open import Cat.Functor.FullSubcategory open import Cat.Diagram.Initial open import Cat.Functor.Adjoint open import Cat.Instances.Comma open import Cat.Instances.Slice open import Cat.Functor.Base open import Cat.Prelude import Cat.Reasoning module Cat.Diagram.Image {o ℓ} (C : Precategory o ℓ) where

# Images🔗

Let
$f : A \to B$
be an ordinary function between sets (or, indeed, arbitrary types). Its
**image**
$\operatorname*{im}f$
can be computed as the subset
$\{ b \in B : (\exists a)\ b = f(a) \}$,
but this description does not carry over to more general categories:
More abstractly, we can say that the image embeds into
$B$,
and admits a map from
$A$
(in material set theory, this is
$f$
itself — structurally, it is called the **corestriction**
of
$f$).
Furthermore, these two maps *factor*
$f$,
in that:

$(A {\xrightarrow{e}} \operatorname*{im}f {\xhookrightarrow{m}} B) = (A {\xrightarrow{f}} B)$

While these are indeed two necessary properties of an *image*,
they fail to accurately represent the set-theoretic construction:
Observe that, e.g. for
$f(x) = 2x$,
we could take
$\operatorname*{im}f = {\mathbb{N}}$,
taking
$e = f$
itself and
$m = {\mathbb{N}} {\xhookrightarrow{{{\mathrm{id}}_{}}}} {\mathbb{N}}$.
This factoring clearly recovers
$f$,
as
${{\mathrm{id}}_{}}\circ f = f$.
But by taking this as the image, we’ve lost the information that
$f$
lands in the evens!

We can refine the abstract definition by saying that, for a mono
$\operatorname*{im}f {\xhookrightarrow{m}} B$
to be *the* image of
$f$,
it must be the *smallest* subobject of
$B$
through which
$f$
factors — given any other factoring of
$f = m' \circ e'$,
we must have
$m \sube m'$
in the proset of subobjects of
$B$,
i.e. there exists some
$k$
such that
$m = m' \circ k$.

In general categories, monomorphisms of
$\mathcal{C}$
may be the wrong notion of “subobject” to use. For example, in topology,
we’d rather talk about the image which admits a *subspace
inclusion* onto
$B$.
We may expand the definition above to work for an arbitrary subclass
$M \sube {\mathrm{Mono}}(\mathcal{C})$
of the monomorphisms of
$C$,
by requiring that the
$M$-image
of
$f$
be the smallest
$M$-subobject
through which
$f$
factors.

Since keeping track of all the factorisations by hand would be fiddly, we formalise the idea of image here using comma categories, namely the idea of universal morphisms as in the construction of adjoints. Fix a morphism $f : a \to b$, and consider it as an object of the slice category $\mathcal{C}/b$.

For a given subclass of monomorphisms
$M$,
there is a full subcategory of
$\mathcal{C}/b$
spanned by those maps in
$M$
— let us call it
$M/b$
— admitting an evident ff inclusion
$F : M/b {\hookrightarrow}\mathcal{C}/b$.
An
**$M$-image
of
$f$**
is a universal morphism from
$f$
to
$F$.

Class-of-monos : ∀ ℓ → Type _ Class-of-monos ℓ = Σ[ M ∈ (∀ {a b} → Hom a b → Type ℓ) ] (∀ {a b} {f : Hom a b} → M f → is-monic f) M-image : ∀ {a b} → Class-of-monos ℓ′ → Hom a b → Type _ M-image {a = a} {b} M f = Universal-morphism (cut f) (Forget-full-subcat {C = Slice C b} {P = (λ o → M .fst (o .map))})

**The** image is the
$M$-image
for
$M$
= the class of all monomorphisms.

Image : ∀ {a b} → Hom a b → Type _ Image {b = b} f = Universal-morphism (cut f) (Forget-full-subcat {C = Slice C b} {P = is-monic ⊙ map})

## Friendly interface🔗

Since this definition is incredibly abstract and indirect, we provide a very thin wrapper module over M-image which unpacks the definition into friendlier terms.

module M-Image {a b} {M : Class-of-monos ℓ′} {f : Hom a b} (im : M-image M f) where

The first thing to notice is that, being an initial object in the comma category $f \swarrow F$, we have an object $(c, c {\xhookrightarrow{m}} b)$ — $c$ is the image object, and $m$ is the inclusion map:

Im : Ob Im = im .bot .y .object .domain Im→codomain : Hom Im b Im→codomain = im .bot .y .object .map

Furthermore, this map is both an inclusion (since $M$ is a class of monomorphisms), and an $M$-inclusion at that:

Im→codomain-is-M : M .fst Im→codomain Im→codomain-is-M = im .bot .y .witness Im→codomain-is-monic : is-monic Im→codomain Im→codomain-is-monic = M .snd Im→codomain-is-M

So far, we’ve been looking at the “codomain” part of the object in the comma category. We also have the “morphism” part, which provides our (universal) factoring of $f$:

corestrict : Hom a Im corestrict = im .bot .map .map image-factors : Im→codomain ∘ corestrict ≡ f image-factors = im .bot .map .commutes

This is also the *smallest* factorisation, which takes quite a
lot of data to express. Let’s see it:

Suppose we have

- Some other object $c$; and,
- An $M$-monomorphism $c {\xhookrightarrow{m}} b$; and,
- A corestriction map $a {\xrightarrow{i}} c$; such that
- $m \circ i = f$.

Then we have a map $k : \operatorname*{im}f \to c$ (written im≤other-image in the code below), and the canonical inclusion $\operatorname*{im}f {\hookrightarrow}B$ factors through $k$:

im≤other-image : ∀ {c} (m : Hom c b) (M-m : M .fst m) (i : Hom a c) → m ∘ i ≡ f → Hom Im c im≤other-image m M i p = im .has⊥ obj .centre .β .map where obj : ↓Obj _ _ obj .x = tt obj .y = restrict (cut m) M obj .map = record { map = i ; commutes = p } im≤other-image-factors : ∀ {c} {m : Hom c b} {M : M .fst m} {i : Hom a c} → {p : m ∘ i ≡ f} → m ∘ im≤other-image m M i p ≡ Im→codomain im≤other-image-factors {m = m} {M} {i} {p} = im .has⊥ _ .centre .β .commutes