open import Cat.Displayed.Base
open import Cat.Prelude

import Cat.Displayed.Cartesian
import Cat.Displayed.Reasoning
import Cat.Displayed.Morphism
import Cat.Reasoning

module Cat.Displayed.Cartesian.Joint
  {o ℓ o' ℓ'}
  {B : Precategory o ℓ} (E : Displayed B o' ℓ')
  where

open Cat.Displayed.Cartesian E
open Cat.Displayed.Reasoning E
open Cat.Displayed.Morphism E
open Cat.Reasoning B
open Displayed E

Jointly cartesian families🔗

A family of morphisms $f_{i} : E_{u_{i}} (A^{'}, B_{i}^{'})$ over $u_{i} : B (A, B_{i})$ is jointly cartesian if it satisfies a familial version of the universal property of a cartesian map.

record is-jointly-cartesian
  {ℓi} {Ix : Type ℓi}
  {a : Ob} {bᵢ : Ix → Ob}
  {a' : Ob[ a ]} {bᵢ' : (ix : Ix) → Ob[ bᵢ ix ]}
  (uᵢ : (ix : Ix) → Hom a (bᵢ ix))
  (fᵢ : (ix : Ix) → Hom[ uᵢ ix ] a' (bᵢ' ix))
  : Type (o ⊔ ℓ ⊔ o' ⊔ ℓ' ⊔ ℓi) where

Explicitly, a family $f_{i}$ is jointly cartesian if for every map $v : B (X, A)$ and family of morphisms $h_{i} : E_{u_{i} \circ v} (X^{'}, B_{i}^{'}),$ there exists a unique universal map $⟨ v, h_{i} ⟩ : E_{v} (X^{'}, A^{'})$ such that $f_{i} \circ ⟨ v, h_{i} ⟩ = h_{i} .$

  no-eta-equality
  field
    universal
      : ∀ {x x'}
      → (v : Hom x a)
      → (∀ ix → Hom[ uᵢ ix ∘ v ] x' (bᵢ' ix))
      → Hom[ v ] x' a'
    commutes
      : ∀ {x x'}
      → (v : Hom x a)
      → (hᵢ : ∀ ix → Hom[ uᵢ ix ∘ v ] x' (bᵢ' ix))
      → ∀ ix → fᵢ ix ∘' universal v hᵢ ≡ hᵢ ix
    unique
      : ∀ {x x'}
      → {v : Hom x a}
      → {hᵢ : ∀ ix → Hom[ uᵢ ix ∘ v ] x' (bᵢ' ix)}
      → (other : Hom[ v ] x' a')
      → (∀ ix → fᵢ ix ∘' other ≡ hᵢ ix)
      → other ≡ universal v hᵢ

  universal'
      : ∀ {x x'}
      → {v : Hom x a} {wᵢ : ∀ ix → Hom x (bᵢ ix)}
      → (p : ∀ ix → uᵢ ix ∘ v ≡ wᵢ ix)
      → (∀ ix → Hom[ wᵢ ix ] x' (bᵢ' ix))
      → Hom[ v ] x' a'
  universal' {v = v} p hᵢ =
    universal v λ ix → hom[ p ix ]⁻ (hᵢ ix)

  commutesp
    : ∀ {x x'}
    → {v : Hom x a} {wᵢ : ∀ ix → Hom x (bᵢ ix)}
    → (p : ∀ ix → uᵢ ix ∘ v ≡ wᵢ ix)
    → (hᵢ : ∀ ix → Hom[ wᵢ ix ] x' (bᵢ' ix))
    → ∀ ix → fᵢ ix ∘' universal' p hᵢ ≡[ p ix ] hᵢ ix
  commutesp p hᵢ ix =
    to-pathp⁻ (commutes _ (λ ix → hom[ p ix ]⁻ (hᵢ ix)) ix)

  universalp
    : ∀ {x x'}
    → {v₁ v₂ : Hom x a} {wᵢ : ∀ ix → Hom x (bᵢ ix)}
    → (p : ∀ ix → uᵢ ix ∘ v₁ ≡ wᵢ ix) (q : v₁ ≡ v₂) (r : ∀ ix → uᵢ ix ∘ v₂ ≡ wᵢ ix)
    → (hᵢ : ∀ ix → Hom[ wᵢ ix ] x' (bᵢ' ix))
    → universal' p hᵢ ≡[ q ] universal' r hᵢ
  universalp p q r hᵢ =
    apd (λ i ϕ → universal' ϕ hᵢ) prop!

  uniquep
    : ∀ {x x'}
    → {v₁ v₂ : Hom x a} {wᵢ : ∀ ix → Hom x (bᵢ ix)}
    → (p : ∀ ix → uᵢ ix ∘ v₁ ≡ wᵢ ix) (q : v₁ ≡ v₂) (r : ∀ ix → uᵢ ix ∘ v₂ ≡ wᵢ ix)
    → {hᵢ : ∀ ix → Hom[ wᵢ ix ] x' (bᵢ' ix)}
    → (other : Hom[ v₁ ] x' a')
    → (∀ ix → fᵢ ix ∘' other ≡[ p ix ] hᵢ ix)
    → other ≡[ q ] universal' r hᵢ
  uniquep p q r {hᵢ} other s =
    to-pathp⁻ (unique other (λ ix → from-pathp⁻ (s ix)) ∙ from-pathp⁻ (universalp p q r hᵢ))

