Image (category theory)

In category theory, a branch of mathematics, the image of a morphism is a generalization of the image of a function.

General definition

Given a category $C$ and a morphism $f : X \to Y$ in $C$ , the image^[1] of $f$ is a monomorphism $m : I \to Y$ satisfying the following universal property:

There exists a morphism $e : X \to I$ such that $f = m e$ .
For any object $I^{'}$ with a morphism $e^{'} : X \to I^{'}$ and a monomorphism $m^{'} : I^{'} \to Y$ such that $f = m^{'} e^{'}$ , there exists a unique morphism $v : I \to I^{'}$ such that $m = m^{'} v$ .

Remarks:

such a factorization does not necessarily exist.
$e$ is unique by definition of $m$ monic.
$m^{'} e^{'} = f = m e = m^{'} v e$ , therefore $e^{'} = v e$ by $m^{'}$ monic.
$v$ is monic.
$m = m^{'} v$ already implies that $v$ is unique.

The image of $f$ is often denoted by $Im f$ or $Im (f)$ .

Proposition: If $C$ has all equalizers then the $e$ in the factorization $f = m e$ of (1) is an epimorphism.^[2]

Proof

Let $α, β$ be such that $α e = β e$ , one needs to show that $α = β$ . Since the equalizer of $(α, β)$ exists, $e$ factorizes as $e = q e^{'}$ with $q$ monic. But then $f = (m q) e^{'}$ is a factorization of $f$ with $(m q)$ monomorphism. Hence by the universal property of the image there exists a unique arrow $v : I \to E q_{α, β}$ such that $m = m q v$ and since $m$ is monic ${id}_{I} = q v$ . Furthermore, one has $m q = (m q v) q$ and by the monomorphism property of $m q$ one obtains ${id}_{E q_{α, β}} = v q$ .

This means that $I \equiv E q_{α, β}$ and thus that ${id}_{I} = q v$ equalizes $(α, β)$ , whence $α = β$ .

Second definition

In a category $C$ with all finite limits and colimits, the image is defined as the equalizer $(I m, m)$ of the so-called cokernel pair $(Y ⊔_{X} Y, i_{1}, i_{2})$ , which is the cocartesian of a morphism with itself over its domain, which will result in a pair of morphisms $i_{1}, i_{2} : Y \to Y ⊔_{X} Y$ , on which the equalizer is taken, i.e. the first of the following diagrams is cocartesian, and the second equalizing.^[3]

Remarks:

Finite bicompleteness of the category ensures that pushouts and equalizers exist.
$(I m, m)$ can be called regular image as $m$ is a regular monomorphism, i.e. the equalizer of a pair of morphisms. (Recall also that an equalizer is automatically a monomorphism).
In an abelian category, the cokernel pair property can be written $i_{1} f = i_{2} f \Leftrightarrow (i_{1} - i_{2}) f = 0 = 0 f$ and the equalizer condition $i_{1} m = i_{2} m \Leftrightarrow (i_{1} - i_{2}) m = 0 m$ . Moreover, all monomorphisms are regular.

Theorem—If $f$ always factorizes through regular monomorphisms, then the two definitions coincide.

Proof

First definition implies the second: Assume that (1) holds with $m$ regular monomorphism.

Equalization: one needs to show that $i_{1} m = i_{2} m$ . As the cokernel pair of $f, i_{1} f = i_{2} f$ and by previous proposition, since $C$ has all equalizers, the arrow $e$ in the factorization $f = m e$ is an epimorphism, hence $i_{1} f = i_{2} f \Rightarrow i_{1} m = i_{2} m$ .
Universality: in a category with all colimits (or at least all pushouts) $m$ itself admits a cokernel pair $(Y ⊔_{I} Y, c_{1}, c_{2})$

Moreover, as a regular monomorphism,

(I, m)

is the equalizer of a pair of morphisms

b_{1}, b_{2} : Y ⟶ B

but we claim here that it is also the equalizer of

c_{1}, c_{2} : Y ⟶ Y ⊔_{I} Y

.

Indeed, by construction

b_{1} m = b_{2} m

thus the "cokernel pair" diagram for

m

yields a unique morphism

u^{'} : Y ⊔_{I} Y ⟶ B

such that

b_{1} = u^{'} c_{1}, b_{2} = u^{'} c_{2}

. Now, a map

m^{'} : I^{'} ⟶ Y

which equalizes

(c_{1}, c_{2})

also satisfies

b_{1} m^{'} = u^{'} c_{1} m^{'} = u^{'} c_{2} m^{'} = b_{2} m^{'}

, hence by the equalizer diagram for

(b_{1}, b_{2})

, there exists a unique map

h^{'} : I^{'} \to I

such that

m^{'} = m h^{'}

.

