module Cat.Morphism.StrongEpi {o ℓ} (C : Precategory o ℓ) where

open Cat.Reasoning C

A strong epimorphism is an epimorphism which is, additionally, left orthogonal to every monomorphism. Unfolding that definition, for to be a strong epimorphism means that, given any mono, and arbitrarily fit into a commutative diagram like

there is a contractible space of maps which make the two triangles commute. In the particular case of strong epimorphisms, it actually suffices to have any lift: they are automatically unique.

is-strong-epi : ∀ {a b} → Hom a b → Type _
is-strong-epi f = is-epic f × ∀ {c d} (m : c ↪ d) → m⊥m C f (m .mor)

  : ∀ {a b} {f : Hom a b}
  → is-epic f
  → ( ∀ {c d} (m : c ↪ d) {u} {v} → v ∘ f ≡ m .mor ∘ u
    → Σ[ w ∈ Hom b c ] ((w ∘ f ≡ u) × (m .mor ∘ w ≡ v)))
  → is-strong-epi f
lifts→is-strong-epi epic lift-it = epic , λ {c} {d} mm sq →
  contr (lift-it mm sq) λ { (x , p , q) → Σ-prop-path!
    (mm .monic _ _ (sym (q ∙ sym (lift-it mm sq .snd .snd)))) }

To see that the uniqueness needed for orthogonality against a monomorphism is redundant, suppose you’d had two fillers as in

Since is a monomorphism, it suffices to have but by commutativity of the lower triangles we have


The proofs here are transcribed from (Borceux 1994, vol. 1, sec. 4.3). Strong epimorphisms are closed under composition, for suppose that and are strong epics, and is the monomorphism to lift against. Fit them in a skewed commutative rectangle like

By considering most of the right half as a single, weirdly-shaped square (the commutative “square”), we get an intermediate lift such that and — such that and organise into the faces of a lifting diagram, too$ Since is a strong epic, we have a second lift now satisfying and A quick calculation, implicit in the diagram, shows that is precisely the lift for against

  : ∀ {a b c} (g : Hom a b) (f : Hom b c)
  → is-strong-epi g
  → is-strong-epi f
  → is-strong-epi (f ∘ g)
strong-epi-compose g f (g-e , g-s) (f-e , f-s) =
  lifts→is-strong-epi epi fg-s
  epi : is-epic (f ∘ g)
  epi α β p = f-e α β $ g-e (α ∘ f) (β ∘ f) $
    sym (assoc _ _ _) ·· p ·· assoc _ _ _
  fg-s : ∀ {c d} (m : c ↪ d) {u v} → v ∘ f ∘ g ≡ m .mor ∘ u → _
  fg-s z {u} {v} vfg=zu =
      (w , wg=u , zw=vf) = g-s z (sym (assoc _ _ _) ∙ vfg=zu) .centre
      (t , tf=w , zt=v)  = f-s z (sym zw=vf) .centre
    in t , pulll tf=w ∙ wg=u , zt=v

Additionally, there is a partial converse to this result: If the composite is a strong epi, then is, too! Still thinking of the same diagram, suppose the whole diagram is a strong epi, and you’re given to lift against We don’t have a as before, but we can take to get a lift

  : ∀ {a b c} (f : Hom b c) (g : Hom a b)
  → is-strong-epi (f ∘ g)
  → is-strong-epi f
strong-epi-cancel-l g f (gf-epi , gf-str) = lifts→is-strong-epi epi lifts where
  epi : is-epic g
  epi α β p = gf-epi α β (extendl p)

  lifts : ∀ {c d} (m : c ↪ d) {u} {v} → v ∘ g ≡ m .mor ∘ u → _
  lifts {α} {β} mm {u} {v} sq = lifted .fst , lemma , lifted .snd .snd where
    lifted = gf-str mm {u = u ∘ f} {v = v} (extendl sq) .centre
    lemma = mm .monic _ _ (pulll (lifted .snd .snd) ∙ sq)

As an immediate consequence of the definition, a monic strong epi is an isomorphism. This is because, being left orthogonal to all monos, it’d be, in particular, left orthogonal to itself, and the only self-orthogonal maps are isos.

  : ∀ {a b} {f : Hom a b} → is-monic f → is-strong-epi f → is-invertible f
strong-epi-mono→is-invertible mono (_ , epi) =
  self-orthogonal→is-iso C _ (epi (record { monic = mono }))

Regular epis are strong🔗

Suppose that is a regular epimorphism, that is, it identifies as some quotient of — the stuff of is that of but with new potential identifications thrown in. Since we’re taking “strong epimorphism” as the definition of “well-behaved epimorphism”, we’d certainly like regular epis to be strong epis!