  uniquep₂
    : ∀ {x x'}
    → {v₁ v₂ : Hom x a} {wᵢ : ∀ ix → Hom x (bᵢ ix)}
    → (p : ∀ ix → uᵢ ix ∘ v₁ ≡ wᵢ ix) (q : v₁ ≡ v₂) (r : ∀ ix → uᵢ ix ∘ v₂ ≡ wᵢ ix)
    → {hᵢ : ∀ ix → Hom[ wᵢ ix ] x' (bᵢ' ix)}
    → (other₁ : Hom[ v₁ ] x' a')
    → (other₂ : Hom[ v₂ ] x' a')
    → (∀ ix → fᵢ ix ∘' other₁ ≡[ p ix ] hᵢ ix)
    → (∀ ix → fᵢ ix ∘' other₂ ≡[ r ix ] hᵢ ix)
    → other₁ ≡[ q ] other₂
  uniquep₂ p q r {hᵢ = hᵢ} other₁ other₂ α β =
    cast[] $
      other₁          ≡[]⟨ uniquep p refl p other₁ α ⟩≡[]
      universal' p hᵢ ≡[]⟨ symP (uniquep r (sym q) p other₂ β) ⟩≡[]
      other₂          ∎

private variable
  ℓi ℓi' ℓj : Level
  Ix Ix' : Type ℓi
  Jx : Ix → Type ℓj
  a b c : Ob
  bᵢ cᵢ : Ix → Ob
  cᵢⱼ : (i : Ix) → Jx i → Ob
  a' b' c' : Ob[ a ]
  bᵢ' cᵢ' : (ix : Ix) → Ob[ bᵢ ix ]
  cᵢⱼ' : (i : Ix) → (j : Jx i) → Ob[ cᵢⱼ i j ]
  u v : Hom a b
  uᵢ uⱼ vᵢ vⱼ : ∀ (ix : Ix) → Hom a (bᵢ ix)
  f g : Hom[ u ] a' b'
  fᵢ fⱼ fᵢ' gᵢ gⱼ : ∀ (ix : Ix) → Hom[ uᵢ ix ] a' (bᵢ' ix)

Warning

Some sources ((Adámek, Herrlich, and Strecker 2004), (Dubuc and Español 2006)) refer to jointly cartesian families as “initial families”. We opt to avoid this terminology, as it leads to unnecessary conflicts with initial objects.

At first glance, jointly cartesian families appear very similar to cartesian maps. However, replacing a single map with a family of maps has some very strong consequences: cartesian morphisms typically behave as witnesses to base change operations, whereas jointly cartesian families act more like a combination of base-change maps and projections out of a product. This can be seen by studying prototypical examples of jointly cartesian families:

If we view the category of topological spaces as a displayed category, then the jointly cartesian maps are precisely the initial topologies.
Jointly cartesian maps in the subobject fibration of $Sets$ arise from pulling back a family of subsets $A_{i} \subset Y_{i}$ along maps $u_{i} : X \to Y_{i},$ and then taking their intersection.
When $B$ is considered as a displayed category over the terminal category, the jointly cartesian families are precisely the indexed products. In contrast, the cartesian maps are the invertible maps.

Relating cartesian maps and jointly cartesian families🔗

Every cartesian map can be regarded as a jointly cartesian family over a contractible type.

cartesian→jointly-cartesian
  : is-contr Ix
  → is-cartesian u f
  → is-jointly-cartesian {Ix = Ix} (λ _ → u) (λ _ → f)
cartesian→jointly-cartesian {u = u} {f = f} ix-contr f-cart = f-joint-cart where
  module f = is-cartesian f-cart
  open is-jointly-cartesian

  f-joint-cart : is-jointly-cartesian (λ _ → u) (λ _ → f)
  f-joint-cart .universal v hᵢ =
    f.universal v (hᵢ (ix-contr .centre))
  f-joint-cart .commutes v hᵢ ix =
    f ∘' f.universal v (hᵢ (ix-contr .centre)) ≡⟨ f.commutes v (hᵢ (ix-contr .centre)) ⟩≡
    hᵢ (ix-contr .centre)                      ≡⟨ ap hᵢ (ix-contr .paths ix) ⟩≡
    hᵢ ix                                      ∎
  f-joint-cart .unique other p =
    f.unique other (p (ix-contr .centre))

Conversely, if the constant family $λi . f$ is a jointly cartesian $I -indexed$ family over a merely inhabited type $I,$ then $f$ is cartesian.

const-jointly-cartesian→cartesian
  : ∥ Ix ∥
  → is-jointly-cartesian {Ix = Ix} (λ _ → u) (λ _ → f)
  → is-cartesian u f
const-jointly-cartesian→cartesian {Ix = Ix} {u = u} {f = f} ∥ix∥ f-joint-cart =
  rec! f-cart ∥ix∥
  where
    module f = is-jointly-cartesian f-joint-cart
    open is-cartesian

    f-cart : Ix → is-cartesian u f
    f-cart ix .universal v h = f.universal v λ _ → h
    f-cart ix .commutes v h = f.commutes v (λ _ → h) ix
    f-cart ix .unique other p = f.unique other (λ _ → p)

Jointly cartesian families over empty types act more like codiscrete objects than pullbacks, as the space of maps into the shared domain of the family is unique for any $v : E B (X, A)$ and $X^{'} X .$ In the displayed category of topological spaces, such maps are precisely the codiscrete spaces.

empty-jointly-cartesian→codiscrete
  : ∀ {uᵢ : (ix : Ix) → Hom a (bᵢ ix)} {fᵢ : (ix : Ix) → Hom[ uᵢ ix ] a' (bᵢ' ix)}
  → ¬ Ix
  → is-jointly-cartesian uᵢ fᵢ
  → ∀ {x} (v : Hom x a) → (x' : Ob[ x ]) → is-contr (Hom[ v ] x' a')
empty-jointly-cartesian→codiscrete ¬ix fᵢ-cart v x' =
  contr (fᵢ.universal v λ ix → absurd (¬ix ix)) λ other →
    sym (fᵢ.unique other λ ix → absurd (¬ix ix))
  where
    module fᵢ = is-jointly-cartesian fᵢ-cart

