module Data.Set.Surjection where

Here we prove that surjective maps are exactly the regular
epimorphisms in the category of sets: Really, we prove that
surjections are regular epimorphisms (this is straightforward), then we
prove that every epimorphism is a
surjection (this takes a detour). Putting these together, we get
*every epimorphism is regular* as a corollary.

## Surjections are epic🔗

This is the straightforward direction: We know (since
$\mathbf{Sets}$
has pullbacks) that if a morphism is going to be a regular epimorphism,
then it must be an *effective* epimorphism, too: so if it is to
coequalise any morphisms, it certainly coequalises its kernel pair.

surjective→regular-epi : ∀ {ℓ} (c d : n-Type ℓ 2) (f : ∣ c ∣ → ∣ d ∣) → (∀ x → ∥ fibre f x ∥) → is-regular-epi (Sets ℓ) {c} {d} f surjective→regular-epi c _ f x .r = el! (Σ ∣ c ∣ λ x → Σ ∣ c ∣ λ y → f x ≡ f y) surjective→regular-epi _ _ f x .arr₁ = λ (y , _ , _) → y surjective→regular-epi _ _ f x .arr₂ = λ (_ , y , _) → y

That’s our choice of cofork: it remains to show that $f$ coequalises this. Given any map $e' : c \to F$, element $x : d$, mere fibre $f^*(x)$, and proof $p$ that $e'$ is constant on the kernel pair of $f$ (i.e. that it also coequalises the kernel pair cofork), we can construct an element of $F$: since $e'$ is suitably constant, we can use the elimination principle for $\| f^*x \| \to F$, since $F$ is a set.

surjective→regular-epi c d f surj .has-is-coeq = coeqs where go : ∀ {F} (e' : ∣ c ∣ → ∣ F ∣) p (x : ∣ d ∣) → ∥ fibre f x ∥ → ∣ F ∣ go e' p x = ∥-∥-rec-set hlevel! (λ x → e' (x .fst)) (λ x y → p $ₚ (x .fst , y .fst , x .snd ∙ sym (y .snd)))

After a small amount of computation to move the witnesses of surjectivity out of the way, we get what we wanted.

coeqs : is-coequaliser (Sets _) _ _ _ coeqs .coequal i (x , y , p) = p i coeqs .universal {F} {e'} p x = go {F = F} e' p x (surj x) coeqs .factors {F} {e'} {p = p} = funext λ x → ∥-∥-elim {P = λ e → go {F} e' p (f x) e ≡ e' x} (λ x → hlevel!) (λ e → p $ₚ (e .fst , x , e .snd)) (surj (f x)) coeqs .unique {F} {e'} {p} {colim} comm = funext λ a → ∥-∥-elim {P = λ e → colim a ≡ go {F} e' p a e} (λ x → hlevel!) (λ x → ap colim (sym (x .snd)) ∙ comm $ₚ x .fst) (surj a)

# Epis are surjective🔗

Now *this* is the hard direction. Our proof follows (Rijke and
Spitters 2015, sec. 2.9), so if the exposition below doesn’t
make a lot of sense, be sure to check their paper, too. We define a
higher inductive type standing for the *cofibre* of
$f$,
also known as the *mapping cone*. Strictly speaking, this is the
homotopy pushout

but Cubical Agda makes it very convenient to define a purpose-built
HIT in-line. We use the names “tip”, “base”, and “cone” to evoke the
construction of a cone in solid geometry: the `tip`

is.. the tip,
the `base`

is the circle,
and the `cone`

is the
triangular side which we have rotated around the vertical axis.

data Cofibre {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} (f : A → B) : Type (ℓ ⊔ ℓ') where tip : Cofibre f base : B → Cofibre f cone : ∀ a → tip ≡ base (f a)

What’s important here is that if a map
$f$
has connected
cofibre, then it is a surjection — so our proof that epis are surjective
will factor through showing that epis have connected cofibres^{1}.

connected-cofibre→surjective : ∀ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} (f : A → B) → is-connected (Cofibre f) → ∀ x → ∥ fibre f x ∥ connected-cofibre→surjective {A = A} {B = B} f conn x = transport cen (lift tt) where

We define a family
$P$
of propositions over
$\mathrm{Cofibre}(f)$
which maps `tip`

to the unit
proposition and the base point with coordinate
$x$
to the fibre
$f^x$.
Since
$\mathrm{Prop}$
is a set,
$P$
extends to a map
$P' : \| \mathrm{Cofibre}(f) \|_0 \to \mathrm{Prop}$.