This is fortunately the case. Suppose that is the coequaliser of some maps 1, and that is a monomorphism we want to lift against.

By the universal property of a coequaliser, to “slide over” to a map it suffices to show that it also coequalises the pair i.e. that Since is a mono, it’ll suffice to show that but note that since the square commutes. Then we have

so there is a map such that — that’s commutativity of the top triangle handled. For the bottom triangle, since is a regular epic (thus an epic), to show it’ll suffice to show that But by assumption, and by the universal property! We’re done.

  : ∀ {a b} (f : Hom a b)
  → is-regular-epi C f
  → is-strong-epi f
is-regular-epi→is-strong-epi {a} {b} f regular =
    (λ m x → map m x , r.factors , lemma m x)
    module r = is-regular-epi regular renaming (arr₁ to s ; arr₂ to t)
    module _ {c} {d} (z : c ↪ d) {u} {v} (vf=zu : v ∘ f ≡ z .mor ∘ u) where
      module z = _↪_ z
      map : Hom b c
      map = r.universal {e' = u} $ z.monic _ _ $
        z .mor ∘ u ∘ r.s ≡⟨ extendl (sym vf=zu) ⟩≡
        v ∘ f ∘ r.s      ≡⟨ refl⟩∘⟨ r.coequal ⟩≡
        v ∘ f ∘ r.t      ≡˘⟨ extendl (sym vf=zu) ⟩≡˘
        z .mor ∘ u ∘ r.t ∎
      lemma =→is-epic _ _ $
        sym (vf=zu ∙ pushr (sym r.factors))


Now we come to the raison d’être for strong epimorphisms: Images. The definition of image we use is very complicated, and the justification is already present there, but the short of it is that the image of a morphism is a monomorphism which is universal amongst those through which factors.

Since images have a universal property, and one involving comma categories of slice categories at that, they are tricky to come by. However, in the case where we factor as

and the epimorphism is strong, then we automatically have an image factorisation of on our hands!

  : ∀ {a b im} (f : Hom a b)
  → (a→im : Hom a im) → is-strong-epi a→im
  → (im→b : Hom im b) → is-monic im→b
  → im→b ∘ a→im ≡ f
  → Image C f
strong-epi-mono→image f a→im (_ , str-epi) im→b mono fact = go where
  open Initial
  open /-Obj
  open /-Hom
  open ↓Obj
  open ↓Hom

  obj : ↓Obj (Const (cut f)) (Forget-full-subcat {P = is-monic ⊙ map})
  obj .x = tt
  obj .y = restrict (cut im→b) mono
  obj .map = record { map = a→im ; commutes = fact }

Actually, for an image factorisation, we don’t need that be an epimorphism — we just need it to be orthogonal to every monomorphism. This turns out to be precisely the data of being initial in the relevant comma categories.

  go : Image C f
  go .bot = obj
  go .has⊥ other = contr dh unique where
    module o = ↓Obj other

    the-lifting =
        (record { monic = o.y .witness })
        {u = .map}
        {v = im→b} (sym ( .commutes ∙ sym fact))

    dh : ↓Hom (Const (cut f)) _ obj other
    dh .α = tt
    dh .β .map = the-lifting .centre .fst
    dh .β .commutes = the-lifting .centre .snd .snd
    dh .sq = ext (idr _ ∙ sym (the-lifting .centre .snd .fst))

    unique : ∀ om → dh ≡ om
    unique om = ↓Hom-path _ _ refl $ ext $ ap fst $ the-lifting .paths $
      om .β .map , sym (ap map (om .sq)) ∙ idr _ , om .β .commutes

In the lex case🔗

Suppose that is additionally left exact, or more restrictively, that it has all equalisers. In that case, a map left orthogonal to all monomorphisms is automatically an epimorphism, thus a strong epi. Let’s see how. First, there’s a quick observation to be made about epimorphisms: if is such that there exists a with then is an epimorphism. You can think of this as a special case of “ epic implies epic” or as a short calculation:

  : ∀ {a b} {f : Hom a b} {g : Hom b a}
  → f ∘ g ≡ id
  → is-epic f
retract-is-epi {f = f} {g} p α β q =
  α         ≡⟨ intror p ⟩≡
  α ∘ f ∘ g ≡⟨ extendl q ⟩≡
  β ∘ f ∘ g ≡⟨ elimr p ⟩≡
  β         ∎