In the other direction, let $f : E_{u} (A^{'}, B^{'})$ be some map. If the constant family $λb . f : 2 \to E_{u} (A^{'}, B^{'})$ is jointly cartesian as a family over the booleans, then the type of morphisms $E_{u \circ v} (X^{'}, A^{'})$ is a proposition for every $v : B (X, A)$ and $X^{'} X .$

const-pair-joint-cartesian→thin
  : ∀ {u : Hom a b} {f : Hom[ u ] a' b'}
  → is-jointly-cartesian {Ix = Bool} (λ _ → u) (λ _ → f)
  → ∀ {x} (v : Hom x a) → (x' : Ob[ x ]) → is-prop (Hom[ u ∘ v ] x' b')

Let $g, h : E_{u \circ v} (X^{'}, A^{'})$ be a pair of parallel maps in $E .$ We can view the the pair $(g, h)$ as a Bool indexed family of maps over $u \circ v,$ so by the universal property of jointly cartesian families, there must be a universal map $α$ such that $g = f \circ α$ and $h = f \circ α;$ thus $f = g .$

const-pair-joint-cartesian→thin {b' = b'} {u = u} {f = f} f-cart v x' g h =
  cast[] $
    g                   ≡[]˘⟨ commutes v gh true ⟩≡[]˘
    f ∘' universal v gh ≡[]⟨ commutes v gh false ⟩≡[]
    h                   ∎
  where
    open is-jointly-cartesian f-cart

    gh : Bool → Hom[ u ∘ v ] x' b'
    gh true = g
    gh false = h

As a corollary, if $(i d_{A^{'}}, i d_{A^{'}})$ is a jointly cartesian family, then every hom set $E_{u} (X^{'}, A^{'})$ is a proposition.

id-pair-joint-cartesian→thin
  : is-jointly-cartesian {Ix = Bool} (λ _ → id {a}) (λ _ → id' {a} {a'})
  → ∀ {x} (u : Hom x a) → (x' : Ob[ x ]) → is-prop (Hom[ u ] x' a')
id-pair-joint-cartesian→thin id²-cart u x' f g =
  cast[] $
    f       ≡[]⟨ wrap (sym (idl u)) ⟩≡[]
    hom[] f ≡[]⟨ const-pair-joint-cartesian→thin id²-cart u x' (hom[ idl u ]⁻ f) (hom[ idl u ]⁻ g) ⟩≡[]
    hom[] g ≡[]⟨ unwrap (sym (idl u)) ⟩≡[]
    g ∎

Closure properties of jointly cartesian families🔗

If $g_{i} : X^{'} \to_{v_{i}} Y_{i}^{'}$ is an $I -indexed$ jointly cartesian family, and $f_{i, j} : Y_{i}^{'} \to_{u_{i, j}} Z_{i, j}^{'}$ is an $I -indexed$ family of $J_{i} -indexed$ jointly cartesian families, then their composite is a $Σ (i : I) . J_{i} -indexed$ jointly cartesian family.

jointly-cartesian-∘
  : {uᵢⱼ : (i : Ix) → (j : Jx i) → Hom (bᵢ i) (cᵢⱼ i j)}
  → {fᵢⱼ : (i : Ix) → (j : Jx i) → Hom[ uᵢⱼ i j ] (bᵢ' i) (cᵢⱼ' i j)}
  → {vᵢ : (i : Ix) → Hom a (bᵢ i)}
  → {gᵢ : (i : Ix) → Hom[ vᵢ i ] a' (bᵢ' i)}
  → (∀ i → is-jointly-cartesian (uᵢⱼ i) (fᵢⱼ i))
  → is-jointly-cartesian vᵢ gᵢ
  → is-jointly-cartesian
      (λ ij → uᵢⱼ (ij .fst) (ij .snd) ∘ vᵢ (ij .fst))
      (λ ij → fᵢⱼ (ij .fst) (ij .snd) ∘' gᵢ (ij .fst))

_ = cartesian-∘

Despite the high quantifier complexity of the statement, the proof follows the exact same plan that we use to show that cartesian maps compose.

jointly-cartesian-∘ {Ix = Ix} {uᵢⱼ = uᵢⱼ} {fᵢⱼ = fᵢⱼ} {vᵢ = vᵢ} {gᵢ = gᵢ} fᵢⱼ-cart gᵢ-cart =
  fᵢⱼ∘gᵢ-cart
  where
    module fᵢⱼ (i : Ix) = is-jointly-cartesian (fᵢⱼ-cart i)
    module gᵢ = is-jointly-cartesian gᵢ-cart
    open is-jointly-cartesian

    fᵢⱼ∘gᵢ-cart
      : is-jointly-cartesian
          (λ ij → uᵢⱼ (ij .fst) (ij .snd) ∘ vᵢ (ij .fst))
          (λ ij → fᵢⱼ (ij .fst) (ij .snd) ∘' gᵢ (ij .fst))
    fᵢⱼ∘gᵢ-cart .universal v hᵢⱼ =
      gᵢ.universal v λ i →
      fᵢⱼ.universal' i (λ j → assoc (uᵢⱼ i j) (vᵢ i) v) λ j →
      hᵢⱼ (i , j)
    fᵢⱼ∘gᵢ-cart .commutes w hᵢⱼ (i , j) =
      cast[] $
        (fᵢⱼ i j ∘' gᵢ i) ∘' gᵢ.universal _ (λ i → fᵢⱼ.universal' i _ (λ j → hᵢⱼ (i , j))) ≡[]⟨ pullr[] _ (gᵢ.commutes w _ i) ⟩≡[]
        fᵢⱼ i j ∘' fᵢⱼ.universal' i _ (λ j → hᵢⱼ (i , j))                                  ≡[]⟨ fᵢⱼ.commutesp i _ _ j ⟩≡[]
        hᵢⱼ (i , j) ∎
    fᵢⱼ∘gᵢ-cart .unique {hᵢ = hᵢⱼ} other p =
      gᵢ.unique other $ λ i →
      fᵢⱼ.uniquep i _ _ _ (gᵢ i ∘' other) λ j →
        fᵢⱼ i j ∘' gᵢ i ∘' other   ≡[]⟨ assoc' (fᵢⱼ i j) (gᵢ i) other ⟩≡[]
        (fᵢⱼ i j ∘' gᵢ i) ∘' other ≡[]⟨ p (i , j) ⟩≡[]
        hᵢⱼ (i , j)                ∎