Finally, use the cokernel pair diagram (of

f

) with

j_{1} := c_{1}, j_{2} := c_{2}, Z := Y ⊔_{I} Y

: there exists a unique

u : Y ⊔_{X} Y ⟶ Y ⊔_{I} Y

such that

c_{1} = u i_{1}, c_{2} = u i_{2}

. Therefore, any map

g

which equalizes

(i_{1}, i_{2})

also equalizes

(c_{1}, c_{2})

and thus uniquely factorizes as

g = m h^{'}

. This exactly means that

(I, m)

is the equalizer of

(i_{1}, i_{2})

.

Second definition implies the first:

Factorization: taking $m^{'} := f$ in the equalizer diagram ( $m^{'}$ corresponds to $g$ ), one obtains the factorization $f = m h$ .
Universality: let $f = m^{'} e^{'}$ be a factorization with $m^{'}$ regular monomorphism, i.e. the equalizer of some pair $(d_{1}, d_{2})$ .

Then

d_{1} m^{'} = d_{2} m^{'} \Rightarrow d_{1} f = d_{1} m^{'} e = d_{2} m^{'} e = d_{2} f

so that by the "cokernel pair" diagram (of

f

), with

j_{1} := d_{1}, j_{2} := d_{2}, Z := D

, there exists a unique

u^{″} : Y ⊔_{X} Y ⟶ D

such that

d_{1} = u^{″} i_{1}, d_{2} = u^{″} i_{2}

.

Now, from

i_{1} m = i_{2} m

(m from the equalizer of (i₁, i₂) diagram), one obtains

d_{1} m = u^{″} i_{1} m = u^{″} i_{2} m = d_{2} m

, hence by the universality in the (equalizer of (d₁, d₂) diagram, with f replaced by m), there exists a unique

v : I m ⟶ I^{'}

such that

m = m^{'} v

.

Examples

In the category of sets the image of a morphism $f : X \to Y$ is the inclusion from the ordinary image ${f (x) | x \in X}$ to $Y$ . In many concrete categories such as groups, abelian groups and (left- or right) modules, the image of a morphism is the image of the correspondent morphism in the category of sets.

In any normal category with a zero object and kernels and cokernels for every morphism, the image of a morphism $f$ can be expressed as follows:

im f = ker coker f

In an abelian category (which is in particular binormal), if f is a monomorphism then f = ker coker f, and so f = im f.

Essential Image

A related notion to image is essential image.^[4]

A subcategory $C \subset B$ of a (strict) category is said to be replete if for every $x \in C$ , and for every isomorphism $ι : x \to y$ , both $ι$ and $y$ belong to C.

Let $F : A \to B$ be a functor between categories. The essential image of $F$ is the smallest replete subcategory of the target category $B$ containing the image of $A$ under $F$ .

References

↑ Mitchell, Barry (1965), Theory of categories, Pure and applied mathematics, vol. 17, Academic Press, ISBN 978-0-12-499250-4, MR 0202787 Section I.10 p.12
↑ Mitchell, Barry (1965), Theory of categories, Pure and applied mathematics, vol. 17, Academic Press, ISBN 978-0-12-499250-4, MR 0202787 Proposition 10.1 p.12
↑ Kashiwara, Masaki; Schapira, Pierre (2006), "Categories and Sheaves", Grundlehren der Mathematischen Wissenschaften, vol. 332, Berlin Heidelberg: Springer, pp. 113–114 Definition 5.1.1
↑ "essential image in nLab". ncatlab.org. Retrieved 2024-11-15.

[1] Mitchell, Barry (1965), Theory of categories, Pure and applied mathematics, vol. 17, Academic Press, ISBN 978-0-12-499250-4, MR 0202787 Section I.10 p.12

[2] Mitchell, Barry (1965), Theory of categories, Pure and applied mathematics, vol. 17, Academic Press, ISBN 978-0-12-499250-4, MR 0202787 Proposition 10.1 p.12

[3] Kashiwara, Masaki; Schapira, Pierre (2006), "Categories and Sheaves", Grundlehren der Mathematischen Wissenschaften, vol. 332, Berlin Heidelberg: Springer, pp. 113–114 Definition 5.1.1

[4] "essential image in nLab". ncatlab.org. Retrieved 2024-11-15.

[1]

[2]

[3]

[4]

General definition

Second definition

Examples

Essential Image

See also

References