectively.
4. Suppose that a category is Ab1. If $\im(f)$ and $\coim(f)$ exist for $f:A\to B$ , then there is a unique morphism $\overline{f}:\coim(f)\to \im(f)$ such that
   
   is commutative. The Ab2 Axiom, a la Grothendieck, is the statement that if $\overline{f}$ exists, it is an isomorphism. A category is said to be Ab2 if it is Ab1, and every morphism satisfies the Ab2 Axiom.
5. A category $\mathcal{C}$ is said to have images (dually, has coimages) if the image (coimage) of any morphism exists.

Every abelian category has images and coimages, and $$\im(f)=\ker(\cok(f))\qquad \mbox{ and }\qquad\coim(f)=\cok(\ker(f)),$$ where $\ker$ and $\cok$ are the kernel and cokernel operations. In addition, we have the following important result:

if a morphism $f:A\to B$ can be factored as $$A \goto{f} B = A \goto{h} I \goto{g} B$$ with $g$ a monomorphism and $h$ an epimorphism, then $I$ (with $g$ ) is the image of $f$ and $I$ (with $h$ ) is the coimage of $f$ .