As a nice corollary, we get that jointly cartesian families compose with cartesian maps, as cartesian maps are precisely the singleton jointly cartesian families.

jointly-cartesian-cartesian-∘
  : is-jointly-cartesian uᵢ fᵢ
  → is-cartesian v g
  → is-jointly-cartesian (λ ix → uᵢ ix ∘ v) (λ ix → fᵢ ix ∘' g)

We actually opt to prove this corollary by hand to get nicer definitional behaviour of the resulting universal maps.

jointly-cartesian-cartesian-∘ {uᵢ = uᵢ} {fᵢ = fᵢ} {v = v} {g = g} fᵢ-cart g-cart = fᵢ∘g-cart
  where
    module fᵢ = is-jointly-cartesian fᵢ-cart
    module g = is-cartesian g-cart
    open is-jointly-cartesian

    fᵢ∘g-cart : is-jointly-cartesian (λ ix → uᵢ ix ∘ v) (λ ix → fᵢ ix ∘' g)
    fᵢ∘g-cart .universal w hᵢ =
      g.universal w $ fᵢ.universal' (λ ix → assoc (uᵢ ix) v w) hᵢ
    fᵢ∘g-cart .commutes w hᵢ ix =
      cast[] $
        (fᵢ ix ∘' g) ∘' universal fᵢ∘g-cart w hᵢ             ≡[]⟨ pullr[] _ (g.commutes w _) ⟩≡[]
        fᵢ ix ∘' fᵢ.universal' (λ ix → assoc (uᵢ ix) v w) hᵢ ≡[]⟨ fᵢ.commutesp _ hᵢ ix ⟩≡[]
        hᵢ ix                                                ∎
    fᵢ∘g-cart .unique other pᵢ =
      g.unique other $
      fᵢ.uniquep _ _ _ (g ∘' other) λ ix →
        assoc' (fᵢ ix) g other ∙[] pᵢ ix

Similarly, if $f_{i}$ is a family of maps with each $f_{i}$ individually cartesian, and $g_{i}$ is jointly cartesian, then the composite $f_{i} \circ g_{i}$ is jointly cartesian.

pointwise-cartesian-jointly-cartesian-∘
  : (∀ ix → is-cartesian (uᵢ ix) (fᵢ ix))
  → is-jointly-cartesian vᵢ gᵢ
  → is-jointly-cartesian (λ ix → uᵢ ix ∘ vᵢ ix) (λ ix → fᵢ ix ∘' gᵢ ix)

We again prove this by hand to get better definitional behaviour.

pointwise-cartesian-jointly-cartesian-∘
  {uᵢ = uᵢ} {fᵢ = fᵢ} {vᵢ = vᵢ} {gᵢ = gᵢ} fᵢ-cart gᵢ-cart = fᵢ∘gᵢ-cart where
  module fᵢ ix = is-cartesian (fᵢ-cart ix)
  module gᵢ = is-jointly-cartesian gᵢ-cart
  open is-jointly-cartesian

  fᵢ∘gᵢ-cart : is-jointly-cartesian (λ ix → uᵢ ix ∘ vᵢ ix) (λ ix → fᵢ ix ∘' gᵢ ix)
  fᵢ∘gᵢ-cart .universal v hᵢ =
    gᵢ.universal v λ ix → fᵢ.universal' ix (assoc (uᵢ ix) (vᵢ ix) v) (hᵢ ix)
  fᵢ∘gᵢ-cart .commutes v hᵢ ix =
    cast[] $
      (fᵢ ix ∘' gᵢ ix) ∘' gᵢ.universal v (λ ix → fᵢ.universal' ix _ (hᵢ ix)) ≡[]⟨ pullr[] refl (gᵢ.commutes v _ ix) ⟩≡[]
      fᵢ ix ∘' fᵢ.universal' ix _ (hᵢ ix)                                    ≡[]⟨ fᵢ.commutesp ix (assoc (uᵢ ix) (vᵢ ix) v) (hᵢ ix) ⟩≡[]
      hᵢ ix                                                                  ∎
  fᵢ∘gᵢ-cart .unique other p =
    gᵢ.unique other λ ix →
    fᵢ.uniquep ix _ _ _ (gᵢ ix ∘' other)
      (assoc' (fᵢ ix) (gᵢ ix) other ∙[] p ix)

Like their non-familial counterparts, jointly cartesian maps are stable under vertical retractions.

jointly-cartesian-vertical-retraction-stable
  : ∀ {fᵢ : ∀ (ix : Ix) → Hom[ uᵢ ix ] a' (cᵢ' ix)}
  → {fᵢ' : ∀ (ix : Ix) → Hom[ uᵢ ix ] b' (cᵢ' ix)}
  → {ϕ : Hom[ id ] a' b'}
  → is-jointly-cartesian uᵢ fᵢ
  → has-section↓ ϕ
  → (∀ ix → fᵢ' ix ∘' ϕ ≡[ idr _ ] fᵢ ix)
  → is-jointly-cartesian uᵢ fᵢ'

_ = cartesian-vertical-retraction-stable

The proof is essentially the same as the non-joint case, so we omit the details.