We already know that if lifts exist and the map is epic, then it’s a strong epi, so let’s assume that lifts exist — we’ll have no need for uniqueness, here. Given and to lift against, form their equaliser and arrange them like

  : ∀ {a b} {f : Hom a b}
  → (∀ {a b} (f g : Hom a b) → Equaliser C f g)
  → ( ∀ {c d} (m : c ↪ d) {u} {v} → v ∘ f ≡ m .mor ∘ u
    → Σ[ w ∈ Hom b c ] ((w ∘ f ≡ u) × (m .mor ∘ w ≡ v)))
  → is-strong-epi f
equaliser-lifts→is-strong-epi {f = f} eqs ls = lifts→is-strong-epi epi ls where

By the universal property of since there’s there’s a unique map such that Arranging and the identity(!) into a lifting square like the one above, we conclude the existence of a dotted map satisfying, most importantly, — so that being a retract, is an epimorphism.

  epi : is-epic f
  epi u v uf=vf =
      module ker = Equaliser (eqs u v)
      k = ker.universal uf=vf
      (w , p , q) = ls
        (record { monic = is-equaliser→is-monic C _ ker.has-is-eq })
        {u = k} {v = id}
        (idl _ ∙ sym ker.factors)
      e-epi : is-epic ker.equ
      e-epi = retract-is-epi q

Now, is the universal map which equalises and — so that we have and since we’ve just shown that is epic, this means we have — exactly what we wanted!

    in e-epi u v ker.equal

Extremal epimorphisms🔗

Another well-behaved subclass of epimorphism are the extremal epimorphisms: An epimorphism is extremal if when, given a factorisation through a monomorphism then is an isomorphism. In a finitely complete category, every extremal epimorphism is strong; the converse is immediate.

  : ∀ {a b} {e : Hom a b}
  → Finitely-complete C
  → (∀ {c} (m : c ↪ b) (g : Hom a c) → e ≡ m .mor ∘ g → is-invertible (m .mor))
  → is-strong-epi e
is-extremal-epi→is-strong-epi {a} {b} {e} lex extremal =
  equaliser-lifts→is-strong-epi lex.equalisers λ w → Mk.the-lift w where
    module lex = Finitely-complete lex

We adapt the proof from [Borceux (1994); §4.3.7]. After equaliser-lifts→is-strong-epi, it will suffice to construct some lift for a square with with monic. Pull back along to obtain the square

and obtain the unique factorisation Note that the map is a monomorphism since it results from pulling back a monomorphism.

    module Mk {c d : Ob} (m : c ↪ d) {u : Hom a c} {v : Hom b d}
              (wit : v ∘ e ≡ m .mor ∘ u) where
      module P = Pullback (lex.pullbacks v (m .mor)) renaming (p₁ to q ; p₂ to p)
      r : Hom a P.apex
      r = P.universal {p₁' = e} {p₂' = u} wit

        q-mono : is-monic P.q
        q-mono = is-monic→pullback-is-monic (m .monic) (rotate-pullback P.has-is-pb)

We thus have a factorisation of through a monomorphism which since was assumed extremal, must be an isomorphism. We define the diagonal map to be and compute that it commutes appropriately:

      q-iso : is-invertible P.q
      q-iso = extremal record{ monic = q-mono } r (sym P.p₁∘universal)

      q⁻¹ = q-iso .is-invertible.inv

      the-lift : Σ (Hom b c) λ w → (w ∘ e ≡ u) × (m .mor ∘ w ≡ v)
      the-lift .fst = P.p ∘ q⁻¹
      the-lift .snd .fst = m .monic _ _ $
        m .mor ∘ (P.p ∘ q⁻¹) ∘ e ≡⟨ extendl (pulll (sym P.square)) ⟩≡
        (v ∘ P.q) ∘ q⁻¹ ∘ e      ≡⟨ cancel-inner (q-iso .is-invertible.invl) ⟩≡
        v ∘ e                    ≡⟨ wit ⟩≡
        m .mor ∘ u               ∎
      the-lift .snd .snd = invertible→epic q-iso _ _ $
        (m .mor ∘ (P.p ∘ q⁻¹)) ∘ P.q ≡⟨ pullr (cancelr (q-iso .is-invertible.invr)) ⟩≡
        m .mor ∘ P.p                 ≡˘⟨ P.square ⟩≡˘
        v ∘ P.q                      ∎

  1. If you care, is for “relation” — the intuition is that specifies the relations imposed on to get ↩︎