open import Cat.Instances.Functor
open import Cat.Instances.Product
open import Cat.Diagram.Pullback
open import Cat.Bi.Base
open import Cat.Prelude

import Cat.Reasoning

module Cat.Bi.Instances.Spans {o ℓ} (C : Precategory o ℓ) where

private module C = Cat.Reasoning C
open C

The bicategory of spans🔗

Let $\mathcal{C}$ be a precategory. By a span in $\mathcal{C}$ (from an object $A:\mathcal{C}$ to an object $B:\mathcal{C}\text{),}$ we mean a diagram of the form $A\leftarrow C\to B\text{.}$ Note that the “vertex” of this span, the object $C\text{,}$ is part of the data, so that the collection of “spans in $\mathcal{C}$” will not be a set (unless $\mathcal{C}$ is strict) — so we can not construct a category where $\mathbf{Hom}(a,b)$ is the collection of spans from $a$ to $b\text{.}$

However, we can make spans in $\mathcal{C}$ the objects of a category, and the hom-sets are the maps in $\mathcal{C}$ between the vertices which “commute with the legs”. Diagrammatically, a map between spans is the dashed line in

where both the left and right triangles must commute.

record Span (a b : Ob) : Type (o ⊔ ℓ) where
  constructor span
  field
    apex  : Ob
    left  : Hom apex a
    right : Hom apex b

open Span

record Span-hom {a b : Ob} (x y : Span a b) : Type ℓ where
  no-eta-equality
  field
    map   : Hom (x .apex) (y .apex)
    left  : x .left  ≡ y .left ∘ map
    right : x .right ≡ y .right ∘ map

open Span-hom
unquoteDecl H-Level-Span-hom = declare-record-hlevel  H-Level-Span-hom (quote Span-hom)

instance
  Underlying-Span : ∀ {a b} ⦃ _ : Underlying Ob ⦄ → Underlying (Span a b)
  Underlying-Span = record { ⌞_⌟ = λ S → ⌞ S .apex ⌟ }

Span-hom-path
  : {a b : Ob} {x y : Span a b} {f g : Span-hom x y}
  → f .map ≡ g .map → f ≡ g
Span-hom-path p i .map = p i
Span-hom-path {x = x} {y} {f} {g} p i .left j =
  is-set→squarep (λ i j → Hom-set _ _)
    (λ _ → x .left) (λ j → f .left j) (λ j → g .left j) (λ j → y .left ∘ p j) i j
Span-hom-path {x = x} {y} {f} {g} p i .right j =
  is-set→squarep (λ i j → Hom-set _ _)
    (λ _ → x .right) (λ j → f .right j) (λ j → g .right j) (λ j → y .right ∘ p j) i j

The category ${\mathrm{Spans}}_{\mathcal{C}}(A,B)$ of spans between $A$ and $B$ admits a faithful functor to $\mathcal{C}$ (projecting the vertex and the “middle map”, respectively), so that equality of maps of spans can be compared at the level of maps in $\mathcal{C}\text{.}$

Spans : Ob → Ob → Precategory _ _
Spans x y .Precategory.Ob = Span x y
Spans x y .Precategory.Hom = Span-hom
Spans x y .Precategory.Hom-set _ _ = hlevel 
Spans x y .Precategory.id .map = id
Spans x y .Precategory.id .left = intror refl
Spans x y .Precategory.id .right = intror refl
Spans x y .Precategory._∘_ f g .map = f .map ∘ g .map
Spans x y .Precategory._∘_ f g .left = g .left ∙ pushl (f .left)
Spans x y .Precategory._∘_ f g .right = g .right ∙ pushl (f .right)
Spans x y .Precategory.idr f = Span-hom-path (idr _)
Spans x y .Precategory.idl f = Span-hom-path (idl _)
Spans x y .Precategory.assoc f g h = Span-hom-path (assoc _ _ _)

Now suppose that $\mathcal{C}$ admits pullbacks for arbitrary pairs of maps. Supposing that we have some spans $S:A\to B$ and $S^{\prime }:B\to C\text{,}$ we can fit them in an M-shaped diagram like

so that taking the pullback of the diagram $S\to B\leftarrow S^{\prime }$ gives us an universal solution to the problem of finding a span $A\leftarrow (S{\times }_B{S}^{\prime })\to C\text{.}$ Since pullbacks are universal, this composition operation is functorial, and so extends to a composition operation Span-∘:

module _ (pb : ∀ {a b c} (f : Hom a b) (g : Hom c b) → Pullback C f g) where
  open Functor