jointly-cartesian-vertical-retraction-stable
  {uᵢ = uᵢ} {fᵢ = fᵢ} {fᵢ' = fᵢ'} {ϕ = ϕ} fᵢ-cart ϕ-sect factor
  = fᵢ'-cart
  where
    module fᵢ = is-jointly-cartesian fᵢ-cart
    module ϕ = has-section[_] ϕ-sect

    fᵢ'-cart : is-jointly-cartesian uᵢ fᵢ'
    fᵢ'-cart .is-jointly-cartesian.universal v hᵢ =
      hom[ idl v ] (ϕ ∘' fᵢ.universal v hᵢ)
    fᵢ'-cart .is-jointly-cartesian.commutes v hᵢ ix =
      cast[] $
        fᵢ' ix ∘' hom[] (ϕ ∘' fᵢ.universal v hᵢ) ≡[]⟨ unwrapr (idl v) ⟩≡[]
        fᵢ' ix ∘' ϕ ∘' fᵢ.universal v hᵢ         ≡[]⟨ pulll[] (idr (uᵢ ix)) (factor ix) ⟩≡[]
        fᵢ ix ∘' fᵢ.universal v hᵢ               ≡[]⟨ fᵢ.commutes v hᵢ ix ⟩≡[]
        hᵢ ix ∎
    fᵢ'-cart .is-jointly-cartesian.unique {v = v} {hᵢ = hᵢ} other p =
      cast[] $
        other                                 ≡[]⟨ introl[] _ ϕ.is-section' ⟩≡[]
        (ϕ ∘' ϕ.section') ∘' other            ≡[]⟨ pullr[] _ (wrap (idl v) ∙[] fᵢ.unique _ unique-lemma) ⟩≡[]
        ϕ ∘' fᵢ.universal v hᵢ                ≡[]⟨ wrap (idl v) ⟩≡[]
        hom[ idl v ] (ϕ ∘' fᵢ.universal v hᵢ) ∎

      where
        unique-lemma : ∀ ix → fᵢ ix ∘' hom[ idl v ] (ϕ.section' ∘' other) ≡ hᵢ ix
        unique-lemma ix =
          cast[] $
            fᵢ ix ∘' hom[ idl v ] (ϕ.section' ∘' other) ≡[]⟨ unwrapr (idl v) ⟩≡[]
            fᵢ ix ∘' ϕ.section' ∘' other                ≡[]⟨ pulll[] _ (symP (pre-section[] ϕ-sect (factor ix))) ⟩≡[]
            fᵢ' ix ∘' other                             ≡[]⟨ p ix ⟩≡[]
            hᵢ ix                                       ∎

Cancellation properties of jointly cartesian families🔗

Every jointly cartesian family is a jointly weak monic family; this follows immediately from the uniqueness portion of the universal property.

jointly-cartesian→jointly-weak-monic
  : is-jointly-cartesian uᵢ fᵢ
  → is-jointly-weak-monic fᵢ
jointly-cartesian→jointly-weak-monic {fᵢ = fᵢ} fᵢ-cart {g = w} g h p p' =
  fᵢ.uniquep₂ (λ _ → ap₂ _∘_ refl p) p (λ _ → refl) g h p' (λ _ → refl)
  where module fᵢ = is-jointly-cartesian fᵢ-cart

If $f_{i, j}$ is an $I -indexed$ family of $J_{i} -indexed$ jointly weak monic families and $f_{i, j} \circ g_{i}$ is a $Σ (i : I) . J_{i} -indexed$ jointly cartesian family, then $g_{i}$ must be a $I -indexed$ jointly cartesian family.

jointly-cartesian-weak-monic-cancell
  : {uᵢⱼ : (i : Ix) → (j : Jx i) → Hom (bᵢ i) (cᵢⱼ i j)}
  → {fᵢⱼ : (i : Ix) → (j : Jx i) → Hom[ uᵢⱼ i j ] (bᵢ' i) (cᵢⱼ' i j)}
  → {vᵢ : (i : Ix) → Hom a (bᵢ i)}
  → {gᵢ : (i : Ix) → Hom[ vᵢ i ] a' (bᵢ' i)}
  → (∀ i → is-jointly-weak-monic (fᵢⱼ i))
  → is-jointly-cartesian
      (λ ij → uᵢⱼ (ij .fst) (ij .snd) ∘ vᵢ (ij .fst))
      (λ ij → fᵢⱼ (ij .fst) (ij .snd) ∘' gᵢ (ij .fst))
  → is-jointly-cartesian vᵢ gᵢ

Like the general composition lemma for jointly cartesian families, the statement is more complicated than the proof, which follows from some short calculations.

jointly-cartesian-weak-monic-cancell
  {uᵢⱼ = uᵢⱼ} {fᵢⱼ} {vᵢ} {gᵢ} fᵢⱼ-weak-mono fᵢⱼ∘gᵢ-cart
  = gᵢ-cart
  where
    module fᵢⱼ∘gᵢ = is-jointly-cartesian fᵢⱼ∘gᵢ-cart
    open is-jointly-cartesian

    gᵢ-cart : is-jointly-cartesian vᵢ gᵢ
    gᵢ-cart .universal w hᵢ =
      fᵢⱼ∘gᵢ.universal' (λ (i , j) → sym (assoc (uᵢⱼ i j) (vᵢ i) w)) λ (i , j) →
        fᵢⱼ i j ∘' hᵢ i
    gᵢ-cart .commutes w hᵢ i =
      fᵢⱼ-weak-mono i _ _ refl $ λ j →
      cast[] $
        fᵢⱼ i j ∘' gᵢ i ∘' fᵢⱼ∘gᵢ.universal' _ (λ (i , j) → fᵢⱼ i j ∘' hᵢ i)   ≡[]⟨ assoc' _ _ _ ⟩≡[]
        (fᵢⱼ i j ∘' gᵢ i) ∘' fᵢⱼ∘gᵢ.universal' _ (λ (i , j) → fᵢⱼ i j ∘' hᵢ i) ≡[]⟨ fᵢⱼ∘gᵢ.commutesp _ _ (i , j) ⟩≡[]
        fᵢⱼ i j ∘' hᵢ i                                                        ∎
    gᵢ-cart .unique other p =
      fᵢⱼ∘gᵢ.uniquep _ _ _ other λ (i , j) →
        pullr[] _ (p i)