P : Cofibre f → Prop _ P tip = el (Lift _ ⊤) hlevel! P (base x) = el ∥ fibre f x ∥ hlevel! P (cone a i) = n-ua {X = el (Lift _ ⊤) hlevel!} {Y = el ∥ fibre f (f a) ∥ hlevel!} (prop-ext hlevel! hlevel! (λ _ → inc (a , refl)) λ _ → lift tt) i P' : ∥ Cofibre f ∥₀ → Prop _ P' = ∥-∥₀-elim (λ _ → hlevel!) P

Letting $x$ be an element of the codomain, and since by assumption $f$’s cofibre is connected, we have a path

$\top = P'(\mathrm{tip}) = P'(\mathrm{base}_x) = \| f^x \|\text{,}$

so since the unit type is trivially inhabited, so is the fibre of $f$ over $x$: $f$ is surjective.

cen : Lift _ ⊤ ≡ ∥ fibre f x ∥ cen = ap ∣_∣ (ap P' (is-contr→is-prop conn (inc tip) (inc (base x))))

## Epis have connected cofibre🔗

Now suppose that
$f$
is an epic morphism ^{2} ^{3}. We know that the
set-truncation of
$f$’s
cofibre is inhabited (since every cofibre is inhabited), so it remains
to show that any two points are merely equal.

epi→connected-cofibre : ∀ {ℓ} (c d : n-Type ℓ 2) (f : ∣ c ∣ → ∣ d ∣) → Cr.is-epic (Sets ℓ) {c} {d} f → is-connected (Cofibre f) epi→connected-cofibre c d f epic = contr (inc tip) $ ∥-∥₀-elim (λ _ → is-prop→is-set (squash _ _)) λ w → ∥-∥₀-path.from (hom w) where

Let
$x : d$
— we’ll show that
$\| \mathrm{tip} = \mathrm{base}_x \|$.
Here’s where we
$f$’s
epimorphy: we have a homotopy
$| \mathrm{tip} | = | \mathrm{base}_{(f x)} |$,
namely the `cone`

— and since we
can write its left-hand-side as the composition of
$f$
with a constant function,
$f$
gives us a path
$| \mathrm{tip} | = | \mathrm{base}_x |$
— which, by the characterisation of paths in the set truncation, means
we merely have
$\| \mathrm{tip} = \mathrm{base}_x \|$.

go : ∀ x → ∥ tip ≡ base x ∥ go x = ∥-∥₀-path.to (p $ₚ x) where p = epic {c = el ∥ Cofibre f ∥₀ squash} (λ _ → inc tip) (λ x → inc (base x)) (funext λ x → ap ∥_∥₀.inc (cone x))

An appeal to induction over the cofibre completes the proof: Epimorphisms have connected cofibres. Note that the path case is discharged automatically since we’re mapping into a proposition.

hom : ∀ w → ∥ tip ≡ w ∥ hom tip = inc refl hom (base x) = go x hom (cone a i) = is-prop→pathp (λ i → ∥_∥.squash {A = tip ≡ cone a i}) (inc refl) (go (f a)) i

And, composing this with the previous step, we have what we originally set out to prove: all $\mathbf{Sets}$-epimorphisms are regular, because they’re all surjections!

epi→surjective : ∀ {ℓ} (c d : n-Type ℓ 2) (f : ∣ c ∣ → ∣ d ∣) → Cr.is-epic (Sets ℓ) {c} {d} f → ∀ x → ∥ fibre f x ∥ epi→surjective {ℓ} c d f epi x = connected-cofibre→surjective f (epi→connected-cofibre c d f (λ {x} → epi {x})) x

## References

- Rijke, Egbert, and Bas Spitters. 2015. “Sets in Homotopy Type
Theory.”
*Mathematical Structures in Computer Science*25 (5): 1172–202. https://doi.org/10.1017/s0960129514000553.