  Span-∘ : ∀ {a b c} → Functor (Spans b c ×ᶜ Spans a b) (Spans a c)
  Span-∘ .F₀ (sp1 , sp2) =
    span pb.apex (sp2 .left ∘ pb.p₂) (sp1 .right ∘ pb.p₁)
    where module pb = Pullback (pb (sp1 .left) (sp2 .right))

  Span-∘ .F₁ {x1 , x2} {y1 , y2} (f , g) = res
    where
      module x = Pullback (pb (x1 .left) (x2 .right))
      module y = Pullback (pb (y1 .left) (y2 .right))

      x→y : Hom x.apex y.apex
      x→y = y.universal {p₁' = f .map ∘ x.p₁} {p₂' = g .map ∘ x.p₂} comm
        where abstract
          open Pullback
          comm : y1 .left ∘ f .map ∘ x.p₁ ≡ y2 .right ∘ g .map ∘ x.p₂
          comm = pulll (sym (f .left)) ∙ x.square ∙ pushl (g .right)

      res : Span-hom _ _
      res .map = x→y
      res .left = sym (pullr y.p₂∘universal ∙ pulll (sym (g .left)))
      res .right = sym (pullr y.p₁∘universal ∙ pulll (sym (f .right)))

  Span-∘ .F-id {x1 , x2} = Span-hom-path $ sym $ x.unique id-comm id-comm
    where module x = Pullback (pb (x1 .left) (x2 .right))

  Span-∘ .F-∘ {x1 , x2} {y1 , y2} {z1 , z2} f g =
    Span-hom-path $ sym $ z.unique
      (pulll z.p₁∘universal ∙ pullr y.p₁∘universal ∙ assoc _ _ _)
      (pulll z.p₂∘universal ∙ pullr y.p₂∘universal ∙ assoc _ _ _)
    where
      module x = Pullback (pb (x1 .left) (x2 .right))
      module y = Pullback (pb (y1 .left) (y2 .right))
      module z = Pullback (pb (z1 .left) (z2 .right))

What we’ll show in the rest of this module is that Span-∘ lets us make Spans into the categories of 1-cells of a prebicategory, the (pre)bicategory of spans (of $\mathcal{C}\text{),}$ $\mathrm{Spans}(\mathcal{C})\text{.}$ As mentioned before, this prebicategory has (a priori) no upper bound on the h-levels of its 1-cells, so it is not locally strict. We remark that when $\mathcal{C}$ is univalent, then $\mathrm{Spans}(\mathcal{C})$ is locally so, and when $\mathcal{C}$ is gaunt, then $\mathrm{Spans}(\mathcal{C})$ is strict.

Since the details of the full construction are grueling, we will present only an overview of the unitors and the associator. For the left unitor, observe that the composition $\mathrm{id}\circ S$ is implemented by a pullback diagram like

but observe that the maps $S\stackrel{f}{\to }A$ and $S\stackrel{\mathrm{id}}{\to }S$ also form a cone over the cospan $A\to A\stackrel{f}{\leftarrow }S\text{,}$ so that there is a unique map filling the dashed line in the diagram above such that everything commutes: hence there is an invertible 2-cell $\mathrm{id}\circ S\Rightarrow S\text{.}$ The right unitor is analogous.

  open Prebicategory
  open Pullback

  private
    _¤_ : ∀ {a b c} (x : Span b c) (y : Span a b) → Span a c
    f ¤ g = Span-∘ .F₀ (f , g)

    sλ← : ∀ {A B} (x : Span A B) → Span-hom x (span _ C.id C.id ¤ x)
    sλ← x .map   = pb _ _ .universal id-comm-sym
    sλ← x .left  = sym $ pullr (pb _ _ .p₂∘universal) ∙ idr _
    sλ← x .right = sym $ pullr (pb _ _ .p₁∘universal) ∙ idl _

    sρ← : ∀ {A B} (x : Span A B) → Span-hom x (x ¤ span _ C.id C.id)
    sρ← x .map   = pb _ _ .universal id-comm
    sρ← x .left  = sym $ pullr (pb _ _ .p₂∘universal) ∙ idl _
    sρ← x .right = sym $ pullr (pb _ _ .p₁∘universal) ∙ idr _