As an immediate corollary, we get a left cancellation property for composites of joint cartesian families.

jointly-cartesian-cancell
  : {uᵢⱼ : (i : Ix) → (j : Jx i) → Hom (bᵢ i) (cᵢⱼ i j)}
  → {fᵢⱼ : (i : Ix) → (j : Jx i) → Hom[ uᵢⱼ i j ] (bᵢ' i) (cᵢⱼ' i j)}
  → {vᵢ : (i : Ix) → Hom a (bᵢ i)}
  → {gᵢ : (i : Ix) → Hom[ vᵢ i ] a' (bᵢ' i)}
  → (∀ i → is-jointly-cartesian (uᵢⱼ i) (fᵢⱼ i))
  → is-jointly-cartesian
      (λ ij → uᵢⱼ (ij .fst) (ij .snd) ∘ vᵢ (ij .fst))
      (λ ij → fᵢⱼ (ij .fst) (ij .snd) ∘' gᵢ (ij .fst))
  → is-jointly-cartesian vᵢ gᵢ
jointly-cartesian-cancell fᵢⱼ-cart fᵢⱼ∘gᵢ-cart =
  jointly-cartesian-weak-monic-cancell
    (λ i → jointly-cartesian→jointly-weak-monic (fᵢⱼ-cart i))
    fᵢⱼ∘gᵢ-cart

We also obtain a further set of corollaries that describe some special cases of the general cancellation property.

jointly-cartesian-pointwise-weak-monic-cancell
  : (∀ ix → is-weak-monic (fᵢ ix))
  → is-jointly-cartesian (λ ix → uᵢ ix ∘ vᵢ ix) (λ ix → fᵢ ix ∘' gᵢ ix)
  → is-jointly-cartesian vᵢ gᵢ

jointly-cartesian-jointly-weak-monic-cancell
  : is-jointly-weak-monic fᵢ
  → is-jointly-cartesian (λ ix → uᵢ ix ∘ v) (λ ix → fᵢ ix ∘' g)
  → is-cartesian v g

jointly-cartesian-pointwise-cartesian-cancell
  : (∀ ix → is-cartesian (uᵢ ix) (fᵢ ix))
  → is-jointly-cartesian (λ ix → uᵢ ix ∘ vᵢ ix) (λ ix → fᵢ ix ∘' gᵢ ix)
  → is-jointly-cartesian vᵢ gᵢ

jointly-cartesian-cartesian-cancell
  : is-jointly-cartesian uᵢ fᵢ
  → is-jointly-cartesian (λ ix → uᵢ ix ∘ v) (λ ix → fᵢ ix ∘' g)
  → is-cartesian v g

As before, we opt to prove these results by hand to get nicer definitional behaviour.

jointly-cartesian-pointwise-weak-monic-cancell
  {uᵢ = uᵢ} {fᵢ = fᵢ} {vᵢ = vᵢ} {gᵢ = gᵢ} fᵢ-weak-monic fᵢ∘gᵢ-cart
  = gᵢ-cart
  where
    module fᵢ∘gᵢ = is-jointly-cartesian fᵢ∘gᵢ-cart
    open is-jointly-cartesian

    gᵢ-cart : is-jointly-cartesian vᵢ gᵢ
    gᵢ-cart .universal w hᵢ =
      fᵢ∘gᵢ.universal' (λ ix → sym (assoc (uᵢ ix) (vᵢ ix) w))
        (λ ix → fᵢ ix ∘' hᵢ ix)
    gᵢ-cart .commutes w hᵢ ix =
      fᵢ-weak-monic ix _ _ refl $
      cast[] $
        fᵢ ix ∘' gᵢ ix ∘' fᵢ∘gᵢ.universal' _ (λ ix → fᵢ ix ∘' hᵢ ix)   ≡[]⟨ assoc' _ _  _ ⟩≡[]
        (fᵢ ix ∘' gᵢ ix) ∘' fᵢ∘gᵢ.universal' _ (λ ix → fᵢ ix ∘' hᵢ ix) ≡[]⟨ fᵢ∘gᵢ.commutesp _ _ ix ⟩≡[]
        fᵢ ix ∘' hᵢ ix                                                 ∎
    gᵢ-cart .unique other p =
      fᵢ∘gᵢ.uniquep _ _ _ other (λ ix → pullr[] _ (p ix))

jointly-cartesian-jointly-weak-monic-cancell
  {uᵢ = uᵢ} {fᵢ = fᵢ} {v = v} {g = g} fᵢ-weak-monic fᵢ∘g-cart
  = g-cart
  where
    module fᵢ∘g = is-jointly-cartesian fᵢ∘g-cart
    open is-cartesian

    g-cart : is-cartesian v g
    g-cart .universal w h =
      fᵢ∘g.universal' (λ ix → sym (assoc (uᵢ ix) v w)) (λ ix → fᵢ ix ∘' h)
    g-cart .commutes w h =
      fᵢ-weak-monic _ _ refl λ ix →
      cast[] $
        fᵢ ix ∘' g ∘' fᵢ∘g.universal' _ (λ ix → fᵢ ix ∘' h)   ≡[]⟨ assoc' _ _ _ ⟩≡[]
        (fᵢ ix ∘' g) ∘' fᵢ∘g.universal' _ (λ ix → fᵢ ix ∘' h) ≡[]⟨ fᵢ∘g.commutesp _ _ ix ⟩≡[]
        fᵢ ix ∘' h                                            ∎
    g-cart .unique other p =
      fᵢ∘g.uniquep _ _ _ other (λ ix → pullr[] _ p)

