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[parent] image of a morphism (Definition)

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$:
    $\displaystyle A \buildrel f \over \longrightarrow B = A \buildrel h \over \longrightarrow I \buildrel g \over \longrightarrow B$
  • if $ f$ factors through a monomorphism $ s:J\to B$:
    $\displaystyle A \buildrel f \over \longrightarrow B = A \buildrel t \over \longrightarrow J \buildrel s \over \longrightarrow B,$
    then there is a morphism $ x:I\to J$ such that
    $\displaystyle I \buildrel g \over \longrightarrow B = I \buildrel x \over \longrightarrow J \buildrel s \over \longrightarrow B.$

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$:
    $\displaystyle A \buildrel f \over \longrightarrow B = A \buildrel h \over \longrightarrow C \buildrel g \over \longrightarrow B$
  • if $ f$ factors through an epimorphism $ t:A\to D$:
    $\displaystyle A \buildrel f \over \longrightarrow B = A \buildrel t \over \longrightarrow D \buildrel s \over \longrightarrow B,$
    then there is a morphism $ y:D\to C$ such that
    $\displaystyle A \buildrel h \over \longrightarrow C = A \buildrel t \over \longrightarrow D \buildrel y \over \longrightarrow C.$

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:
    $ \xymatrix@R-=2pt{ & I \ar[dd]^x \ar[dr]^g & \ A\ar[ur]^h \ar[dr]_t && B \ & J \ar[ur]_s & }$
  2. Dually, $ y$ is a uniquely determined epimorphism satisfying the following commutative diagram:
    $ \xymatrix@R-=2pt{ & D \ar[dd]^y \ar[dr]^g & \ A\ar[ur]^t \ar[dr]_h && B \ & C \ar[ur]_s & }$
  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 $ \operatorname{im}(f)$ and $ \operatorname{coim}(f)$ respectively.
  4. Suppose that a category is Ab1. If $ \operatorname{im}(f)$ and $ \operatorname{coim}(f)$ exist for $ f:A\to B$, then there is a unique morphism $ \overline{f}:\operatorname{coim}(f)\to \operatorname{im}(f)$ such that
    $\displaystyle \xymatrix@C+=4pc{ {A}\ar[r]^{f}\ar[d]&{B}\ {\operatorname{coim}(f)}\ar[r]^{\overline{f}}&{\operatorname{im}(f)}\ar[u] } $
    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

$\displaystyle \operatorname{im}(f)=\ker(\operatorname{cok}(f))$    and $\displaystyle \qquad\operatorname{coim}(f)=\operatorname{cok}(\ker(f)),$
where $ \ker$ and $ \operatorname{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
$\displaystyle A \buildrel f \over \longrightarrow B = A \buildrel h \over \longrightarrow I \buildrel g \over \longrightarrow B$
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$.
In other words, the factorization above is uniquely determined, up to isomorphism.



"image of a morphism" is owned by CWoo.
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See Also: exact sequence, categorical sequence

Other names:  have images, have coimages, Ab2 axiom
Also defines:  image, coimage, has images, has coimages, Ab2

This object's parent.

Attachments:
inverse image of a morphism (Definition) by CWoo
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Cross-references: addition, operations, cokernel, kernel, abelian category, commutative, Ab1, isomorphism, commutative diagram, epimorphism, quotient object, equivalent, canonical injection, function, category of sets, factors, monomorphism, subobject, objects, morphism, category
There are 26 references to this entry.

This is version 11 of image of a morphism, born on 2008-06-08, modified 2008-09-04.
Object id is 10684, canonical name is ImageOfAMorphism.
Accessed 1451 times total.

Classification:
AMS MSC: 18A05 (Category theory; homological algebra :: General theory of categories and functors :: Definitions, generalizations)
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