For the associator, while doing the construction in elementary terms is quite complicated, we observe that, diagrammatically, the composite of three morphisms fits into a diagram like

so that, at a high level, there is no confusion as to which composite is meant. From then, it’s just a matter of proving pullbacks are associative (in a standard, but annoying, way), and showing that these canonically-obtained isomorphisms (are natural in all the possible variables and) satisfy the triangle and pentagon identities.

    sα← : ∀ {A B C D} ((f , g , h) : Span C D × Span B C × Span A B)
        → Span-hom ((f ¤ g) ¤ h) (f ¤ (g ¤ h))
    sα← (f , g , h) .map = pb _ _ .universal resp' where
      abstract
        resp : g .left C.∘ pb (f .left) (g .right) .p₂
           C.∘ pb ((f ¤ g) .left) (h .right) .p₁
             ≡ h .right C.∘ pb ((f ¤ g) .left) (h .right) .p₂
        resp = assoc _ _ _ ∙ pb _ _ .square

      shuffle = pb _ _ .universal {p₁' = pb _ _ .p₂ C.∘ pb _ _ .p₁} {p₂' = pb _ _ .p₂} resp

      abstract
        resp' : f .left C.∘ pb (f .left) (g .right) .p₁
            C.∘ pb ((f ¤ g) .left) (h .right) .p₁
              ≡ (g ¤ h) .right C.∘ shuffle
        resp' = sym $ pullr (pb _ _ .p₁∘universal) ∙ extendl (sym (pb _ _ .square))

    sα← (f , g , h) .left = sym $ pullr (pb _ _ .p₂∘universal) ∙ pullr (pb _ _ .p₂∘universal)
    sα← (f , g , h) .right = sym $ pullr (pb _ _ .p₁∘universal) ∙ assoc _ _ _

    sα→ : ∀ {A B C D} ((f , g , h) : Span C D × Span B C × Span A B)
        → Span-hom (f ¤ (g ¤ h)) ((f ¤ g) ¤ h)
    sα→ (f , g , h) .map = pb _ _ .universal resp' where
      abstract
        resp : f .left C.∘ pb (f .left) ((g ¤ h) .right) .p₁
             ≡ g .right C.∘ pb (g .left) (h .right) .p₁
           C.∘ pb (f .left) ((g ¤ h) .right) .p₂
        resp = pb _ _ .square ∙ sym (assoc _ _ _)

      shuffle = pb _ _ .universal {p₁' = pb _ _ .p₁} {p₂' = pb _ _ .p₁ C.∘ pb _ _ .p₂} resp

      abstract
        resp' : (f ¤ g) .left C.∘ shuffle
              ≡ h .right C.∘ pb (g .left) (h .right) .p₂
            C.∘ pb (f .left) ((g ¤ h) .right) .p₂
        resp' = pullr (pb _ _ .p₂∘universal) ∙ extendl (pb _ _ .square)

    sα→ (f , g , h) .left = sym $ pullr (pb _ _ .p₂∘universal) ∙ assoc _ _ _
    sα→ (f , g , h) .right = sym $ pullr (pb _ _ .p₁∘universal) ∙ pullr (pb _ _ .p₁∘universal)