jointly-cartesian-pointwise-cartesian-cancell fᵢ-cart fᵢ∘gᵢ-cart =
  jointly-cartesian-pointwise-weak-monic-cancell
    (λ ix → cartesian→weak-monic (fᵢ-cart ix))
    fᵢ∘gᵢ-cart

jointly-cartesian-cartesian-cancell fᵢ-cart fᵢ∘g-cart =
  jointly-cartesian-jointly-weak-monic-cancell
    (jointly-cartesian→jointly-weak-monic fᵢ-cart)
    fᵢ∘g-cart

Extending jointly cartesian families🔗

This section characterises when we can extend an $I^{'} -indexed$ jointly cartesian family $f_{i}$ to a $I -indexed$ cartesian family along a map $e : I^{'} \to I .$ Though seemingly innocent, being able to extend every family $f_{i} : E_{u_{i}} (A^{'}, B_{i}^{'})$ is equivalent to the displayed category being thin!

For the forward direction, suppose that $E$ is thin. Let $f_{i} : E u_{i} (A^{'}, B_{i}^{'})$ be a family such that the restriction of $f_{i}$ along a map $e : I^{'} \to I$ is jointly cartesian. We can then easily extend the family $f_{i}$ along an arbitrary map by ignoring every single equality, as all hom sets involved are thin.

thin→jointly-cartesian-extend
  : ∀ {u : (i : Ix) → Hom a (bᵢ i)} {fᵢ : (i : Ix) → Hom[ uᵢ i ] a' (bᵢ' i)}
  → (∀ {x} (v : Hom x a) → (x' : Ob[ x ]) → ∀ (i : Ix) → is-prop (Hom[ uᵢ i ∘ v ] x' (bᵢ' i)))
  → (e : Ix' → Ix)
  → is-jointly-cartesian (λ i' → uᵢ (e i')) (λ i' → fᵢ (e i'))
  → is-jointly-cartesian (λ i → uᵢ i) (λ i → fᵢ i)
thin→jointly-cartesian-extend {uᵢ = uᵢ} {fᵢ = fᵢ} uᵢ∘v-thin e fₑᵢ-cart = fᵢ-cart where
  module fₑᵢ = is-jointly-cartesian fₑᵢ-cart
  open is-jointly-cartesian

  fᵢ-cart : is-jointly-cartesian (λ i' → uᵢ i') (λ i' → fᵢ i')
  fᵢ-cart .universal v hᵢ =
    fₑᵢ.universal v (λ i' → hᵢ (e i'))
  fᵢ-cart .commutes {x} {x'} v hᵢ i =
    uᵢ∘v-thin v x' i (fᵢ i ∘' fₑᵢ.universal v (λ i' → hᵢ (e i'))) (hᵢ i)
  fᵢ-cart .unique {x} {x'} {v} {hᵢ} other p =
    fₑᵢ.unique other λ i' → uᵢ∘v-thin v x' (e i') (fᵢ (e i') ∘' other) (hᵢ (e i'))

For the reverse direction, suppose we could extend arbitrary families. In particular, this means that we can extend singleton families to constant families, which lets us transfer a proof that a morphism is cartesian to a proof that a constant family is jointly cartesian.

In particular, this means that the pair $(id, id)$ is jointly cartesian, which means that the hom set is thin!

jointly-cartesian-extend→thin
  : ∀ (extend
    : ∀ {Ix' Ix : Type} {bᵢ : Ix → Ob} {bᵢ' : (i : Ix) → Ob[ bᵢ i ]}
    → {uᵢ : (i : Ix) → Hom a (bᵢ i)} {fᵢ : (i : Ix) → Hom[ uᵢ i ] a' (bᵢ' i)}
    → (e : Ix' → Ix)
    → is-jointly-cartesian (λ i' → uᵢ (e i')) (λ i' → fᵢ (e i'))
    → is-jointly-cartesian (λ i → uᵢ i) (λ i → fᵢ i))
  → ∀ {x} (v : Hom x a) → (x' : Ob[ x ]) → is-prop (Hom[ v ] x' a')
jointly-cartesian-extend→thin extend v x' =
  id-pair-joint-cartesian→thin
    (extend (λ _ → true)
      (cartesian→jointly-cartesian ⊤-is-contr cartesian-id))
    v x'

Universal properties🔗

Jointly cartesian families have an alternative presentation of their universal property: a family $f_{i} : E_{u_{i}} (A^{'}, B_{i}^{'})$ is jointly cartesian if and only if the joint postcomposition map

h \mapsto λi . f_{i} \circ h

is an equivalence.

postcompose-equiv→jointly-cartesian
  : {uᵢ : ∀ ix → Hom a (bᵢ ix)}
  → (fᵢ : ∀ ix → Hom[ uᵢ ix ] a' (bᵢ' ix))
  → (∀ {x} (v : Hom x a) → (x' : Ob[ x ])
    → is-equiv {B = ∀ ix → Hom[ uᵢ ix ∘ v ] x' (bᵢ' ix)} (λ h ix → fᵢ ix ∘' h))
  → is-jointly-cartesian uᵢ fᵢ

jointly-cartesian→postcompose-equiv
  : {uᵢ : ∀ ix → Hom a (bᵢ ix)}
  → {fᵢ : ∀ ix → Hom[ uᵢ ix ] a' (bᵢ' ix)}
  → is-jointly-cartesian uᵢ fᵢ
  → ∀ {x} (v : Hom x a) → (x' : Ob[ x ])
  → is-equiv {B = ∀ ix → Hom[ uᵢ ix ∘ v ] x' (bᵢ' ix)} (λ h ix → fᵢ ix ∘' h)