  open make-natural-iso
  {-# TERMINATING #-}
  Spanᵇ : Prebicategory _ _ _
  Spanᵇ .Ob = C.Ob
  Spanᵇ .Hom = Spans
  Spanᵇ .id = span _ C.id C.id
  Spanᵇ .compose = Span-∘
  Spanᵇ .unitor-l = to-natural-iso ni where
    ni : make-natural-iso (Id {C = Spans _ _}) _
    ni .eta = sλ←
    ni .inv x .map = pb _ _ .p₂
    ni .inv x .left = refl
    ni .inv x .right = pb _ _ .square
    ni .eta∘inv x = Span-hom-path (Pullback.unique₂ (pb _ _) {p = idl _ ∙ ap₂ C._∘_ refl (introl refl)}
      (pulll (pb _ _ .p₁∘universal))
      (pulll (pb _ _ .p₂∘universal))
      (id-comm ∙ pb _ _ .square)
      id-comm)
    ni .inv∘eta x = Span-hom-path (pb _ _ .p₂∘universal)
    ni .natural x y f = Span-hom-path $
      Pullback.unique₂ (pb _ _) {p = idl _ ∙ f .right}
        (pulll (pb _ _ .p₁∘universal) ∙ pullr (pb _ _ .p₁∘universal) ∙ idl _)
        (pulll (pb _ _ .p₂∘universal) ∙ pullr (pb _ _ .p₂∘universal) ∙ idr _)
        (pulll (pb _ _ .p₁∘universal) ∙ sym (f .right))
        (pulll (pb _ _ .p₂∘universal) ∙ idl _)
  Spanᵇ .unitor-r = to-natural-iso ni where
    ni : make-natural-iso (Id {C = Spans _ _}) _
    ni .eta = sρ←
    ni .inv _ .map = pb _ _ .p₁
    ni .inv _ .left = sym (pb _ _ .square)
    ni .inv _ .right = refl
    ni .eta∘inv x = Span-hom-path (Pullback.unique₂ (pb _ _) {p = introl refl}
      (pulll (pb _ _ .p₁∘universal) ∙ idl _)
      (pulll (pb _ _ .p₂∘universal))
      (idr _)
      (id-comm ∙ sym (pb _ _ .square)))
    ni .inv∘eta x = Span-hom-path (pb _ _ .p₁∘universal)
    ni .natural x y f = Span-hom-path $
      Pullback.unique₂ (pb _ _) {p = sym (f .left) ∙ introl refl}
        (pulll (pb _ _ .p₁∘universal) ∙ pullr (pb _ _ .p₁∘universal) ∙ idr _)
        (pulll (pb _ _ .p₂∘universal) ∙ pullr (pb _ _ .p₂∘universal) ∙ idl _)
        (pulll (pb _ _ .p₁∘universal) ∙ idl _)
        (pulll (pb _ _ .p₂∘universal) ∙ sym (f .left))
  Spanᵇ .associator = to-natural-iso ni where
    ni : make-natural-iso _ _
    ni .eta = sα←
    ni .inv = sα→
    ni .eta∘inv x = Span-hom-path $
      Pullback.unique₂ (pb _ _) {p = pb _ _ .square}
      (pulll (pb _ _ .p₁∘universal) ∙ pullr (pb _ _ .p₁∘universal) ∙ pb _ _ .p₁∘universal)
      (pulll (pb _ _ .p₂∘universal) ∙ unique₂ (pb _ _) {p = extendl (pb _ _ .square)}
          (pulll (pb _ _ .p₁∘universal) ∙ pullr (pb _ _ .p₁∘universal) ∙ pb _ _ .p₂∘universal)
          (pulll (pb _ _ .p₂∘universal) ∙ pb _ _ .p₂∘universal)
          refl refl)
      (idr _)
      (idr _)
    ni .inv∘eta x = Span-hom-path $
      Pullback.unique₂ (pb _ _) {p = pb _ _ .square}
      (pulll (pb _ _ .p₁∘universal) ∙ unique₂ (pb _ _) {p = extendl (pb _ _ .square)}
        (pulll (pb _ _ .p₁∘universal) ∙ pb _ _ .p₁∘universal)
        (pulll (pb _ _ .p₂∘universal) ∙ pullr (pb _ _ .p₂∘universal) ∙ pb _ _ .p₁∘universal)
        refl refl)
      (pulll (pb _ _ .p₂∘universal) ∙ pullr (pb _ _ .p₂∘universal) ∙ pb _ _ .p₂∘universal)
      (idr _)
      (idr _)
    ni .natural x y f = Span-hom-path $ Pullback.unique₂ (pb _ _)