The proofs are just shuffling about data, so we shall skip over the details.

postcompose-equiv→jointly-cartesian {a = a} {uᵢ = uᵢ} fᵢ eqv = fᵢ-cart where
  module eqv {x} {v : Hom x a} {x' : Ob[ x ]} = Equiv (_ , eqv v x')
  open is-jointly-cartesian

  fᵢ-cart : is-jointly-cartesian uᵢ fᵢ
  fᵢ-cart .universal v hᵢ =
    eqv.from hᵢ
  fᵢ-cart .commutes v hᵢ ix =
    eqv.ε hᵢ ·ₚ ix
  fᵢ-cart .unique {hᵢ = hᵢ} other p =
    sym (eqv.η other) ∙ ap eqv.from (ext p)

jointly-cartesian→postcompose-equiv {uᵢ = uᵢ} {fᵢ = fᵢ} fᵢ-cart v x' .is-eqv hᵢ =
  contr (fᵢ.universal v hᵢ , ext (fᵢ.commutes v hᵢ)) λ fib →
    Σ-prop-pathp! (sym (fᵢ.unique (fib .fst) (λ ix → fib .snd ·ₚ ix)))
  where
    module fᵢ = is-jointly-cartesian fᵢ-cart

Jointly cartesian lifts🔗

A jointly cartesian lift of a family of objects $Y_{i}^{'} Y_{i}$ along a family of maps $u_{i} : B (X, Y_{i})$ is an object $⋂_{i : I} u_{i}^{*} Y_{i}$ equipped with a jointly cartesian family $π_{i} : E_{u_{i}} (⋂_{i : I} u_{i}^{*} Y_{i}, Y_{i}) .$

record Joint-cartesian-lift
  {ℓi : Level} {Ix : Type ℓi}
  {x : Ob} {yᵢ : Ix → Ob}
  (uᵢ : (ix : Ix) → Hom x (yᵢ ix))
  (yᵢ' : (ix : Ix) → Ob[ yᵢ ix ])
  : Type (o ⊔ ℓ ⊔ o' ⊔ ℓ' ⊔ ℓi)
  where
  no-eta-equality
  field
    {x'} : Ob[ x ]
    lifting : (ix : Ix) → Hom[ uᵢ ix ] x' (yᵢ' ix)
    jointly-cartesian : is-jointly-cartesian uᵢ lifting

  open is-jointly-cartesian jointly-cartesian public

A $κ$ jointly cartesian fibration is a displayed category that joint cartesian lifts of all $κ -small$ families.

Jointly-cartesian-fibration : (κ : Level) → Type _
Jointly-cartesian-fibration κ =
  ∀ (Ix : Type κ)
  → {x : Ob} {yᵢ : Ix → Ob}
  → (uᵢ : (ix : Ix) → Hom x (yᵢ ix))
  → (yᵢ' : (ix : Ix) → Ob[ yᵢ ix ])
  → Joint-cartesian-lift uᵢ yᵢ'

module Jointly-cartesian-fibration {κ} (fib : Jointly-cartesian-fibration κ) where

The following section defines some nice notation for jointly cartesian fibrations, but is a bit verbose.

  module _
    (Ix : Type κ) {x : Ob} {yᵢ : Ix → Ob}
    (uᵢ : (ix : Ix) → Hom x (yᵢ ix))
    (yᵢ' : (ix : Ix) → Ob[ yᵢ ix ])
    where
    open Joint-cartesian-lift (fib Ix uᵢ yᵢ')
      using ()
      renaming (x' to ∏*)
      public

  module _
    {Ix : Type κ} {x : Ob} {yᵢ : Ix → Ob}
    (uᵢ : (ix : Ix) → Hom x (yᵢ ix))
    (yᵢ' : (ix : Ix) → Ob[ yᵢ ix ])
    where
    open Joint-cartesian-lift (fib Ix uᵢ yᵢ')
      using ()
      renaming (lifting to π*)
      public

  module π*
    {Ix : Type κ} {x : Ob} {yᵢ : Ix → Ob}
    {uᵢ : (ix : Ix) → Hom x (yᵢ ix)}
    {yᵢ' : (ix : Ix) → Ob[ yᵢ ix ]}
    where
    open Joint-cartesian-lift (fib Ix uᵢ yᵢ')
      hiding (x'; lifting)
      public

Every jointly cartesian fibration has objects that act like codiscrete spaces arising from lifts of empty families.

  opaque
    Codisc* : ∀ (x : Ob) → Ob[ x ]
    Codisc* x = ∏* (Lift _ ⊥) {yᵢ = λ ()} (λ ()) (λ ())

    codisc*
      : ∀ {x y : Ob}
      → (u : Hom x y) (x' : Ob[ x ])
      → Hom[ u ] x' (Codisc* y)
    codisc* u x' = π*.universal u (λ ())

    codisc*-unique
      : ∀ {x y : Ob}
      → {u : Hom x y} {x' : Ob[ x ]}
      → (other : Hom[ u ] x' (Codisc* y))
      → other ≡ codisc* u x'
    codisc*-unique other = π*.unique other (λ ())

References

Adámek, Jiří, Horst Herrlich, and George E. Strecker. 2004. Abstract and Concrete Categories: The Joy of Cats. Dover Books on Mathematics. Mineola, NY: Dover Publ.
Dubuc, Eduardo J., and Luis Español. 2006. “Topological Functors as Familiarly-Fibrations,” no. arXiv:math/0611701 (November). https://doi.org/10.48550/arXiv.math/0611701.