      {p₁' = f .fst .map C.∘ pb _ _ .p₁ C.∘ pb _ _ .p₁}
      {p₂' = pb _ _ .universal
        {p₁' = f .snd .fst .map C.∘ pb _ _ .p₂ C.∘ pb _ _ .p₁}
        {p₂' = f .snd .snd .map C.∘ pb _ _ .p₂}
        (pulll (sym (f .snd .fst .left)) ∙ assoc _ _ _ ∙ pb _ _ .square ∙ pushl (f .snd .snd .right))}
      {p = sym $ pullr (pb _ _ .p₁∘universal) ∙ pulll (sym (f .snd .fst .right)) ∙ extendl (sym (pb _ _ .square)) ∙ pushl (f .fst .left)}
      (pulll (pb _ _ .p₁∘universal) ∙ pullr (pb _ _ .p₁∘universal))
      (pulll (pb _ _ .p₂∘universal) ∙ pb _ _ .unique
        (pulll (extendl (pb _ _ .p₁∘universal)) ∙ pullr (pullr (pb _ _ .p₂∘universal)) ∙ ap₂ C._∘_ refl (pb _ _ .p₁∘universal))
        (pulll (extendl (pb _ _ .p₂∘universal)) ∙ pullr (pullr (pb _ _ .p₂∘universal)) ∙ ap₂ C._∘_ refl (pb _ _ .p₂∘universal)))
      (pulll (pb _ _ .p₁∘universal) ∙ pullr (pb _ _ .p₁∘universal) ∙ pulll (pb _ _ .p₁∘universal) ∙ sym (assoc _ _ _))
      (pulll (pb _ _ .p₂∘universal) ∙ pb _ _ .unique
        (pulll (pb _ _ .p₁∘universal) ∙ pullr (pb _ _ .p₁∘universal) ∙ extendl (pb _ _ .p₂∘universal))
        (pulll (pb _ _ .p₂∘universal) ∙ pb _ _ .p₂∘universal))
  Spanᵇ .triangle f g = Span-hom-path $
    pb _ _ .unique
      (pulll (pb _ _ .p₁∘universal) ∙ pullr (pb _ _ .p₁∘universal) ∙ pb _ _ .p₁∘universal ∙ introl refl)
      (pulll (pb _ _ .p₂∘universal) ∙ pullr (pb _ _ .p₂∘universal) ∙ eliml refl)
  Spanᵇ .pentagon f g h i = Span-hom-path $
    Pullback.unique₂ (pb _ _)
      {p = pullr (pulll (pb _ _ .p₂∘universal) ∙ pullr (pulll (pb _ _ .p₂∘universal) ∙ pullr (pb _ _ .p₂∘universal)) ∙ ap₂ C._∘_ refl (pulll (pb _ _ .p₁∘universal)))
         ∙ ap₂ C._∘_ refl (extendl (pb _ _ .p₂∘universal)) ∙ sym (ap₂ C._∘_ refl (idl _ ∙ extendl (pb _ _ .p₂∘universal)) ∙ extendl (sym (pb _ _ .square)))}
      (pulll (pb _ _ .p₁∘universal) ∙ pullr (pulll (pb _ _ .p₁∘universal)))
      (pulll (pb _ _ .p₂∘universal) ∙ pullr (pulll (pb _ _ .p₂∘universal) ∙ pullr (pb _ _ .p₂∘universal)))
      (pulll (pb _ _ .p₁∘universal)
      ∙ Pullback.unique₂ (pb _ _) {p = pullr (pb _ _ .p₂∘universal) ∙ extendl (pb _ _ .square) ∙ sym (assoc _ _ _)}
          (pulll (pb _ _ .p₁∘universal) ∙ pb _ _ .p₁∘universal)
          (pulll (pb _ _ .p₂∘universal) ∙ pullr (pb _ _ .p₂∘universal))
          (pulll (pb _ _ .p₁∘universal) ∙ pb _ _ .unique
            (pulll (pb _ _ .p₁∘universal) ∙ pulll (pb _ _ .p₁∘universal) ∙ pb _ _ .p₁∘universal ∙ idl _)
            (pulll (pb _ _ .p₂∘universal) ∙ pulll (pullr (pb _ _ .p₂∘universal)) ∙ pullr (pullr (pb _ _ .p₂∘universal) ∙ pulll (pb _ _ .p₁∘universal)) ∙ pulll (pb _ _ .p₁∘universal)))
          (pulll (pb _ _ .p₂∘universal) ∙ pullr (pulll (pb _ _ .p₂∘universal) ∙ pullr (pb _ _ .p₂∘universal))
          ∙ ap₂ C._∘_ refl (pulll (pb _ _ .p₁∘universal)) ∙ pulll (pb _ _ .p₂∘universal) ∙ sym (assoc _ _ _)))
      ( pulll (pb _ _ .p₂∘universal)
      ·· pullr (pb _ _ .p₂∘universal)
      ·· sym (idl _ ·· pulll (pb _ _ .p₂∘universal) ·· sym (assoc _ _ _)))